北京大学：中级微观经济学（上）：2

北京大学经济学院 1 Chapter Three Consumer’s Behavior Theory: Further analysis(1) ? 2005 北京大学经济学院 Slide2 Chapter 3 includes: ? 3.1 Revealed Preference Theory(显示性偏好理论) ? 3.2 Choice Under Uncertainty(不确定条件下的 选择理论) 北京大学经济学院 Slide3 Outline of Today’s Class ? Direct Preference Revelation ? Indirect Preference Revelation ? Two Axioms of Revealed Preference （显示性偏好的两个公理) – Weak Axioms of Revealed Preference. （显示性偏好弱公理） – Strong Axioms of Revealed Preference。（显示性偏好强公理） ? Recovering Indifference Curves ? Revealed Preference Theory Application: Index Numbers(指数) 北京大学经济学院 Slide4 Readings about the part of this chapter ? Zhang: Chapter 4—4.1；4.2 ? Nicholson: Chapter 5—P136-139 北京大学经济学院 Slide5 Revealed Preference Theory: Introduction ? Suppose we observe the demands (consumption choices) that a consumer makes for different budgets. ? This reveals information about the consumer’s preferences. We can use this information to ... 北京大学经济学院 Slide6 Revealed Preference Analysis – Test the behavioral hypothesis that a consumer chooses the most preferred bundle from those available. – Discover the consumer’s preference relation. 北京大学经济学院 Slide7 Assumptions on Preferences ? Preferences – do not change while the choice data are gathered. – are strictly convex. – are monotonic. – Rational consumer always maximizes his utility. ? Together, convexity and monotonicity imply that the most preferred affordable bundle is unique. 北京大学经济学院 Slide8 Assumptions on Preferences x 2 x 1 x 1 * x 2 * If preferences are convex and monotonic (i.e. well-behaved) then the most preferred affordable bundle is unique. 北京大学经济学院 Slide9 Direct Preference Revelation （直接显示性偏好） ? Suppose that the bundle x * is chosen when the bundle y is affordable. ? Then x* is revealed directly as preferred to y (otherwise y would have been chosen). 北京大学经济学院 Slide10 Direct Preference Revelation x 2 x 1 x * y The chosen bundle x * is revealed directly as preferred to the bundles y and z. z 北京大学经济学院 Slide11 Direct Preference Revelation ? That x is revealed directly as preferred to y will be written as x y D p 北京大学经济学院 Slide12 Indirect Preference Revelation （间接显示性偏好） ? Suppose x is revealed directly preferred to y, and y is revealed directly preferred to z. Then, by transitivity, x is revealed indirectly as preferred to z. Write this as ? so x y and y z x z. D p D p I p ZX I f 北京大学经济学院 Slide13 Indirect Preference Revelation x 2 z is not affordable when x* is chosen, z* is not affordable when y* is chosen. So x* and z cannot be compared directly. x* y* z But x* y* and y* z so x* z. D p D p I p x 1 北京大学经济学院 Slide14 Two Axioms of Revealed Preference (显示性偏好的两个公理) ? To apply revealed preference analysis, choices must satisfy two criteria -- the Weak and the Strong Axioms of Revealed Preference.（显示性偏好弱 和强公理） 北京大学经济学院 Slide15 The Weak Axiom of Revealed Preference (WARP) ? If the bundle x is revealed directly as preferred to the bundle y then it is never the case that y is revealed directly as preferred to x; i.e. x y not (y x). D p D p 北京大学经济学院 Slide16 The Weak Axiom of Revealed Preference (WARP)（显示性偏好弱公理） ? Choice data which violate the WARP are inconsistent with economic rationality. ? The WARP is a necessary condition for applying economic rationality to explain observed choices. 北京大学经济学院 Slide17 The Weak Axiom of Revealed Preference (WARP) x 2 x y These statements are inconsistent with each other. x is chosen when y is available, so x y. y is chosen when x is available, so y x. D p D p x 1 北京大学经济学院 Slide18 The Strong Axiom of Revealed Preference (SARP)（显示性偏好强公理） ? If the bundle x is revealed (directly or indirectly) as preferred to the bundle y and x ≠ y, then it is never the case that the y is revealed (directly or indirectly) as preferred to x; i.e. x y or x y not ( y x or y x ). D p D p I p I p 北京大学经济学院 Slide19 The Strong Axiom of Revealed Preference ? That the observed choice data satisfy the SARP is a condition necessary and sufficient for there to be a well-behaved preference relation that “rationalizes” the data. 北京大学经济学院 Slide20 From Revealed Preference to Indifference Curves ? Suppose we have the choice data satisfy the SARP. ? Then we can discover approximately where are the consumer’s indifference curves. ? How? 北京大学经济学院 Slide21 From Revealed Preference to Indifference Curves ? Suppose we observe: A: (p 1 ,p 2 ) = ($1,$1) & (x 1 ,x 2 ) = (15,15) B: (p 1 ,p 2 ) = ($2,$1) & (x 1 ,x 2 ) = (10,20) C: (p 1 ,p 2 ) = ($1,$2) & (x 1 ,x 2 ) = (20,10) D: (p 1 ,p 2 ) = ($2,$5) & (x 1 ,x 2 ) = (30,12) E: (p 1 ,p 2 ) = ($5,$2) & (x 1 ,x 2 ) = (12,30). ? Where lies the indifference curve containing the bundle A = (15,15)? 北京大学经济学院 Slide22 From Revealed Preference to Indifference Curves x 2 x 1 A B E C D 北京大学经济学院 Slide23 From Revealed Preference to Indifference Curves x 2 x 1 A A is directly revealed preferred to any gray bundle. A 北京大学经济学院 Slide24 From Revealed Preference to Indifference Curves x 2 x 1 B A is directly revealed preferred to B and B is directly revealed preferred to all gray bundles. By transitivity, A is indirectly revealed preferred to all gray bundles A 北京大学经济学院 Slide25 From Revealed Preference to Indifference Curves x 2 x 1 B so A is now revealed preferred to all bundles in the union. A 北京大学经济学院 Slide26 From Revealed Preference to Indifference Curves x 2 x 1 C The same argument with C A is revealed preferred to all bundles in the union. B A Therefore the indifference curve containing A must lie everywhere else above this shaded set. 北京大学经济学院 Slide27 From Revealed Preference to Indifference Curves x 2 x 1 D D is directly revealed preferred to A. Well-behaved preferences are convex so all bundles on the line between A and D are also preferred to A. A 北京大学经济学院 Slide28 From Revealed Preference to Indifference Curves x 2 x 1 D Preference are monotonic: all bundles containing the same amount of 2 and more of 1 than D are preferred to D and hence to A. A 北京大学经济学院 Slide29 From Revealed Preference to Indifference Curves x 2 x 1 D A bundles revealed to be strictly preferred to A because of D 北京大学经济学院 Slide30 From Revealed Preference to Indifference Curves x 2 x 1 E D A bundles revealed to be strictly preferred to A because of D and E 北京大学经济学院 Slide31 From Revealed Preference to Indifference Curves x 2 x 1 A bundles revealed to be strictly preferred to A bundles revealed to be less preferred to A Possible indifference curve 北京大学经济学院 Slide32 Revealed Preference Theory Application: Index Numbers(指数) ? Over time, many prices change. Are consumers better or worse off “overall” as a consequence? ? Utility cannot be directly observed. ? Index numbers give approximate answers to such questions. 北京大学经济学院 Slide33 Quantity Index Numbers ? A quantity index is a price-weighted average of quantities demanded; ? (p 1 ,p 2 ) can be base period prices (p 1 b ,p 2 b ) or current period prices (p 1 t ,p 2 t ). I px px px px q tt bb = + + 11 22 11 22 北京大学经济学院 Slide34 Quantity Index Numbers ? If (p 1 ,p 2 ) = (p 1 b ,p 2 b ) then we have the Laspeyres quantity index(拉氏指数); L px px px px q bt bt bb bb = + + 11 22 11 22 北京大学经济学院 Slide35 Quantity Index Numbers ? If (p 1 ,p 2 ) = (p 1 t ,p 2 t ) ，then we have the Paasche quantity index（帕氏指 数）; P px px px px q tt tt tb tb = + + 11 22 11 22 北京大学经济学院 Slide36 Quantity Index Numbers ? If then so consumers overall were better off in the base period than they are now in the current period. L px px px px q bt bt bb bb = + + < 11 22 11 22 1 px px px px bt bt bb bb 11 22 11 22 +<+ 北京大学经济学院 Slide37 Quantity Index Numbers ? If then so consumers overall are better off in the current period than in the base period. P px px px px q tt tt tb tb = + + > 11 22 11 22 1 px px px px tt tt tb tb 11 22 11 22 +>+ 北京大学经济学院 Slide38 Price Index（价格指数） ? A price index is a quantity-weighted average of price; ? (x 1 ,x 2 ) can be base period quantities (x 1 b ,x 2 b ) or current period quantities (x 1 t ,x 2 t ). 21 21 21 21 xpxp xpxp I bb tt q + + = 北京大学经济学院 Slide39 Price Index Numbers ? If (x 1 ,x 2 ) = (x 1 b ,x 2 b ) then we have the Laspeyres price index(拉氏价格指数); bbbb btbt p xpxp xpxp L 2211 2211 + + = 北京大学经济学院 Slide40 Price Index Numbers ? If (x 1 ,x 2 ) = (x 1 t ,x 2 t ) ，then we have the Paasche price index（帕氏价格指 数）; tbtb tttt p xpxp xpxp P 2211 2211 + + = 北京大学经济学院 Slide41 Price Index Numbers ? Define the expenditure ratio（支出比 率） M px px px px tt tt bb bb = + + 11 22 11 22 北京大学经济学院 Slide42 Price Index Numbers ? If then so consumers overall are better off in the current period. L px px px px p tb tb bb bb = + + 11 22 11 22 < + + = px px px px M tt tt bb bb 11 22 11 22 px px px px tb tb tt tt 11 22 11 22 +<+ 北京大学经济学院 Slide43 Price Index Numbers ? If then so consumers overall were better off in the base period. P px px px px p tt tt bt bt = + + 11 22 11 22 > + + = px px px px M tt tt bb bb 11 22 11 22 px px px px bt bt bb bb 11 22 11 22 +<+ 北京大学经济学院 Slide44 Indexation（指数化）? ? Changes in price indices are sometimes used to adjust wage rates or transfer payments. This is called “indexation”（指数化）. ?“indexation” occurs when the wages or payments are increased at the same rate as the price index used to measure inflation rate. 北京大学经济学院 Slide45 Full Indexation? ? A common proposal is to index fully Social Security payments, with the intention of preserving for the elderly the “purchasing power” of these payments. 北京大学经济学院 Slide46 Indexation? ? The usual price index proposed for indexation is the Paasche quantity index. ? What will be the consequence? 北京大学经济学院 Slide47 Indexation? P px px px px q tt tt tb tb = + + 11 22 11 22 Notice that this index uses current period prices to weight both base and current period consumptions. 北京大学经济学院 Slide48 Full Indexation? x 2 x 1x 1 b Base period budget constraint Base period choice Current period choice after indexation Current period budget constraint before indexation x 1 t Current period budget constraint after indexation x 2 b x 2 t 北京大学经济学院 Slide49 Full Indexation? x 2 x 1x 1 b x 1 t (x 1 t ,x 2 t ) is revealed preferred to (x 1 b ,x 2 b ) so full indexation makes the recipient strictly better off x 2 b x 2 t 北京大学经济学院 50 The End 北京大学经济学院 51 Last Revised: Oct. 8, 2005 北京大学经济学院 1 Chapter Three Consumer’s Behavior Theory: Further analysis ? 2005 北京大学经济学院 Slide2 Chapter 3 includes: ? 3.1 Revealed Preference Theory(显示性偏好理论) ? 3.2 Choice Under Uncertainty(不确定条件下的 选择理论) 北京大学经济学院 Slide3 Outline of Today’s Class ? Probability and Expected Value(期望值) ? The Expected Utility（期望效用） ? St. Petersburg Paradox（圣彼得堡悖论）and its solution ? von Neumann-Morgenstern utility function ? Attitudes Towards Risk（对待风险的态度） ? Measuring Risk Aversion ? Risk Aversion and Insurance 北京大学经济学院 Slide4 Readings about the part of this chapter ? Zhang: Chapter 15,p438-458 ? Nicholson: Chapter 8, P198-224 北京大学经济学院 Slide5 Types of Decision-Making Environments ? Type 1: Decision-making under certainty – The decision-maker knows with certainty the consequences of every alternative or decision choice ? Type 2: Decision-making under risk – The decision-maker does know the probabilities(概率) of the various outcomes ? Type 3:Decision-making under uncertainty – The decision-maker does not know the probabilities of the various outcomes 北京大学经济学院 Slide6 Types of Uncertainty ? Uncertainty in prior knowledge E.g., some causes of a disease are unknown and are not represented in the background knowledge of a medical-assistant agent ? Uncertainty in actions E.g., actions are represented with relatively short lists of preconditions, while these lists are in fact arbitrary long ? Uncertainty in perception E.g., sensors do not return exact or complete information about the world; a robot never knows exactly its position 北京大学经济学院 Slide7 Questions ??? ? How to represent uncertainty in knowledge? ? How to perform inferences with uncertain knowledge? ? Which action to choose under uncertainty? 北京大学经济学院 Slide8 Decision-Making Under Risk Expected Monetary Value (EMV) nature.states of number of where n )P(SSative)EMV(Altern j n j j = ×= ∑ =1 北京大学经济学院 Slide9 Investment Decision Table for Mineral Water Plant State of Nature A lternative Favorable Market Unfavorable Market Construct a large plant $200,000 -$180,000 Construct a sm all plant $100,000 -$20,000 D o nothing $0 0 北京大学经济学院 Slide10 Decision Table for Mineral Water Plant Favorable Market Unfavorable Market Alternative State of Nature EMV Construct a large plant $200,000 -$180,000 $10,000 Construct a small plant $100,000 -$20,000 $40,000 Do nothing $0 0 probability 0.50 0.50 北京大学经济学院 Slide11 Random Variables（随机变量） （随机变量） ? A proposition that takes the value True with probability πand False with probability 1-πis a random variable with distribution (π,1-π) ? The (probability) distribution（概率分布） of a random variable X with n values x 1 , x 2 , …, x n is: (π 1 , π 2 , …, π n ) with π(X=x i ) = π i and Σ i=1,…,n π i = 1 北京大学经济学院 Slide12 Expected Value（期望值） （期望值） ? Random variable X with n values x 1 ,…,x n and distribution (π 1 , π 2 , …, π n ) E.g.: X is the state reached after doing an action A under uncertainty ? The expected value of U after doing A is E[U] = Σ i=1,…,n π i U(x i ) 北京大学经济学院 Slide13 Fair Game（公平游戏） ? Suppose that Jones and Smith agree to flip a coin. If a head comes up,Jones will pay Smith $1;if a tail,Smith will pay Jones $1. ? The expected value of the game for Smith is 0)1$( 2 1 )1($ 2 1 =?+ 北京大学经济学院 Slide14 Fair Game ? Now if a head comes up, Jones will pay Smith $10;if a tail, Smith will pay Jones $1. ? The new expected value of the game for Smith is 5.4$)1$( 2 1 )10($ 2 1 =?+ 北京大学经济学院 Slide15 Fair Game ? If this game is played many times, Smith will certainly end up the big winner. ? And Smith might be willing to pay Jones something for the privilege of playing the game, here, Games which are the expected values ($0 or $4.5) for the right to play are called Fair Games ? But in many situation, people refuse to participate in actuarially fair game 北京大学经济学院 Slide16 St. Petersburg Paradox （圣彼得堡悖论） ? Suppose that there is a game, A coin is flipped until a head appears. If a head first appears on the nth,the player is paid $2 n . ? If X i represents the prize awarded when the first head appears on the ith trial, then n n XXXX 2$,.....8$,4$,2$ 321 ==== 北京大学经济学院 Slide17 St. Petersburg Paradox Outcome Payoff Probability H $1 .5 TH $2 .25 TTH $4 .125 TTTH $8 .0625 . . . T..TH $2 n .5 n+1 (n T's) 北京大学经济学院 Slide18 St. Petersburg Paradox ? The probability of getting a head for the first time on the ith trial is ? The probability of prizes are ? The expected value of the St. Petersburg Paradox game is infinite: i ? ? ? ? ? ? 2 1 n n 2 1 ,......, 8 1 , 4 1 , 2 1 321 ==== ππππ ∑∑ ∞ = ∞ = ∞=+++=== 11 ...111 2 1 2)( ii i i ii XXE π 北京大学经济学院 Slide19 St. Petersburg Paradox ? No player would pay very much (much less than infinity) to play this game. ? So in some sense Bernoulli’s game is not worth its (infinite) expected dollar value. This is called St. Petersburg Paradox 北京大学经济学院 Slide20 Bernoulli’s (贝努里) Solution to St. Petersburg Paradox ? Bernoulli thought individuals do not care directly about the dollar prizes of a game; rather they respond to the utility these dollars provide. ? Suppose that MU of income declines as income increases,the St. Petersburg Paradox game may converge to a finite expected utility value that players would be willing to pay for the right to play. 北京大学经济学院 Slide21 Bernoulli’s Solution to St. Petersburg Paradox ? The utility of each prize in the St. Petersburg Paradox is ? The expected utility value of this game is )ln()( ii XXU = ∑∑ ∞ = ∞ = === 11 39.1)2ln( 2 1 )ln()( ii i i ii XXEU π 北京大学经济学院 Slide22 von Neumann-Morgenstern utility function(冯诺伊曼效用函数) ? L = (x 1 ,x 2 ,π) ? U(L) = πu(x 1 ) + [1 -π]u(x 2 ) U(L) is called a von Neumann- Morgenstern utility function ? In words, the psychic value of a lottery is given by the expected utility of the dollar payoffs 北京大学经济学院 Slide23 Expected utility hypothesis ? U(L) = E(u(x)) = Expected utility hypothesis ? In words, lotteries(彩票) are evaluated by the expected utility they yield. 北京大学经济学院 Slide24 The Expected Utility Hypothesis: ? The expected utility hypothesis says that among available lotteries, the decision maker chooses the lottery that maximizes expected utility. The virtue of taking this route in modeling decisions under risk is that it allows us to evaluate lotteries when there are more than two states. 北京大学经济学院 Slide25 The Axiomatic Approach ? Question: What assumptions do you need to get EU theory? 北京大学经济学院 Slide26 Von Neumann - Morgenstern Axioms ? 1.Completeness either X > Y, Y > X, or X ~ Y ? 2.Transitivity if X >~ Y and Y >~ Z, then X >~Z 北京大学经济学院 Slide27 Von Neumann - Morgenstern Axioms 3. Continuity: Suppose X is preferred to Y and Y is preferred to Z. L = (X,Z,p). Then for some p, 0 < p < 1, Y is indifferent to L 北京大学经济学院 Slide28 Von Neumann - Morgenstern Axioms 4. Independence（独立性）: Suppose X is indifferent to Y. Let L X = (X,Z,p) and L Y = (Y,Z,p). Then for all Z and p, L X is indifferent to L Y 5. Unequal Probability（不相等公理）: Suppose X is preferred to Y. Let Lp = (X,Y,p) and Lq = (X,Y,q). Then Lp is preferred to Lq if and only if p > q. 北京大学经济学院 Slide29 Von Neumann - Morgenstern Axioms 6. Compound Lottery（复合赌博公 理）: Let L 1 = (X,Y,p 1 ), L 2 = (L 3 ,L 4 ,p 2 ) and L 3 = (X,Y,P 3 ), L 4 = (X,Y,P 4 ) Consider the compound lottery: L 2 = (L 3 ,L 4 ,p 2 ) , If P 1 =P 2 P 3 +(1-P 2 )P 4， L 2 is indifferent to L 1 北京大学经济学院 Slide30 Is there a von Neumann- Morgenstern utility function? ? Under these axioms, a von Neumann- Morgenstern utility function exists 北京大学经济学院 Slide31 Is there a von Neumann- Morgenstern utility function? ? If the axioms are satisfied, then there exists a utility function U(X) such that the ordering of lotteries by utilities is equivalent to the ordering of preferences, and U(X) is interval. It can have any monotonic shape. ? Then we can measure U(X) - Certainty equivalent method - Probability equivalent method 北京大学经济学院 Slide32 Important point: ? The v-N-M utility function has an element of cardinality to it. (Recall that the utility function which represents certainty preferences is ordinal.) As we have seen, the curvature(曲率) of u is what determines the attitude towards risk. ? If v(x) = au(x) + b, where both a and b are positive, then v represents the same risk preferences as u. 北京大学经济学院 Slide33 Risky Choices ? When there is uncertainty, people do not know what event will occur – they also do not know the actual utility they will get ? The Expected utility can be derived from the Utility of Wealth – It is the Expectation of the utilities associated with possible wealth outcomes 北京大学经济学院 Slide34 Attitudes Towards Risk （对风险的态度） ? Risk Aversion – is associated with Diminishing Marginal Utility of Wealth。 – A risk averse person prefers a given amount of wealth with certainty to an equivalent level of Expected Wealth from risky outcomes – tends to avoid risk. 0 2 2 < dW Ud 北京大学经济学院 Slide35 Risk Aversion Total utility (units) Wealth (thousands of dollars) 65 80 95 036912 5 85 EW = $6,000 )9( 2 1 )3( 2 1 9 2 1 3 2 1 UUU ?+?> ? ? ? ? ? ? ?+? 北京大学经济学院 Slide36 Risk Aversion ? Risk Aversion also means – The certainty equivalent is less than the expected wealth – The cost of risk or Risk premium is positive 北京大学经济学院 Slide37 Risk Aversion Total utility (units) Wealth (thousands of dollars) 65 80 95 036912 5 85 Cost of risk = $1,000 > 0 北京大学经济学院 Slide38 Risk Neutrality（风险中性者） ? A risk-neutral person cares only about expected wealth and doesn’t mind how much uncertainty there is. 北京大学经济学院 Slide39 Suppose Mr. Wang is Risk Neutral 0 2 2 = dW Ud Wealth (thousands of dollars) Total utility (units) 50 75 100 036 912 25 Utility of wealth )9( 2 1 )3( 2 1 9 2 1 3 2 1 UUU ?+?= ? ? ? ? ? ? ?+? 北京大学经济学院 Slide40 Risk Lover（风险喜好者）? )9( 2 1 )3( 2 1 9 2 1 3 2 1 UUU ?+?< ? ? ? ? ? ? ?+? Wealth (thousands of dollars) Total utility (units) 50 75 100 036912 25 0 2 2 > dW Ud 北京大学经济学院 Slide41 Measuring Risk Aversion ? In the 1960s,J.M.Pratt developed a quantitative measure of risk aversion: ? If A person is risk aversion, his r(W)>0 ? If A person is risk neutrality , his r(W)=0 ? If A person is risk lover, his r(W)<0 )( )( )( ' " WU WU Wr ?= 北京大学经济学院 Slide42 The Certainty Equivalent(确定性等值) and the Cost of Risk（风险成本） ?Certainty Equivalent（CE） –The amount of money (wealth) obtained with certainty, which gives the same utility as the expected utility of an uncertain outcome ?The Cost of Risk _Risk Premium(风险升水) –The difference between the Expected Wealth of an action and the Certainty Equivalent )()1()()( 21 WUPWPUCEU ?+= CEWPWPRP ???+?= ])1([ 21 北京大学经济学院 Slide43 Risk Aversion and Certainty Equivalents - Diagram $ W 2 $0 EWW 1 U($) )]([ WEU ( ) 2 WU ( )CEU ( ) 1 WU CE ( ) )()1()( 21 WUPWPUCEU ?+= Risk Premium: RP=EW-CE Utility 北京大学经济学院 Slide44 Wealth (thousands of dollars) Total utility (units) 65 80 95 03 6.9 912 4.2 89 71 7.8 Certainty Equivalent and Risk Premium_ An Example RP= $900 $6,900 is the CERTAINTY EQUIVALENT of telemarketing ρ = 0.2 EW = $7.8 EU = 89 北京大学经济学院 Slide45 Risk Aversion and Fair Bet(公平打赌) ? W* is an individual current wealth. ? U=U(W),U”(W)<0. ? Two Fair game: – A 50-50 chance of winning or losing $h; – A 50-50 chance of winning or losing $2h ? The expected utility for game 1 is ? The expected utility for game 2 is )( 2 1 )( 2 1 )( *** hWUhWUWU h ?++= )2( 2 1 )2( 2 1 )( ***2 hWUhWUWU h ?++= 北京大学经济学院 Slide46 Risk Aversion and Fair Bet(公平打赌) 北京大学经济学院 Slide47 Risk Aversion and Fair Bet(公平打 赌) ? From the figure,we can see: ? Conclusion: – This person will prefer current wealth to that wealth combined with a fair game and will prefer a small game to a large one. ? Reason: – Winning a fair bet adds to enjoyment less than losing hurts )()()( *2** WUWUWU hh >> 北京大学经济学院 Slide48 Risk Aversion and Insurance ? Why do people want to buy insurance? ? In fact, the person might be willing to pay some amount to avoid participating in any game at all. ? the amount to be paid for this game is WW ? * 北京大学经济学院 Slide49 Risk Aversion and Insurance ? Definition: – An individual who always refuses fair bet is said to be risk aversion, if individuals exhibit a diminishing marginal utility of wealth, they will be risk averse. ? As a consequence, they will be willing to pay something to avoid taking fair bets 北京大学经济学院 Slide50 An example: ? Consider a person with a current wealth of $100,000 faces the prospect of a 25% chance of losing his automobile through theft in a year. ? His vNM Utility index is U(W)=ln(W) 北京大学经济学院 Slide51 An Example cont. ? Without insurance, – E[U(W)]=0.75U(100000)+0.25U(80000)=11.45714 ? A fair insurance premium would be $5000(20000X25%),if this person completely insures the care, his wealth would be $95000 – U(95000)=ln(95000)=11.46163 – This person is clearly better off when he purchases fair insurance. ? If he purchases a fair insurance, the maximum amount insurance fee is x, that is – E[U(100000-x)]=U(100000-x) – E[U(100000-x)]=EU(W) – U(100000-X)=11.45714 – X=5426 北京大学经济学院 52 The End 北京大学经济学院 53 Last Revised: Oct. 10, 2005 北京大学经济学院 Chapter Four Production and Cost Function ? 2005 MOL 北京大学经济学院 Chapter Four includes: ? 4.1 Production Function ? 4.2 Cost Function 北京大学经济学院 Outline of Today’s Class ? Definition of Product Function ? The Short Run and Long Run ? TP,AP and MP, Diminishing MP ? The Three Stages of Production 北京大学经济学院 Readings about this Part ? Zhang: Chapter 5 ? Nicholson: Chapter 11, P267-294 北京大学经济学院 北京大学经济学院 Symmetry Between Consumer and Firm Theory （消费者理论和生产者理论的对称关系） ? Use similar tools from consumer theory to analyse the firm. In general: Consumer Firm Maximizing utility Max (Min) profits (cost) Subject to Subject to Budget constraint(s) Technology (output) constraint(s) “Demand functions” for goods “Demand functions” for inputs Parallel concepts 北京大学经济学院 Production Technology(生产技术): Introduction ? Production Technology ? Process by which inputs are converted to outputs ? Inputs also called “factors of production”(生产要素) ? Input（1）: Labor (L) (skilled, unskilled) ? Input（2）: Physical capital (K)（有形资本） (equipment, structures, inventories（存货）, land), intangibles(无形资本)) ? Main issue: Which technology is best? 北京大学经济学院 Production Process（生产过程） Inputs: Labour, Capital Outputs: Goods and Services Production: The Firm 北京大学经济学院 Definition of Production Function ? x i - the amount used (level) of input i ? Input bundle(投入组合) -vector inputs (x 1 , x 2 , … , x n ) ? q -output level（产出水平） ? Production function ? The technology’s production function states the maximum amount of output possible from an input bundle. 1 (, , ) n qfx x= L 北京大学经济学院 Production functions– specific forms ? q=f(L, K) is a general form of the production function with two inputs ? Often we need to work with more specific functional forms. Here are four common ones ? Linear: q=a+bL+cK ? Fixed proportions: q=min(aL, bK) ? Cobb-Douglas: q=AL a K b ? CES: ? In each case q, L, K are variables and a, b, c ,ρε(and A) parameters (,)qfKL K L ε ρρ ρ ? ?==+ ? ? 北京大学经济学院 Short Run vs. Long Run（短期与长期） ? The short run is defined as the period of time when the plant size is fixed. In the short run, at least one factor of production is fixed (unchangeable) ? The long run is defined as the time period necessary to change the plant size. In the long run, all factors of production can be changed… are “variable” ? The duration of the long run (and thus that of the short run) depends on the nature of the production process… 北京大学经济学院 The Short-run Production Function(短期生产函数) Production When Only One Input is Variable 北京大学经济学院 Production When Only One Input is Variable ? There are three important ways to measure the productivity of inputs in short-run: ?Total product (TP) ?Average product (AP) ?Marginal product (MP) 北京大学经济学院 Total Product (TP)（总产量） ? Total Product represents the relationship between the number of workers (or Capitals) and the TOTAL number of units of output produced holding all other factors of production (the plant size) constant. (, ) (, ) L K TP f L K TP f L K = = 北京大学经济学院 The Production Function: TP increases with L 1 234 8 20 25 27 Units Produced Units of Labor q,TP L Inflexion (拐点) Max. Point 北京大学经济学院 0 50,000 0 1,000 LfQ Product, Total )(= 0 2 2 > dL Qd 0< dL dQ 0> dL dQ 0 2 2 < dL Qd Inflection Point (拐点) Negative Marginal Returns Variable Input TP,q Increasing Marginal Returns Diminishing Marginal Returns 北京大学经济学院 Average Product (AP)(平均产量) ? This function represents the average amount of output produced by each unit of labor. ? Output per unit of input (,) (,) L K Output q f KL AP LaborInput L L Output q f KL AP CapitalInput K K === === 北京大学经济学院 AP: slope of ray from origin… q L 10 150 units TP Slope = 150/10 = 15 AP (of 10 workers) = 15 L q=TP 北京大学经济学院 AP: Slope of ray from origin… q q 斜率最 大的点 斜率最 大的点 AP (slope of ray) Decreases after L 0 AP (slope of ray) Increases up to L 0 Lo Lo 北京大学经济学院 AP: Increases, reaches a maximum and decreases. AP Lo AP (slope of ray) Increases up to L 0 AP (slope of ray) Decreases after L 0 L 北京大学经济学院 Marginal Product (MP)(边际产量) ? The additional output that can be produced by adding one more unit of labor (or Capital), holding everything else constant. ? The slope of the Total Product Function LL KK q MP f L q MP f K ? == ? ? = = ? 北京大学经济学院 MP: Slope of the Production Function L TP MP = 10 30 3 Slope = 30/3 = 10 q 160 units 130 units 9 12 北京大学经济学院 The Relationship between TP and MP MP TP L Changes concavity MP is max MP L 北京大学经济学院 The Law of Diminishing Return (边际报酬递减法则). ? What happens to output as we hire more and more workers? ? Remember that the plant size is fixed. ? That means that you will be hiring more workers to SHARE the EXISTING EQUIPMENT and the EXISTING SPACE. 北京大学经济学院 The Law of Diminishing Return (边际报酬递减法则). ? As more of a variable input (labor) is added to a fixed input (plant), additions to output get smaller and smaller.. Note that adding workers increases output but the increases become smaller and smaller as more workers are hired. 北京大学经济学院 Diminishing Marginal Product () () Diminishing MP 0 0 K KK L LL MP f K MP f L ? = < ? ? =< ? 北京大学经济学院 Marginal Products: Diminishing MP Example 3/2 2 3/2 11 )3/1( xxMP ? = 3/1 2 3/1 12 )3/2( ? = xxMP and 1/3 2/3 12 qxx= ? ? MP x xx 1 1 1 53 2 23 2 9 0=? < ? // so 0 9 2 3/4 2 3/1 1 2 2 <?= ? xx x MP ? ? and Both marginal products are diminishing 北京大学经济学院 The Reasons of Diminishing Returns ? Reasons L MP Increasing Returns Teamwork(团队协作) and Specialization（专业化） Diminishing Returns Begins Fewer opportunities for teamwork and specialization MP 北京大学经济学院 The Relationship between AP and MP MP TP L AP is max MP is max Slope of ray is max Changes concavity MP,AP AP L 北京大学经济学院 Relationship between MP and AP MP A P i n c r e a s i n g MP below AP MP above AP A P d e c r e asi n g MP AP MP = AP when AP is max AP 北京大学经济学院 The Relationship between AP and MP ? If MP > AP, then the Average Product increases. ? If MP < AP, then the AP will decrease. ? If MP = AP, then the AP is not increasing or decreasing: it is at the maximum point. 北京大学经济学院 Proof of Relationships between AP and MP (,) L qfKL AP LL == = From , we can get: 2 LL dAP L MP q dL L ? ? = So, at a maximum L, L·MP L =q or MP L =AP L 北京大学经济学院 Three Stages of Production （生产的三个阶段） 北京大学经济学院 0 50,000 0 1,000 -100 0 100 0 1,000 Marginal Product, dq dL Average Product, q L Total Product, q ( ) f L= “Intensive” Margin “Extensive” Margin q , L dq dL Stage II Stage I Stage III Diminishing Marginal Returns Increasing Marginal Returns Negative Marginal Returns Inflection Point Three Stages of Production（生产的三个阶段） Variable Input Variable Input q 北京大学经济学院 The Three Stages of Production ? Stage I ? From zero units of the variable input to where AP is maximized ? Stage II ? From the maximum AP to where MP=0 ? Stage III ? From where MP=0 on 北京大学经济学院 The Three Stages of Production ? In the short run, rational firms should only be operating in Stage II. ? Why Stage II? ?Why not Stage III? ?Firm uses more variable inputs to produce less output! ?Why not Stage I? ?Underutilizing （未充分使用）fixed capacity. ?Can increase output per unit by increasing the amount of the variable input. 北京大学经济学院 The End 北京大学经济学院 Last Revised: Oct. 21, 2005 北京大学经济学院1 Chapter Four Production and Cost Function ? 2005 MOL 北京大学经济学院2 Chapter Four includes: ? 4.1 Production Function ? 4.2 Cost Function 北京大学经济学院3 Overview of Last Class ? Definition of Product Function ? The Short Run and Long Run ? TP,AP and MP, Diminishing MP ? The Three Stages of Production 北京大学经济学院4 Outline of Today’s Class ? Isoquant ? Diminishing MRTS ? Optimal Inputs in the Long Run ? Returns to Scale: CRS,IRS,DRS ? Elasticities of Output ? Elasticity of Productivity ? Elasticity of Substitution 北京大学经济学院5 Readings about the part of this chapter ? Zhang: Chapter 5 ? Nicholson: Chapter 11, P267-294 北京大学经济学院6 The Long-Run Production Function(长期生产函数) Production Function with All Inputs are Variable 北京大学经济学院7 Production Function with All Inputs are Variable _Long-run Analysis ? The two input, one output case ? Input levels are x 1 and x 2 ? Output level is q ? q=f(x 1 , x 2 ) 北京大学经济学院8 Production Function : Multiple Inputs ? Suppose production function is Cobb-Douglas 1/3 1/3 12 1 2 (, ) 2qfxx xx== ? Max output from inputs (x 1 , x 2 ) = (1, 8) is 1/3 1/3 1/3 1/3 12 22182124qxx==××=×= ? Max output from inputs (x 1 ,x 2 ) = (8,8) is 1/3 1/3 1/3 1/3 12 2288228qxx==××=×= 北京大学经济学院9 Production Function with Multiple Inputs Output, y x 1 x 2 (8,1) (8,8) x 1 x 2 q 北京大学经济学院10 Output Mountain 北京大学经济学院11 Isoquants in 3-D Output, y x 1 x 2 y ≡8 y ≡4 Add third axis for output level q=f(x 1 , x 2 ) 北京大学经济学院12 Isoquants with Two Variable Inputs y ≡8 y=4 x 1 x 2 q=4 q=8 北京大学经济学院13 Isoquants with Two Variable Inputs Output, y x 1 x 2 y ≡8 y ≡4 y ≡6 y ≡2 北京大学经济学院14 Isoquants with Two Variable Inputs y ≡8 y ≡4 x 1 x 2 y ≡6 y ≡2 More isoquants tell us more about the technology 北京大学经济学院15 Production Function with Multiple Inputs ? Isoquant Curves（等产量曲线） ? Set of all input bundles yielding output level q ? Analogous(类似) to indifference curve ? Typically implies substitution across inputs possible 北京大学经济学院16 Production Function with Multiple Inputs ? Isoquant map ? complete isoquant collection ? is equivalent to production function ? Can be represented in 2-D or 3-D ? For example, if 1/3 1/3 12 1 2 (, ) 2qfxx xx== 北京大学经济学院17 Production Function with Multiple Inputs x 1 x 2 y 北京大学经济学院18 Production Function with Multiple Inputs x 1 x 2 y 北京大学经济学院19 Production Function with Multiple Inputs x 1 x 2 y 北京大学经济学院20 Production Function with Multiple Inputs x 1 x 2 y 北京大学经济学院21 Production Function with Multiple Inputs x 1 x 2 y 北京大学经济学院22 Production Function with Multiple Inputs x 1 x 2 y 北京大学经济学院23 Production Function with Multiple Inputs x 1 y 北京大学经济学院24 Production Function with Multiple Inputs x 1 y 北京大学经济学院25 Production Function with Multiple Inputs x 1 y 北京大学经济学院26 Production Function with Multiple Inputs x 1 y 北京大学经济学院27 Production Function with Multiple Inputs x 1 y 北京大学经济学院28 Production Function with Multiple Inputs x 1 y 北京大学经济学院29 Production Function with Multiple Inputs x 1 y 北京大学经济学院30 Production Function with Multiple Inputs x 1 y 北京大学经济学院31 Production Function with Multiple Inputs x 1 y 北京大学经济学院32 Production Function with Multiple Inputs x 1 y 北京大学经济学院33 Example: Cobb-Douglas Production Function ? General Cobb-Douglas production function is 12 12 n aaa n qAxx x=××L ? For example, if n=2, A=1, a 1 =1/3, a 2 =1/3 1/3 1/3 12 qxx= 北京大学经济学院34 Cobb-Douglas Technologies: Graph x 2 x 1 All isoquants are hyperbolic （双曲线）,asymptoting（ 渐近线）to, but never touching, any axis 北京大学经济学院35 Example: Fixed-Proportions Technologies ? A fixed-proportions production function is of form 11 22 min{ , , , } nn qaxaxax= L ? For example, with n=2, a 1 =1, a 2 =2, 12 min{ , 2 }qxx= 北京大学经济学院36 Fixed-Proportions Technologies: Graph x 2 x 1 min{x 1 ,2x 2 } = 14 4814 2 4 7 min{x 1 ,2x 2 } = 8 min{x 1 ,2x 2 } = 4 x 1 = 2x 2 12 min{ , 2 }qxx= 北京大学经济学院37 Example: Perfect-Substitutes Technologies ? A perfect-substitutes production function is of form 11 22 nn qax ax ax= +++L ? For example, with n=2, a 1 =1 and a 2 =3, 12 3qx x= + 北京大学经济学院38 Perfect-Substitution Technologies: Graph 9 3 18 6 24 8 x 1 x 2 x 1 + 3x 2 = 9 x 1 + 3x 2 = 18 x 1 + 3x 2 = 24 All isoquants linear and parallel 12 3qx x= + 北京大学经济学院39 The Marginal Rate of Technical Substitution (MRTS) （边际技术替代率） " 1 x " 2 x 2 12 1 x MRTS x ? =? ? x 2 x 1 q=100 x 2 ' x 1 ' Definition: The rate at what rate can a firm substitute one input for another without changing output. A B 1 x? 2 x? 北京大学经济学院40 The Marginal Rate of Technical Substitution : Graph x 2 x 1 q≡100 The Marginal Rate of Technical Substitution = Slope of isoquant = Rate at which x 2 must be given up as x 1 is increased to keep q constant x 2 ' x 1 ' 1 22 12 0 11 lim x xdx MRTS xdx ?→ ?? ? =?=? ?? ? ?? 北京大学经济学院41 The Marginal Rate of Technical Substitution ? Output q is constant along isoquant ? Production function q = f(x 1 , x 2 ) ? A small change (dx 1 , dx 2 ) in input bundle causes a change to output level q of 12 12 0 qq dq dx dx xx ? ? ?? =+= 北京大学经济学院42 The Marginal Rate of Technical Substitution 12 12 211 12 122 0 / / qq dq dx dx xx dx q x MP MRTS dx q x MP ?? ?? =+= ?? =? = = ?? 北京大学经济学院43 MRTS: Cobb-Douglas Example 12 12 (, ) ab qfxx xx== 1 12 1 ab q ax x x ? ? ? = 1 12 2 ab q bx x x ? ? ? = MRTS 12 is 1 2122 12 1 1211 / / ab ab dx q x ax x ax MRTS dx q x bx x bx ?? ?? ? ? =? = = = 北京大学经济学院44 MTRS: Cobb-Douglas Example x 2 x 1 222 12 111 (1/ 3) (2/3) 2 ax x x MRTS bx x x == = 1/3 2/3 12 ;1/3;2/3qxx a b=== 北京大学经济学院45 MRTS: Cobb-Douglas Example x 1 8 4 2 1 8 1 224 x MRTS x = == × x 2 北京大学经济学院46 MRTS: Cobb-Douglas Example x 1 6 12 2 1 61 22124 x MRTS x = == × x 2 北京大学经济学院47 Law of Diminishing Marginal Rate of Technical Substitution（边际技术替代率递减法则） 1 2 3 4 5 Isoquants are downward sloping and convex like indifference curves. 1 1 1 1 2 1 2/3 1/3 q 1 =55 q 2 =75 q 3 =90 Capital per year Labor per month 12345 北京大学经济学院48 Diminishing MRTS(边际技术替代率 递减法则) ? Usually assume that MRTS KL is diminishing ? Follows from the fact that MP of capital and labour is decreasing. Thus, K L q L ? ? = MP L , gets smaller as we increase L when we substitute L for K, while q K ? ? = MP k gets bigger as K gets smaller. 北京大学经济学院49 Diminishing MRTS So as L gets bigger and K gets smaller, the top of the line goes down while the bottom goes up, so dK/dL gets smaller as L gets bigger That is, Isoquants are Quasi ‘convex’ K L q dK L q dL K ? ? ?= ? ? 北京大学经济学院50 () () () 22 Proof: 0 Total Differential L K LK LK LK LK KL KL KK dMRTS dL f MRTS f ff ff dMRTS dL dK ff ff ff ff LL KK dL dK < = ?????? =+ ???? ???? ?? ???? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? Reason for Diminishing MRTS 北京大学经济学院51 Reason for Diminishing MRTS ()()() () 22 2 2 Given that _ K LL L KL K LK L KK KK L KL LK K K LL L KL K LK L KK K LL KLL LKL KKL LKK KK K dMRTS f fffdLff ffdK dL f dL f dL dK f and f f dL f dK dK f f ff f f ff dMRTS dL dL dL f ff ff ff ff ff f f ??? ? ?? =+ ??? ? ??? ? ?? ?? ?= = ???? ?? ?? ?+ ? = ?? ?? ?+ ? ? ?? ?? ?? ?? = = 22 2 2 2 K LL K LKL K LKL L KK K KLL KLKL LKK K fff ff ff f ff fff ff f ?? ??+ ?? ?? ?+ = 北京大学经济学院52 If 0 then 0 KL dMRTS f dL >< ? ? ( ) 0 LMP K ( ) 1 LMP K Where L 1 > L 0 K MP KK 0 北京大学经济学院53 MRTS different from diminishing marginal product ? Note MRTS different from diminishing marginal product ? As we noted above, ‘Law’ of diminishing marginal product says dq/dL gets smaller as L gets bigger holding all other inputs constant y x i 北京大学经济学院54 MRTS different from diminishing marginal product But in this exercise we are reducing K as we increase L, so all other things are not constant So MRTS is not the same as Diminishing Marginal Product, though they are related. x 2 x 1 q dK L q dL K ? ? = ? ? 北京大学经济学院55 Three Distinct Concepts about Production Function ? 1. Diminishing Marginal Product（边际报 酬递减法则） ? 2. Diminishing Marginal Rate of … …Technical Substitution（边际技术替代 率递减法则） ? 3. Returns to Scale（规模报酬） 北京大学经济学院56 Ridge Lines(脊线) and Economic Region L K Positive slope Positive slope horizontal Ver- tical Negative Slope A B C D E RIDGE LINES 北京大学经济学院57 Ridge Lines(脊线) and Economic Region 北京大学经济学院58 Well-Behaved Technologies: Introduction ? A well-behaved technology is ? Monotonic （单调性）– more inputs yield more output ? Convex (凸性) ? Suppose input bundles x’ and x” both provide q units ? Suppose 0 < t < 1 ? Then the mixture tx’ + (1-t)x” provides at least q units of output 北京大学经济学院59 Well-Behaved Technologies - Convexity x 2 x 1 x 2 ' x 1 ' x 2 " x 1 " q=100 北京大学经济学院60 Well-Behaved Technologies - Convexity x 2 x 1 x 2 ' x 1 ' x 2 " x 1 " ( ) tx t x tx t x 1122 '"'" (), ()+? +? q≡100 北京大学经济学院61 Well-Behaved Technologies - Convexity x 2 x 1 x 2 ' x 1 ' x 2 " x 1 " q≡100 q≡120 ( ) tx t x tx t x 1122 '"'" (), ()+? +? 北京大学经济学院62 Well-Behaved Technologies - Convexity x 2 x 1 x 2 ' x 1 ' x 2 " x 1 " Convexity implies that the MRTS becomes smaller and smaller as x 1 increases 北京大学经济学院63 Well-Behaved Technologies x 2 x 1 q 2 ≡100 q 1 ≡50 q 3 ≡200 Higher output 北京大学经济学院64 Production Function Utility Function Output from inputs Preference level from purchases Derived from technologies Derived from preferences Cardinal(Defn: given amount of inputs yields a unique and specific amount of output) Ordinal Marginal Product Marginal Utility 北京大学经济学院65 Isoquant(Defn: all possible combinations of inputs that just suffice to produce a given amount of output) Indifference Curve Marginal Rate of Technical Substitution Marginal Rate of Substitution Production Function Utility Function 北京大学经济学院66 The Optimal Choice In the Long Run The optimal choices includes: ? 1.Maximization of output for a given cost ? 2.Minimization of cost for a given output 北京大学经济学院67 The Optimal Choice In the Long Run Maximization of output for a given cost is one of the optimal choices Isocost Lines(等成本线) Various combinations of inputs that a firm can buy with the same level of expenditure (Cost) wL + rK = C where C is a given money outlay (Cost). 北京大学经济学院68 L K 0 C 0 /r C 0 /w Slope = -w /r L r w r C K ??= 0 北京大学经济学院69 Maximization of output for given cost Labor Capital 0 100 200 300 E 北京大学经济学院70 Condition for the optimal choices: MP L /w = MP K /r Labor Capital 0 100 200 300 E L LK K LK MPw MRTS MPr MP MP wr = = ?= 北京大学经济学院71 The Optimal Choice In the Long Run ? Production Function with Two Inputs ? Cost Lines ? Modeling Maximization of output for given cost (, )q f LK= CwLrK= ?+? 0 .: ( , ) ..: Max q f L K St C w L r K = = ?+? 北京大学经济学院72 The Optimal Choice In the Long Run ? The condition for The Optimal Choice In the Long Run ? Best inputs for this model: LLK K MPwMPMP MPr w r =? = 0 0 (,, ) (,, ) LLwrC KKwrC ? ? = = 北京大学经济学院73 Units of capital (K) O Units of labour (L) EXPANSION PATH（扩展线） TC 1 100 At an output of 100 The firm’s expansion path is the locus of cost- minimizing tangencies. On the assumption of fixed input prices, the curve shows how input use increases as output increases. 北京大学经济学院74 EXPANSION PATH Units of capital (K) O Units of labour (L) TC 1 100 TC 2 200 At an output of 200 北京大学经济学院75 Units of capital (K) O L TC 1 TC 2 TC 3 TC 4 TC 5 TC 6 TC 7 100 200 300 400 500 600 700 Note: increasing returns to scale up to 400 units; decreasing returns to scale above 400 units EXPANSION PATH 北京大学经济学院76 Units of capital (K) O L TC 1 TC 2 TC 3 TC 4 TC 5 TC 6 TC 7 100 200 300 400 500 600 700 Expansion path 北京大学经济学院77 Returns-to-Scale(规模报酬): Introduction ? Marginal product ? change in output as single input changes ? Returns-to-scale ? Change in output as all input change proportionally ? e.g. all input levels doubled, or halved 北京大学经济学院78 Type of Returns-to-Scale ? Constant Returns to Scale（规模报酬固 定） ? Decreasing Returns-to-Scale（规模报酬递 减） ? Increasing Returns-to-Scale（规模报酬递 增） 北京大学经济学院79 Constant Returns to Scale（规模报酬固定） If, for any input bundle (x 1 ,…,x n ), f kx kx kx kf xx x nn (,,, ) (,,,) 12 12 LL= the technology exhibits constant returns-to-scale E.g. (k = 2) doubling all inputs, doubles output 北京大学经济学院80 Constant Returns to Scale Input q = f(x) x’ q’ 2x’ 2q’ Output 北京大学经济学院81 Further Explanation for CRS Constant returns to scale: output doubles when all inputs are doubled ?Size does not affect productivity ?May have a large number of producers ?Isoquants are equidistant(等距离) apart 北京大学经济学院82 Further Explanation for CRS Labor (hours) Capital (machine hours) Constant Returns: Isoquants are equally spaced 10 20 30 15510 2 4 0 A 6 北京大学经济学院83 Decreasing Returns-to-Scale（规模报酬递减） If, for any input bundle (x 1 ,…,x n ), f kx kx kx kf xx x nn (,,, ) (,,,) 12 12 LL< technology has decreasing returns-to-scale E.g. (k = 2) doubling all input levels less than doubles the output level 北京大学经济学院84 Decreasing Returns-to-Scale q = f(x) x’ f(x’) 2x’ f(2x’) 2f(x’) Input Output 北京大学经济学院85 Further Explanation for DRS Decreasing returns to scale: output less than doubles when all inputs are doubled ?Decreasing efficiency with large size ?Reduction of entrepreneurial abilities(企业家才能) ?Isoquants become farther apart 北京大学经济学院86 Further Explanation for DRS Labor (hours) Capital (machine hours) 10 20 30 Decreasing Returns: The isoquants move farther apart 510 2 4 0 A 北京大学经济学院87 REASONS FOR DRS ? Problems of coordination(协调) and control（控 制）as it is hard to send and receive information as the scale rises. ? Other disadvantages of large size: ? slow decision ladder ? inflexibility ? capacity limitations on entrepreneurial skills (there are diminishing returns to the C.E.O. which cannot be completely delegated). 北京大学经济学院88 Increasing Returns-to-Scale（规模报酬递增） If, for any input bundle (x 1 ,…,x n ), f kx kx kx kf xx x nn (,,, ) (,,,) 12 12 LL> then technology has increasing returns-to-scale E.g. (k = 2) doubling all input levels more than doubles the output level 北京大学经济学院89 Increasing Returns-to-Scale y = f(x) x’ f(x’) 2x’ f(2x’) 2f(x’) Input Output 北京大学经济学院90 Further Explanation for IRS Increasing returns to scale: output more than doubles when all inputs are doubled ?Larger output associated with lower cost (autos) ?One firm is more efficient than many (utilities) ?The isoquants get closer together 北京大学经济学院91 Further Explanation for IRS Labor (hours) Capital (machine hours) 10 20 30 Increasing Returns: The isoquants move closer together 510 2 4 0 A 北京大学经济学院92 REASONS FOR IRS ? Specialization in the use of capital and labor. Labor becomes more skilled at tasks, or the equipment is more specialized, as scale increases. ? Other advantages include: avoid inherent lumpiness in the size of equipment, quantity discounts, technical efficiencies in building larger volume equipment. 北京大学经济学院93 Different Returns to Scale (RTS) q = f(x) Decreasing returns-to-scale Increasing returns-to-scale A single technology can ‘locally’ exhibit different returns-to-scale Input Output 北京大学经济学院94 Examples of RTS: Perfect Substitutes The perfect-substitutes production function is 11 22 nn qax ax ax=+++L Expand all input levels proportionately by k: 11 22 11 2 2 '()() () () nn nn q a kx a kx a kx kax ax ax ky =+++ =+++ = L L The perfect-substitutes production function is CRS 北京大学经济学院95 Examples of RTS: Perfect Complements Perfect-complements production function is 11 22 min{ , , , } nn qaxaxax= L Expand all input levels proportionately by k: 1122 11 2 2 'min{ (),(),,()} (min{ , , , }) nn nn qakxakxakx kaxaxax kq = = = L L The perfect-complements production is CRS 北京大学经济学院96 Examples of RTS: Cobb-Douglas The Cobb-Douglas production function is 12 12 n aaa n qxx x= L Expand all input levels proportionately by k: 12 12 12 12 12 1 12 12 '()() () n nn nn n aaa n aaaa aa aa a aaa n aa qkxkx kx kk k xx x kxxx kq +++ ++ = = = = L L L LL L 北京大学经济学院97 Examples of RTS: Cobb-Douglas 1 ' n aa qk q ++ = L The Cobb-Douglas technology’s RTS: constant if a 1 + … + a n = 1 increasing if a 1 + … + a n > 1 decreasing if a 1 + … + a n < 1 北京大学经济学院98 Returns-to-Scale and Diminishing MP ? RTS and MP ? RTS refers to change in all inputs ? MP refers to change in one input, holding all others constant ? Declining MP reflects each new input having less of others to “work with” and becoming less productive ? With RTS, each input has same amount of other inputs to “work with” so RTS need not diminish ? Illustration ? Can have increasing RTS and diminishing MP 北京大学经济学院99 Returns-to-Scale and Diminishing MP 12 2/3 2/3 12 12 1 2 ;4/3, aa qxx xx aa soIRS== += 0)9/2(;)3/2( 3/2 2 3/4 1 1 1 3/2 2 3/1 11 <?= ? ? = ?? xx x MP xxMP 0)9/1(;)3/2( 3/4 2 3/2 1 2 2 3/1 2 3/2 11 <?= ? ? = ?? xx x MP xxMP So, IRS, but both MP diminishing 北京大学经济学院100 Homogenous Production Function A production function is homogeneous of degree α if f(λK, λL) = λ α f(K,L) Ifα=1,this product function is CRS If α>1, this product function is IRS If α<1, this product function is DRS Note: Not all production functions are homogeneous. 北京大学经济学院101 Properties of Production Function ? 1. Elasticity of Output（产出弹性）: L L L K K K q qL MP q e L Lq AP L q qK MP q e K Kq AP K ? ? ==?= ? ? ? ? ==?= ? ? 北京大学经济学院102 Elasticities of Output: e L ? When MP L > AP L , then the labor elasticity, e L > 1. A 1 percent increase in labor will increase output by more than 1 percent. ? When MP L < AP L , then the labor elasticity, e L < 1. A 1 percent increase in labor will increase output by less than 1 percent. 北京大学经济学院103 Properties of Production Function ? 2.Elasticity of Productivity（生产力弹性）: L dq dq x q e dx dx q x = =? is the percentage of all factor changes dx dL dK xLK == Proposition 12 ..... Pn eee e= ++ + 北京大学经济学院104 Elasticity of Productivity and Return to Scales The percentage change in output resulting from 1 percent increase in all inputs. ? e p > 1 ==> increasing returns ? e p < 1 ==> decreasing returns ? e p = 1 ==> constant returns 北京大学经济学院105 Properties of Production Function ? 3.Elasticity of Substitution（替代弹性） ? The Elasticity of Substitution is the ratio of the proportionate change in factor proportions to the proportionate change in the slope of the isoquant. ? Intuition: If a small change in the slope of the isoquant leads to a large change in the K/L ratio then capital and labour are highly substitutable. 北京大学经济学院106 Elasticity of Substitution A small change in the MRTS will lead to Large change in K/L. So, we can get High σ,then K and L are highly substitutable for each other K L 北京大学经济学院107 Elasticity of Substitution A large change in the MRTS will lead to Small change in K/L So,Low σ,then K and L are not highly substitutable for each other K L 北京大学经济学院108 Elasticity of Substitution % Change in K/L % Change in Slope of Isoquant % Change in K/L / ln / % Change in MRTS / ln dK L MRTS K L dMRTS K L MRTS σ σ = ? ==?= ? 北京大学经济学院109 Elasticity of Substitution ( ) () () () (/) / (/) L K L K L K L K dKL dK L KL KL dMRTS M P d MRTS M P MP MP MP K d MP L K MP d L MP σ == ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? =? ???? ?? ?? ?? ?? 北京大学经济学院110 Elasticity of Substitution ? In equilibrium,MRTS = w/r and so the formula for σ reduces to, ( ) () () () (/) / (/) dKL dK L KL K L wdMRTS d r MRTS w r Kw d Lr wK d rL σ == ?? ?? ?? ?? ?? ?? ???? ???? ???? =? ???? ???? ???? 北京大学经济学院111 Properties of the Cobb-Douglas Production Function q=AK α L β The elasticity of substitution = 1 For the Cobb-Douglas, of σ=1 means that a 10% change in the factor price ratio leads to a 10% change in the opposite direction in the factor input ratio. 北京大学经济学院112 "The shape of the isoquant indicates the degree of substitutability of the inputs…" Example: The Elasticity of Substitution L K 0 σ = 0 σ = 1 σ = 5 σ = ∞ 北京大学经济学院113 Properties of Production Function 4.Euler’s theorem(欧拉定理): () KL MPK MPL qε+ = Where ε is the degree of homogeneity 北京大学经济学院114 Properties of the Cobb-Douglas Production Function q=AK α L β The Cobb-Douglas is homogeneous of degree ε = (α+ β). 北京大学经济学院115 Properties of the Cobb-Douglas Production Function Given q=K α L β now introduce t q’=(tK) α (tL) β = t α K α t β L β =t α+ β K α L β = t α+ β q So, q’=t ε q, as ε=α+β If ε =1 (α+β=1) then CRS If ε >1 (α+β>1) then IRS If ε <1 (α+β<1) then DRS 北京大学经济学院116 Properties of the Cobb-Douglas Production Function Output Elasticity q=AK α L 1- α . K qK e Kq α ? == ? .1 L qL e Lq α ? ==? ? For Capital For Labour 北京大学经济学院117 Properties of the Cobb-Douglas Production Function Y=AK α L 1- α Marginal Product of Capital k AP.α Marginal Product of Labour ( ) L AP.1 α? 北京大学经济学院118 Properties of the Cobb-Douglas Production Function Y=AK α L 1- α Marginal Rate of Technical Substitution (MRTS) ( ) L K α α?1 北京大学经济学院119 The End 北京大学经济学院120 Last Revised: October 24, 2005 北京大学经济学院1 Chapter Four Production and Cost Function ? 2005 MOL 北京大学经济学院2 Chapter Four includes: ? 4.1 Production Function ? 4.2 Cost Function 北京大学经济学院3 Overview of Last Week ? Production Function ? TP,AP and MP ? Isoquant and Diminishing MRTS ? Optimal Inputs in the Long Run ? Returns to Scale: CRS,IRS,DRS ? Elasticities of Output, Productivity ? Elasticity of Substitution 北京大学经济学院4 Outline of Today’s Class ? Definition of Cost Function ? opportunity costs ? Explicit and Implicit Costs ? From Cost Minimization to the Cost Function ? Cost Functions In the Short Run ? STC,VC,FC;SAC,SAVC,AFC;SMC 北京大学经济学院5 Readings about the part of this chapter ? Zhang: Chapter 6,P191-216 ? Nicholson: Chapter 12, P297-330 北京大学经济学院6 Measuring Cost: Which Costs Matter? ? Accounting Cost(会计成本) ? Actual expenses plus depreciation charges(折旧费) for capital equipment ? Economic Cost（经济成本） ? Cost which a firm utilize all economic resources in production, including opportunity cost Economic Cost vs. Accounting Cost Economic Cost vs. Accounting Cost 北京大学经济学院7 ? The Opportunity cost of using some resource in a particular way is defined as the value of that resource in its next best alternative use . What is Opportunity Cost? 北京大学经济学院8 ? An Example ? A firm owns its own building and pays no rent for office space ? Does this mean the cost of office space is zero? ? Answer: No What is Opportunity Cost? 北京大学经济学院9 ? Sunk Cost（沉没成本） ? Expenditure that has been made and cannot be recovered ? Should not influence a firm’s decisions. 北京大学经济学院10 Explicit and Implicit Costs ? A firm’s cost of production include explicit costs and implicit costs. ?Explicit costs（显性成本）involve a direct money outlay for factors of production which are bought from input markets. ?Implicit costs（隐性成本）are those costs associated with the use of the firm’s own resources and reflect the fact that these resources could be employed elsewhere. 北京大学经济学院11 Economic Profit versus Accounting Profit ? Economists measure a firm’s economic profit （经济利润）as total revenue minus all the opportunity costs (explicit and implicit). ? Accountants measure the accounting profit（会 计利润）as the firm’s total revenue minus only the firm’s explicit costs. In other words, they ignore the implicit costs. 北京大学经济学院12 Economic Profit versus Accounting Profit ?When total revenue exceeds both explicit and implicit costs, the firm earns economic profit. ?Economic profit is smaller than accounting profit. 北京大学经济学院13 Economic Profit versus Accounting Profit Revenue Total opportunity costs How an Economist Views a Firm Explicit costs Economic profit Implicit costs Explicit costs Accounting profit How an Accountant Views a Firm Revenue 北京大学经济学院14 Definition of Cost Function ? Cost Function ? shows the relation between the minimum cost to produce a given level of output and output level, )(qCC = 北京大学经济学院15 Short-Run and Long-Run Cost Functions ? In the short run, some factors of production are fixed: short-run cost function gives the minimum cost to produce a given level of output, only adjusting the variable factors of production. ? In the long run all factors are variable: long run cost function gives the minimum cost to produce a given level of output, adjusting all factors of production. 北京大学经济学院16 From Cost Minimization to the Cost Function ? Cost-minimizing firm ? produces q ≥ 0 at min total cost ? Min cost for each q yields: ? The Total cost function - c(q) ? With input prices w = (w 1 ,w 2 ,…,w n ), ? The Total cost function is C=c(w 1 ,…,w n ,q) 北京大学经济学院17 The Cost-Minimization Problem ? Consider a firm using 2 inputs to make 1 output ? Production function is q = f(x 1 ,x 2 ) ? Take output level q ≥ 0 as given. ? Given input prices w 1 and w 2 , ? Total cost of input bundle (x 1 ,x 2 ) is w 1 x 1 + w 2 x 2 北京大学经济学院18 The Cost-Minimization Problem ? So, for given w 1 , w 2 and q, firm’s cost- minimization problem is to solve 12 11 2 2 ,0 min xx wx wx ≥ + S. t. 12 (, )fxx q= 北京大学经济学院19 The Cost-Minimization Problem ? Solution x 1 *(w 1 ,w 2 ,q) and x 2 *(w 1 ,w 2 ,q) are firm’s conditional demands for inputs 1, 2 ? Smallest possible total cost for producing q is )q,w,w(xw )q,w,w(xw)q,w,w(c 21 * 22 21 * 1121 + = 北京大学经济学院20 The Cost-Minimization Problem: Graph x 1 x 2 Cost min bundle that will produce q’: f(x 1 ,x 2 ) ≡ q’ x 1 * x 2 * 北京大学经济学院21 The Cost-Minimization Problem: Graph x 1 x 2 f(x 1 ,x 2 ) ≡ q’ x 1 * x 2 * At an interior cost-min input bundle: (a) f(x 1 *,x 2 *) = q’ (b) slope of isocost = slope of isoquant )x,x(at MP MP MRTS w w * 2 * 1 2 1 2 1 == ? 北京大学经济学院22 Cobb-Douglas Example of Cost Min. ? Production function is 1/ 3 2/ 3 12 1 2 (, )qfxx xx== ? Input prices are w 1 and w 2 11 22 * 2/3 * 2/3 * 12 2 * 1/3 * 1/3 * 12 1 / / (1/ 3)( ) ( ) (2/3)( ) ( ) 2 wqx wqx xx x xx x ? ? ?? ? ? = == 北京大学经济学院23 A Cobb-Douglas Ex. of Cost Minimization * 1/3 * 2/3 12 ()()qx x= * 1 * 2 2 1 x2 x w w = (a) (b) From (b), * 1 2 1 * 2 x w w2 x = Now substitute into (a) to get 2/3 2/3 *1/3 * * 11 11 1 22 22 () ww qx x x ???? == ???? ???? 2/3 * 2 1 1 2 w xq w ?? = ?? ?? So conditional demand 北京大学经济学院24 A Cobb-Douglas Ex. of Cost Minimization x w w x 2 1 2 1 2 ** = 2/3 * 2 1 1 2 w xq w ?? = ?? ?? Firm’s conditional demand for input 2 is Since and 2/3 1/3 * 12 1 2 21 2 22 2 ww w xqq ww w ?? ?? == ?? ?? ?? ?? 北京大学经济学院25 A Cobb-Douglas Ex. of Cost Minimization So, cost min input bundle for y units of output () ** 112 212 2/3 1/3 21 12 (, ,), (, ,) 2 , 2 xwwq xwwq ww qq ?? ?? ?? = ?? ?? ?? 北京大学经济学院26 x 2 x 1 Fixed w 1 and w 2 Conditional Input Demand Curves(条件投 入需求曲线) q 1 q 2 q 3 北京大学经济学院27 Conditional Input Demand Curves *3 1 ()xq *3 2 ()xq *1 1 ()xq *2 1 ()xq *2 2 ()xq *1 2 ()xq x 2 x 1 1 q 2 q 3 q x 2 * x 1 * y y 1 3 q 2 q 1 q *3 2 ()xq *1 2 ()xq *2 2 ()xq 3 q 2 q 1 q *1 1 ()xq *2 1 ()xq *3 1 ()xq 北京大学经济学院28 Conditional Input Demand Curves *3 1 ()xq *3 2 ()xq *1 1 ()xq *2 1 ()xq *2 2 ()xq *1 2 ()xq x 2 x 1 1 q 2 q 3 q x 2 * x 1 * y y 1 3 q 2 q 1 q *3 2 ()xq *1 2 ()xq *2 2 ()xq 3 q 2 q 1 q *1 1 ()xq *2 1 ()xq *3 1 ()xq output expansion path Cond. demand for input 2 Cond. demand for input 1 北京大学经济学院29 Cobb-Douglas Example of Cost Min q 4 ww 3 3/1 2 21 ? ? ? ? ? ? = qww2qww 2 1 3/2 2 3/1 1 3/13/2 2 3/1 1 3/2 + ? ? ? ? ? = q w w2 wq w2 w w 3/1 2 1 2 3/2 1 2 1 ? ? ? ? ? + ? ? ? ? ? = )q,w,w(xw)q,w,w(xw)q,w,w(c 21 * 2221 * 1121 += So the firm’s long run total cost function is 北京大学经济学院30 Cost function in the Short Run ? In the short run, there are some variable inputs and some fixed inputs in the production. ? So cost function is ),,(),,( qrwKrqrwLwC ?+?= 北京大学经济学院31 ? Therefore, the short run total cost of production equals the fixed cost (the cost of the fixed inputs) plus the variable cost (the cost of the variable inputs), or… STC FC VC= + Cost Functions In the Short Run 北京大学经济学院32 ? Fixed Cost（固定成本） ? those costs that do not vary with the quantity of output produced ? Variable Cost（变动成本） ? those costs that do change as the firm alters the quantity of output produced. Cost Functions In the Short Run Fixed and Variable Costs Fixed and Variable Costs 北京大学经济学院33 Family of Total Costs in the Short Run ? Total Fixed Costs (TFC)（总固定成本） ? Total Variable Costs (TVC)（总变动成本） ? Total Costs (TC)（总成本） TC = TFC + TVC 北京大学经济学院34 From TP L to TVC 北京大学经济学院35 Cost Curves for a Firm Output Cost ($ per year) 100 200 300 400 0 12345678910 1 12 13 TVC Variable cost increases with production and the rate varies with increasing & decreasing returns. TC Total cost is the vertical sum of FC and VC. TFC 50 Fixed cost does not vary with output 北京大学经济学院36 Family of Total Costs Quantity Total Cost Fixed Cost Variable Cost 0 $ 3.00 $3.00 $ 0.00 1 3.30 3.00 0.30 2 3.80 3.00 0.80 3 4.50 3.00 1.50 4 5.40 3.00 2.40 5 6.50 3.00 3.50 6 7.80 3.00 4.80 7 9.30 3.00 6.30 8 11.00 3.00 8.00 9 12.90 3.00 9.90 10 15.00 3.00 12.00 北京大学经济学院37 Total-Cost Curve... $0.00 $2.00 $4.00 $6.00 $8.00 $10.00 $12.00 $14.00 $16.00 0 24681012 Quantity of Output (glasses of lemonade per hour) Total Cost TC TFC TVC 北京大学经济学院38 Family of Average Costs in the Short Run ? Average Fixed Costs (AFC)（平均固定成 本） ? Average Variable Costs (AVC)（平均变 动成本） ? Average Total Costs (ATC)（平均成本） ATC = AFC + AVC 北京大学经济学院39 Family of Average Costs Fixed cost FC AFC= = Q u a n tity q Variable cost TVC AVC= = Q u a n tity q Total cost TC ATC= = Q u a n tity q 北京大学经济学院40 $3.00 Family of Average Costs Quantity AFC AVC ATC 0——— 1 $0.30 $3.30 2 1.50 0.40 1.90 3 1.00 0.50 1.50 4 0.75 0.60 1.35 5 0.60 0.70 1.30 6 0.50 0.80 1.30 7 0.43 0.90 1.33 8 0.38 1.00 1.38 9 0.33 1.10 1.43 10 0.30 1.20 1.50 北京大学经济学院41 ATC AVC MC Average-Cost and Marginal-Cost Curves... $0.00 $0.50 $1.00 $1.50 $2.00 $2.50 $3.00 $3.50 0 2 4 6 8 10 12 Quantity of Output (glasses of lemonade per hour) Costs AFC 北京大学经济学院42 Marginal Cost（边际成本） ? Marginal cost (MC) measures the amount total cost rises when the firm increases production by one unit. 北京大学经济学院43 Marginal Cost (Change in total cost) MC= (Change in quantity) TC = q lim q TC dTC MC qdq ?→∞ ? ? ? == ? 北京大学经济学院44 Marginal Cost Quantity Total Cost Marginal Cost Quantity Total Cost Marginal Cost 0$3.0 — 1 3.30 $0.30 6 $7.80 $1.30 2 3.80 0.50 7 9.30 1.50 3 4.50 0.70 8 11.00 1.70 4 5.40 0.90 9 12.90 1.90 5 6.50 1.10 10 15.00 2.10 北京大学经济学院45 ATC AVC M C Average-Cost and Marginal-Cost Curves... $0.00 $0.50 $1.00 $1.50 $2.00 $2.50 $3.00 $3.50 0 2 4 6 8 10 12 Quantity of Output (glasses of lemonade per hour) Costs AFC 北京大学经济学院46 Relationship Between Marginal Cost and Average Total Cost ? Whenever marginal cost is less than average total cost, average total cost is falling. ? Whenever marginal cost is greater than average total cost, average total cost is rising. 北京大学经济学院47 Relationship Between Marginal Cost and Average Total Cost ? The marginal-cost curve crosses the average-total-cost curve at the efficient scale. ?Efficient scale is the quantity that minimizes average total cost. 北京大学经济学院48 C q MC(q) AC(q) () ()MCq ACq > = < as () 0 dAC q dq > = < 北京大学经济学院49 Proof of Relationship between MC and AC Since () () , TC q AC q q = 2 () () () 1 () . dACq d TCq q MCq TCq dq dq q q ??×?× == ?? ?? Therefore, () 0 dAC q q? > = < () ().qMCq TCq > ×= < as as () 0 dAC q dq > = < () ().MCq ACq > = < 北京大学经济学院50 C q AVC(q) MC(q) () () () 0 dAVC q MC q AVC q dq = ?= The short-run MC curve intersects the short-run AVC curve from below at the AVC curve’s minimum. Relationship Between Marginal Cost and Average Variable Cost 北京大学经济学院51 Proof of Relationship between MC and AVC Since () () , VC q AVC q q = 2 () () () 1 () . dAVC q d VC q q MC q VC q dq dq q q ??×?× == ?? ?? Therefore, () 0 dAVC q q? > = < () ().qMCqVCq > ×= < as as () 0 dAVC q dq > = < () ().MC q AVC q > = < 北京大学经济学院52 Summary: Marginal & Average Cost Functions ? The short-run MC curve intersects the short- run AVC curve from below at the AVC curve’s minimum. ? And, similarly, the short-run MC curve intersects the short-run ATC curve from below at the ATC curve’s minimum. 北京大学经济学院53 C q AVC(q) MC(q) AC(q) 北京大学经济学院54 Summary: Three Important Properties of Cost Curves ? Marginal cost eventually rises with the quantity of output. ? The average-total-cost curve is U- shaped. ? The marginal-cost curve crosses the average-total-cost curve at the minimum of average total cost. 北京大学经济学院55 Summary: TC TVC AC、AVC and MC 北京大学经济学院56 Product Curves and Cost Curves How are the product curves related to the cost curves? 北京大学经济学院57 Cost in the Short Run ? For Example: Assume the wage rate (w) is fixed relative to the number of workers hired. Then: TVC w L= ? 北京大学经济学院58 Cost in the Short Run ? Continuing: 11 L TVC w L AVC w w q qq AP L ? = = =?=? 11 L dTVC dw L dL MC w w w dq dq dq dq MP dL ? = = =? =? =? 北京大学经济学院59 AP MP Product Curves and Cost Curves Labor Average product and marginal product 01.52.0 2 4 6 Rising MP and falling MC: rising AP and falling AVC Falling MP and rising MC: rising AP and falling AVC Falling MP and rising MC: falling AP and rising AVC 北京大学经济学院60 AVC MC Product Curves and Cost Curves Labor Average product and marginal product 06.5 10 3 6 9 12 Maximum AP and minimum AVC Maximum MP and minimum MC 北京大学经济学院61 MP and MC、AP and AC 北京大学经济学院62 Cost Functions In the Short Run: An algebra examples In our example, the short-run cost function was: Variable costs Fixed costs () () 5 3 1 3 70 3,000 3, 000 STC c q q ?? ?? == + ?? ?? ( ) SVC q F 北京大学经济学院63 Short-Run Cost Curve () cq q 000,3 0 ( ) STC q ()SVC q F 北京大学经济学院64 Average Cost Function The short run average cost function: () () 2 3 1 3 70 3,000 () 3, 000 STC q SAC q q qq ?? ?? == + ?? ?? ()SAVC q ()AFC q 北京大学经济学院65 Average Fixed Cost Curve q ()AFC q 3,000 ()AFC q q = 0 北京大学经济学院66 Average Variable Cost Curve ()AVC q 0 ()AVC q q () 2 3 1 3 70 () 3000 AVC q q ? ? ? ? = ? ? ? ? ? ? 北京大学经济学院67 Why is AVC Increasing in q? ? Production function and AVC: () () 1 1 0.6 0.2 0.6 1 3 70 (3,000) ( ) 3000 l q x AVC q q ? ?? ?? =??→= ?? ?? 北京大学经济学院68 Average Cost Curve ()ACq 0 60 124 ()ACq y () 2 3 1 3 70 3,000 () 3000 AC q q q ?? ?? =+ ?? ?? 北京大学经济学院69 Marginal Cost Function ? The short run cost function: ? The short run marginal cost function: () 2 3 1 3 570 () 3 3, 000 MCq q ? ? ? ? = ? ? ? ? ? ? ( ) ( ) () cq TVCq MC q qq ?? == ?? 北京大学经济学院70 Marginal and Average Variable Cost Curves (), () AVC q MCq 0 ()MCq ()AVC q q () 2 3 1 3 570 () 3 3,000 MCq q ? ? ? ? = ? ? ? ? ? ? () 2 3 1 3 70 () 3000 AVC q q ? ? ? ? = ? ? ? ? ? ? 北京大学经济学院71 Marginal and Average Cost Curves (), (), () ACq AVC q MCq 0 60 124 ()ACq ()MCq ()AVC q q 北京大学经济学院72 Variable Costs and the Marginal Cost Curve ()MCq 0 ()MCq q () 2 3 1 3 570 () 3 3,000 MCq q ? ? ? ? = ? ? ? ? ? ? 1 q ( ) 1 TVC q 北京大学经济学院73 The End 北京大学经济学院74 Last Revised: October 24, 2005 北京大学经济学院 Chapter Four Production and Cost Function ? 2005 MOL 北京大学经济学院 Chapter Four includes: ? 4.1 Production Function ? 4.2 Cost Function 北京大学经济学院 Overview of Last Class ? Definition of Cost Function ? opportunity costs ? Explicit and Implicit Costs ? TC, AC and MC ? From Cost Minimization to the Long Run Cost Function ? Cost Functions In the Short Run ? STC,VC,FC;SAC,SAVC,AFC;SMC 北京大学经济学院 Outline of Today’s Class ? How to get the Long Run Cost Function ? An Graphic Approach ? An Algebraic Approach ? LTC,LAC,LMC ? Economies of Scale and Diseconomies of Scale 北京大学经济学院 Readings about the part of this chapter ? Zhang: Chapter 6,P191-216 ? Nicholson: Chapter 12, P297-330 北京大学经济学院 Introduction: Long Run Cost ? In the long run there are no fixed factors of production ? Firm can freely adjust inputs ? Production costs are lower in the long run ? In order to distinguish the Long-run and short-run cost, we rewrite the term of the costs: ? In the Long-Run Cost: LTC,LAC,LMC. ? In the Short-Run: STC,STVC,STFC;SAC,SAVC,SAFC;SMC 北京大学经济学院 Long-Run Cost ? Long-run cost is the cost of production when a firm uses the economically efficient quantities of labor and capital. ? Explain the relationship between a firm’s output and costs in the long run ? Derive and explain a firm’s long-run total cost curve, long-run average cost curve and Long-run marginal cost curve. 北京大学经济学院 Introduction: LTC and LAC ? The long-run total cost (LTC) is the minimum total cost of production in the long run when a firm is perfectly flexible in its choice of inputs and can choose a production facility of any size. ? The long-run average cost of production (LAC) is the long-run total cost divided by the quantity of output produced. 北京大学经济学院 From Expansion Path to Long-Run Total Cost (LTC) 北京大学经济学院 How to get LTC for STC by graphic approach LTC 北京大学经济学院 LTC curve is the envelope of the STC curves ? Suppose the firm wishes to produce q 1 , in the short run, some inputs are fixed, and this prevent the firm from producing q 1 at the lowest possible cost. ? There may be too much or too little of the fixed factors. Only in the very special case of just the right level of the fixed inputs would cost be at the lowest level possible. 北京大学经济学院 LTC curve is the envelope of the STC curves ? The lowest level of cost possible for producing q 1 is precisely the long-run cost, as in the long run, the firm can vary all factors to minimize cost. ? So the LTC must always be no larger than STC. ? LTC is always below the STC curves at every level of output. Each STC curve just touches the LTC at at least one point. 北京大学经济学院 LTC curve is the envelope of the STC curves ? At each level of q, there is a cheapest way of producing that output in the long run. ? Thus each point on the LTC coincides with a point on some STCs. ? LTC curve is the envelope of the STC curves 北京大学经济学院 Summary: LTC is the ‘envelope’ of STC ? The long-run total cost (LTC) is the minimum total cost of production in the long run when a firm is perfectly flexible in its choice of inputs. ? LTC is a set of min. STC at any given outputs. ? So LTC is often referred to as being the ‘envelope’ of the short-run cost curve. 北京大学经济学院 How to get LTC from STCs? An Algebraic Approach ? In the long run, all inputs are variable, and the scale of production can change. ? Assume production function is ? Short-run cost function is: ? ? Holding w 1 ,w 2 constant, we can get: 12 (, ,)qfxxk= 11 2 2 ()STC w x w x kφ= ++ (,) ()STC C q k kφ= + 北京大学经济学院 How to get LTC from STC? ? Let: ? ? The partial derivative of G(C,q,k) is ? We can get: ? Put k=k(q) into G(C,q,k), ? LTC=LTC(q) (,) () ( ,,) 0CCqk k GCqkφ? ?= = (,,) (,,) 0 k GCqk GCqk k ? = = ? ()kkq= 北京大学经济学院 An example: ? Suppose A group of STC is showed by the following form ? k=1,2,3….. ? What is LTC? 32 2 0.04 0.9 (11 ) 5Cqq kqk=?+?+ 32 0.1 0.04 0.95 11 kq LTC q q q = =?+ 北京大学经济学院 How to get The Long-Run Average Cost Curve The long-run average total cost curve is derived from the short-run average total cost curves. The segment of the short-run average total cost curves along which average total cost is the lowest make up the long-run average total cost curve. 北京大学经济学院 Short-Run Costs of Four Different Plants 北京大学经济学院 Long-Run Average Cost Curve 北京大学经济学院 How to get LAC 0 ? ? ? SAC(q,K 1 ) SAC(q,K 2 ) SAC(q,K 3 ) LAC is the envelope of many SACs C LAC(q) q 1 q 2 q 3 q 北京大学经济学院 How to get LMC? 北京大学经济学院 Why is LAC “U” shaped. ? In the long-run: ? Firms experience increasing and decreasing returns to scale and therefore long-run average cost is “U” shaped. 北京大学经济学院 Returns to scale and long run average cost ? Economies of scale(规模经济) a cost function exhibits economies of scale if average cost falls as output expands ? Diseconomies of scale（规模不经济） a cost function exhibits diseconomies of scale if average cost remains increases as output expands 北京大学经济学院 Reasons for Economies of Scale ? Specialisation of labour and capital ? Indivisibilities(不可分割) in plant size ? Marketing（营销）economies ? Transport and storage economies ? Bulk purchase of inputs ? Lower borrowing（借款）costs 北京大学经济学院 Diseconomies of Scale ? A firm experiences diseconomies of scale when an increase in output leads to an increase in long-run average cost—the LAC curve becomes positively sloped. 北京大学经济学院 Reasons for Diseconomies of Scale ? Plant size too big to manage ? Maximum technical size of plant reached ? Organisation too bureaucratic ? Input prices may rise due to scarcity ? Labour relations deteriorate（恶化） 北京大学经济学院 Definition of MES: The quantity of output at which the long run average cost curve attains its minimum point is called the minimum efficient scale(最低成本的有效率规模). 北京大学经济学院 Minimum Efficient Scale ? The minimum efficient scale describes the output at which the long-run average cost curve becomes horizontal. ? Once the minimum efficient scale has been reached, an increase in output no longer decreases the long-run average cost. 北京大学经济学院 AC q* = MES LAC(q) Example: Minimum Efficient Scale 0q(units/yr) 北京大学经济学院 Output Elasticity of Total Cost ? Output Elasticity of Total Cost: ? E c = percent change in cost from a 1% increase in output c LTC LTC q SMC LTC E q qLTCLAC q ? ? == = ? ? LMC 北京大学经济学院 Measuring Economies of Scale ? Therefore, the following is true: ? E C < 1: LMC < LAC LAC will fall. ? economies of scale ? E C = 1: LMC = LAC LAC will be in MES (Minimum Efficient Scale) ? constant economies of scale ? E C > 1: LMC > LAC LAC will rise ? diseconomies of scale 北京大学经济学院 When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale so that LAC(Q) decreases with Q. 北京大学经济学院 ?When the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale so that LAC(Q) increases with Q. ?When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output. 北京大学经济学院 Long-Run Cost with Constant Returns to Scale Output Cost ($ per unit of output) Q 3 SAC 3 SMC 3 Q 2 SAC 2 SMC 2 Q 1 SAC 1 SMC 1 LAC = LMC With many plant sizes with minimum SAC = $10 the LAC = LMC and is a straight line $10 北京大学经济学院 Long-Run Cost with Economies and Diseconomies of Scale Output Cost ($ per unit of output SMC 1 SAC 1 SAC 2 SMC 2 LMC If the output is Q 1 a manager would chose the small plant SAC 1 and SAC $8. Point B is on the LAC because it is a least cost plant for a given output. $10 Q 1 $8 B A LAC SAC 3 SMC 3 北京大学经济学院 Learning Effect and LAC L ΣX Cumulative output X ΣX=500 ΣX=5000 累积性的产品批量 L每批产品的劳动投入量 Output per period (output per period constant) 北京大学经济学院 Learning Effect(学习效应) LABN β? =+ L is the amount of Labor per output,N is a cumulative outputs,A,B>0 If β=0, L=A+B (constant ), no Learning Effect. If β=1, L=A+B/N, N→∞,L→A, Learning Effect is enough. β is the measure index of Learning Effect 北京大学经济学院 Economies of scope（范围经济）and LAC ? There may be cost savings if a given production unit produces multiple outputs compared to the situation where the outputs are produced in separate producing units ? i.e. TC(q 1 ,q 2 )<TC(q 1 )+TC(q 2 ) ? E.g. teaching and research in the department of economics ? Passenger and freight on a given railway line, etc 北京大学经济学院 Economies of scope（范围经济）and LAC ? We say that there are economies of scope if it is less expensive to produce goods jointly than separately.A measure of economies of scope is ),( ),(),0()0,( 21 2121 qqC qqCqCqC SC ?+ = 北京大学经济学院 The End 北京大学经济学院 Last Revised: October 29, 2005 1 Chapter Five Perfect Competitive Market（1） 完全竞争市场 ? 2005 MOL 2 Chapter Five includes: ? 5.1 The Optimal Output Decision in the Short Run ? 5.2 The Optimal Output Decision in the Long Run 3 Outline of Today’s Class ? Perfectly Competitive Markets ? Profit Maximization ? Marginal Revenue, Marginal Cost, and Profit Maximization ? Choosing Output in the Short-Run ? The Competitive Firm’s Short-Run Supply Curve ? Short-Run Market Supply 4 Readings about the part of this chapter ? Zhang: Chapter 7,P238-262 ? Nicholson: Chapter 13, P334-354 ? Chapter 14,P368-397 The Determinant of Firm Supply ? How does a firm decide how much to supply at a given price? This depends upon the firm’s ? technology; ? market environment (Structure); ? goals; ? competitors’ behaviors. Market Environment (Structure) ? Are there many other firms? ? How do other firms’ decisions effect the firm’s payoffs? Market Environment (Structure) ? Monopoly(垄断市场): Just one seller that determines the quantity supplied/the market-clearing price. ? Oligopoly（寡头市场）: A small number of firms, the decisions of each influencing the payoffs（得益）of the other firms. Market Environment (Structure) ? Monopolistic Competition（垄断竞争市 场）: Many firms each making a slightly different product. Each firm’s output level is small relative to the total. ? Perfect Competition（完全竞争市场）: Many firms, all making the same product. Each firm’s output level is small relative to the total output level. 9 Monopoly Oligopoly Monopolistic Competition Perfect Competition ? Tap water （自来水） ? Cable TV ? Tennis balls ? Crude oil ? Novels ? Movies ? Wheat ? Milk Number of Firms? Type of Products? Many firms One firm Few firms Differentiated products（异质 产品） Identical products （同质产品） The Four Types of Market Structure Profit Maximization （利润最大化） ? Each firm is a profit-maximizer ? Each firm choose its output level by solving 0 max ( ) ( ) ( ) () () q qTRqTCq pq q TCq ≥ Π =? =?? ()p pq= is the inverse demand function（反 需求函数） 11 Profit Maximization ? The necessary condition for choosing the value of q that maximizes profits is: ? The first-order condition for a maximum is that: () () '( ) 0 d dTR q dTC q q dq dq dq π π= =?= dTR dTC MRMC dq dq === 12 Profit Maximization ? The second condition is that: 2 ** 2 2 * 2 '( ) 0 () () 0 '( ) '( ) qq qq qq ddq dq dq ddMRqdMCq dq dq dq MR q MC q ππ π == = =< = ?< ∴< 13 Profit Maximization: An Graphic Expression ? Comparing TR(q) and TC(q) ? Question: Why is profit negative when output is zero? TR(q) 0 Cost, Revenue, Profit $ (per year) Output (units per year) TC(q) A B q 0 q * )(qπ C D 14 Marginal Revenue, Marginal Cost, and Profit Maximization ? Comparing TR(q) and TC(q) ? Output levels: q 0 - q * ? TR’(q)> TC’(q) ? MR > MC ? Indicates higher profit at higher output ? Profit is increasing TR(q) 0 Cost, Revenue, Profit $ (per year) Output (units per year) TC(q) A B q 0 q * )(qπ 15 Marginal Revenue, Marginal Cost, and Profit Maximization TR(q) 0 Cost, Revenue, Profit $ (per year) Output (units per year) TC(q) A B q 0 q * )(qπ ? Comparing TR(q) and TC(q) ? Output levels beyond q * : ? TR’(q)< TC’(q) ? MR < MC ? Profit is decreasing 16 Marginal Revenue, Marginal Cost, and Profit Maximization ? Question ? Why is profit reduced when producing more or less than q*? TR(q) 0 Cost, Revenue, Profit $ (per year) Output (units per year) TC(q) A B q 0 q * )(qπ 17 Marginal Revenue, Marginal Cost, and Profit Maximization ? Comparing TR(q) and TC(q) ? Output level: q * ? TR’(q)= TC’(q) ? MR = MC ? Profit is maximized TR(q) 0 Cost, Revenue, Profit $ (per year) Output (units per year) TC(q) A B q 0 q * )(qπ 18 Marginal Revenue, Marginal Cost, and Profit Maximization TR(q) 0 Cost, Revenue, Profit $ (per year) Output (units per year) TC(q) A B q 0 q * )(qπ ? Therefore, it can be said: ? Profits are maximized when MC = MR. 19 The Marginal Revenue/Marginal Cost Rule ? At the profit maximizing level of output Marginal Revenue = Marginal Cost or MR = MC. ? Firms, starting at zero output, can expand output so long as marginal revenue exceeds marginal cost, but don’t go beyond the point where these two are equal. 20 TR(总收益),AR（平均收益） and MR（边际收益） , () 1 (1 ) qp TR TR q pq TR pq AR p qq dTR MR p dq e = = === ==? 21 Perfectly Competitive Markets Assumptions of Perfect Competition ? There are many buyers and sellers（有许多买者 和卖者）, i.e. ? no one firm can dominate/influence the market ? firms are price takers ? Homogeneous product （同质产品） ? Freedom of entry and exit（进入与退出自由） ? Perfect information（完全信息） 23 Perfectly Competitive Markets ? (Assumption 1) There are many buyers and sellers ? The individual firm sells a very small share of the total market output and, therefore, cannot influence market price. ? The individual consumer buys too small a share of industry output to have any impact on market price. 24 Perfectly Competitive Markets ? (Assumption 1) ? Each seller is too small to affect the price ? sellers can sell all they can produce at the market price. ? sellers can sell nothing above the market price. ? sellers have no incentive to offer anything below market price. 25 Perfectly Competitive Markets ? (Assumption 2) Product Homogeneity (产品同 质性) ? The products of all firms are perfect substitutes. ? buyers don’t care where product comes from ? Examples ? Agricultural products (wheat, rice and corn), oil, copper, iron, lumber 26 Perfectly Competitive Markets ? (Assumption 3) Free Entry and Exit（自 由进入与退出） ? Buyers can easily switch from one supplier to another. ? Suppliers can easily enter or exit a market. 27 Perfect Competition ? (Assumption 4) Perfect information (完全 信息) ? Perfect knowledge on all parts of buyers and Sellers. 28 Perfectly Competitive Markets ? Discussion Questions ? What are some barriers to entry and exit? ? Are all markets competitive? ? When is a market highly competitive? Perfect Competition the demand curve faced by the firm The Demand Curve of Perfectly Competitive Markets P Q Market Supply Market Demand p e The Demand Curve of Perfectly Competitive Markets Q P Market Supply p e p’ At a price of p’, zero is demanded from the firm. Market Demand The Demand Curve of Perfectly Competitive Markets Q P Market Supply p e p’ p” At a price of p” the firm faces the entire market demand. At a price of p’, zero is demanded from the firm. Market Demand 33 From Market Demand Curve to the demand curve faced by the firm Market Supply Market Demand P P* P* q the demand curve faced by the firm: P=P* P AR=MR=P* Q Firm’s Demand Curve 34 Marginal Revenue in the competitive firm ? A price taker is a firm or individual whose decisions regarding buying or selling have no effect on the prevailing market price of a good or service. ? For a price taking firm AR=MR = P. 35 Marginal Revenue, Marginal Cost, and Profit Maximization in the competitive firm ? The competitive firm’s demand ? P = D = MR = AR 36 ?The Competitive Firm’s Optimal Decision In the Short Run 37 Choosing Output in the Short Run ? We will combine production and cost analysis with demand to determine output and profitability. Max. Profit When the Firm is a price-taker 0 max ( ) ( ) q qpqcq ≥ Π =? qq* () () ( ) 0 dq ipSMCq dq Π = ?= 2 * 2 () () 0 dq ii at q q dq Π < = F.O.C. S.O.C. Π(q) Max. Profit When the Firm is a price-taker The first-order maximum profit condition is () () 0 dq pMCq dq Π = ?= That is, MCp = So at a profit maximum with q* > 0, the market price p equals the marginal cost of production at q = q*. 40 Choosing Optimal Output in the Short Run: TR and TC TR,TC Profit MC MR = P q 0 q * TR TC q0 q0 MC > MR MC < MR MC = MR MC > MR 41 q 0 Lost profit for q q < q * Lost profit for q 2 > q * q 1 q 2 Choosing Optimal Output in the Short Run: MR and MC 10 20 30 40 Price ($ per unit) 01234567891011 50 60 SMC SAVC SAC AR=MR=P Output q * At q * : MR = SMC and P > SAC ABCDor qx AC) -(P * = π D A B C q 1 : MR > SMC and q 2 : SMC > MR and q 0 : SMC = MR but SMC falling 42 The Firm’s Optimal Decision and its Excess Profit 43 SAVC SAC SMC A q Price 0 MR = P B q 1 Excess Profit P 1 >SAC→π>0 (Excess Profit 超额利润) P 1 C 44 SAVC SAC SMC A Price 0 AR=MR = P q 2 P 2 =SAC→π=0 (but there is normal profit) P 2 q 45 SAVC SAC SMC A q Price 0 MR = P B Loss SAVC<P 3 <SAC→π<0 (there is loss, but P 3 can cover all average variable cost and a part of FC, So the firm will continue) C P 3 q 3 46 SAVC SAC SMC A Price 0 MR = P B q 4 Loss P 4 =SAVC→π<0 (A point is shutdown（停工）) C P 4 q 47 SAVC SAC SMC A q Price 0 MR = P B q 5 Loss P 5 <SAVC→π<0 (there is great loss, and P 5 can not cover a part of average variable cost and all FC, So the firm will produce no output) C P 5 48 The Shutdown Decision(停工决策): An Algebra Method ? STC=SFC+STVC ? Profits are given by ? If q=0,STVC and Total Revenues are 0, So ? π=-SFC ? The firm will produce something only if π>- SFC, But that means that TR STC pq SFC STVCπ = ?=?? 49 The Shutdown Decision ? The firm will opt for q > 0 providing ? The price must exceed average variable cost. or, dividing by q, . Pq STVC STVC P SAVC q ? ≥ ≥= 50 The Shutdown Decision ? The shutdown price is the price below which the firm will choose to produce no output in the short-run. It is equal to minimum average variable costs. 51 Summary: Choosing Output in the Short Run ? Profit is maximized when SMC = MR ? If P > SAC the firm is making profits. ? If SAVC < P < SAC the firm should produce at a loss. ? If P < SAVC < SAC the firm should shut- down. 52 The Conditions for Choosing Optimal Output in the Short Run ?P=AR=MR=MC ?MR’<MC’ ?P≥min SAVC 53 SAVC SAC SMC q Price 0 Shut Down Point P1 P2 P3 P4 54 A Competitive Firm’s Short-Run Supply Curve ? Observations: ? P = MR ? MR = MC——MAX π ? P = MC→P=MC(q)——短期供给曲线 ? Supply is the amount of output for every possible price. Therefore: ? If P = P 1 , then q = q 1; If P = P 2 , then q = q 2 ? If P = P 3 , then q = q 3; If P = P 4 , then q = q 4 55 SAVC SAC SMC P1 P2 P3 P4 q Price 0 Shut Down Point Firm’s Supply 56 A Competitive Firm’s Short-Run Supply Curve ? In above Figure, the shutdown price is P 4 . ? For all P ≥ P 4 ,the firm will follow the P = MC rule, so the supply curve will be the short-run marginal cost curve. 57 The Firm’s Short-Run Supply Curve _Summary ? The firm’s short-run supply curve is the relationship between price and quantity supplied by a firm in the short-run. ? For a price-taking firm, this is the positively sloped portion of the short-run marginal cost curve. ? For all possible prices, the marginal cost curve shows how much output the firm should supply. 58 A Competitive Firm’s Short-Run Supply Curve ? Observations: ? Supply is upward sloping due to diminishing returns ? Higher price compensates the firm for higher cost of additional output and increases total profit because it applies to all units. 59 How to get Supply Function: An Example ? Assume a firm’s cost function is ? What is this firm’s SR supply curve 32 0.1 2 15 10STC q q q= ?++ 412 2 ,5 0.6 0 P P S +? ≥ ? ? = ? ? ? P<5 60 From the SR individual Supply Curve to SR Market Supply Curve ? The short-run market supply curve shows the amount of output that the industry will produce in the short-run for every possible price. 61 SR Market Supply Curve: Hold Input Price constant when output increases 62 Short Run perfectly Competitive equilibrium ** ** 12 n S()S()..S() ()PP PDP+++= D S Q(P*) SAVC q* Q P q P P* S 63 64 The Short-Run Market Supply Curve ? Elasticity of Market Supply(Zhang,P45-49) ?衡量市场供给对价格变化的敏感度，正值。 , , 0 lim S sp s SS sp p s s Q Qp Q e p pQ p QpdQp e p QdpQ ?→ ? ? ==? ? ? ?? ? =?=? ?? ? ?? 65 Classification of Elasticity of Supply 66 Geometric Meaning of e s,P 67 Geometric Meaning of e s,P ? eS=供给曲线上某点到原点的斜率×供给曲线 斜率的倒数； 9①A点切线通过原点时，供给弹性等于1　 9②B点切线与纵轴相交时，供给弹性大于1 9③C点切线相交于横轴时，供给弹性小于1 68 Producer Surplus（生产者剩余）in the Short Run ? Producer Surplus in the Short Run ? The producer surplus is the sum over all units produced of the difference between the market price of the good and the marginal cost of production. ? RECALL consumer surplus: 消費者愿意付出 的价格与实际付出的价格之间的差額 69 Producer Surplus for a Firm A D B C Producer Surplus Alternatively, TVC is the sum of SMC or ODCq * . TR is P x q * or OABq * . Producer surplus = TR - TVC or ABCD. Price ($ per unit of output) Output AVCMC 0 P q * At q * MC = MR. Between 0 and q , MR > MC for all units. 70 The Short-Run Market Supply Curve ? Producer Surplus in the Short-Run Producer Surplus TR - TVC= Profit TR -TVC - FC PS= FC π π = = + 71 Producer Surplus * * 0 0 [* ()] (* () ** (*)[*0 (0)] * q q PS P MC q dq P q TC q P q TC q P TC FCπ =? =? =???? =+ ∫ 72 Producer Surplus for a Market D P * Q * Producer Surplus Market producer surplus is the difference between P* and S from 0 to Q * . Price ($ per unit of output) Output S 73 The End 74 Last Revised: October 31, 2005 1 Chapter Five Profit Maximization and Competitive Supply ? 2005 MOL 2 Chapter Five includes: ? 5.1 The Optimal Output Decision in the Short Run ? 5.2 The Optimal Output Decision in the Long Run 3 Overview of Last Class ? Perfectly Competitive Markets ? Profit Maximization ? Marginal Revenue, Marginal Cost, and Profit Maximization ? Choosing Output in the Short-Run ? The Competitive Firm’s Short-Run Supply Curve 4 Outline of Today’s Class ? Choosing Output in the Long-Run ? The Industry’s Long-Run Supply Curve ? Evaluating the Gains and Losses from Government Policies--Consumer and Producer Surplus ? The Efficiency of a Competitive Market ? Applications 5 Readings about the part of this chapter ? Zhang: Chapter 7,P238-263 ? Nicholson: Chapter 13, P334-354 ? Chapter 14,P368-397 6 Choosing Output in the Long Run ? In the long run, a firm can alter all its inputs, including the size of the plant. ? We assume free entry and free exit. Choosing Output in the Long Run ? A competitive firm’s long-run profit function is ? The long-run cost LTC(q) of producing q units of output consists only of variable costs since all inputs are variable in the long-run. )()( qLTCpqq ?=Π Choosing Output in the Long Run The firm’s long-run supply level decision is to maximize its economic profit: 0 )( .. .. )()( > = ?=Π dq qdLMC COS LMCp COF qLTCpqq 9 The Firm’s Long-Run Decision to Exit or Enter a Market ? In the long-run, the firm exits if the revenue it would get from producing is less than its total cost. Exit if TR < LTC Exit if TR/Q < LTC/Q Exit if P < LAC 10 The Firm’s Long-Run Decision to Exit or Enter a Market ? A firm will enter the industry if such an action would be profitable. Enter if TR > LTC Enter if TR/Q > LTC/Q Enter if P > LAC 11 Output Choice in the Long Run q 1 A B C D In the short run, the firm is faced with fixed inputs. P = $40 > SAC. Profit is equal to ABCD. Price ($ per unit of output) Output P = MR $40 SAC 1 SMC 1 In the long run, the plant size will be increased and output increased to q 3 .Long-run profit, EFGD > short run profit ABCD. q 3 q 2 G F $30 LAC E LMC SAC 2 SMC 2 12 Choosing Output in the Long Run ? Economic Profit ? Economic profit = TR -wL - rK ?wl = labor cost ?rk = opportunity cost of capital )(π 13 Choosing Output in the Long Run ? Zero-Profit ? If TR > wL + rk, economic profits are positive, other firm will enter this industry. ? If TR < wl + rk, consider going out of business ? If TR = wL + rk, zero economic profits, but the firms is earning a normal profit; indicating the industry is competitive. Long-Run Competitive Equilibrium Long-Run Competitive Equilibrium 14 ? Entry and Exit ? The long-run response to short-run profits is to increase output and profits. ? Profits will attract other producers. ? More producers increase industry supply which lowers the market price. Long-Run Competitive Equilibrium Long-Run Competitive Equilibrium Choosing Output in the Long Run S 1 Output Output $ per unit of output $ per unit of output $40 LAC LMC D S 2 P 1 Q 1 q 2 Firm Industry $30 Q 2 P 2 ?Profit attracts firms ?Supply increases until profit = 0 Long-Run Competitive Equilibrium 16 q 1 A B C D Output Choice in the Long Run Price ($ per unit of output) Output P = MR $40 SAC SMC Question: Is the producer making a profit after increased output lowers the price to $30? q 3 q 2 G F $30 LAC E LMC 17 Choosing Output in the Long Run ? Long-Run Competitive Equilibrium 1) SMC=SAC=LMC =P=LAC=MR =AR ? P = LAC, No incentive to leave or enter ? Profit = 0 2) LMC’(q)>0 3) Equilibrium Market Price(S=D) 18 The Industry’s Long-Run Supply Curve ? The shape of the long-run supply curve depends on the extent to which changes in industry output affect the prices the firms must pay for inputs 19 The Industry’s Long-Run Supply Curve ? To determine long-run supply, we assume: ? All firms have access to the available production technology. ? Output is increased by using more inputs, not by invention. Long-Run Supply in a Constant-Cost Industry(成本固定产业的长期供给曲线) A P 1 AC P 1 MC q 1 D 1 S 1 Q 1 C D 2 P 2 P 2 q 2 B S 2 Q 2 Economic profits attract new firms. Supply increases to S 2 and the market returns to long-run equilibrium. Output Output $ per unit of output $ per unit of output S L Q 1 increase to Q 2 . Long-run supply = S L = LRAC. Change in output has no impact on input cost. 21 Long-Run Supply in a Constant-Cost Industry ? In a constant-cost industry, long-run supply is a horizontal line at a price that is equal to the minimum average cost of production. Long-Run Supply in an Increasing-Cost Industry (成本递增产业的长期供给曲线) Output Output $ per unit of output $ per unit of output S 1 D 1 P 1 LAC 1 P 1 SMC 1 q 1 Q 1 A S L P 3 SMC 2 Due to the increase in input prices, long-run equilibrium occurs at a higher price. LAC 2 B S 2 P 3 Q 3 q 2 P 2 P 2 D 2 Q 2 23 Long-Run Supply in a Increasing-Cost Industry ? In a increasing-cost industry, long-run supply curve is upward sloping. Long-Run Supply in a Decreasing-Cost Industry (成本递减产业的长期供给曲线) S 2 B S L P 3 Q 3 SMC 2 P 3 LAC 2 Due to the decrease in input prices, long-run equilibrium occurs at a lower price. Output Output $ per unit of output $ per unit of output P 1 P 1 SMC 1 A D 1 S 1 Q 1 q 1 LAC 1 Q 2 q 2 P 2 P 2 D 2 25 Long-Run Supply in a Decreasing-Cost Industry ? In a decreasing-cost industry, long-run supply curve is downward sloping. 26 Further Analysis ? Evaluating the Gains and Losses from Government Policies--Consumer and Producer Surplus ? The Efficiency of a Competitive Market ? Applications 27 Evaluating the welfare from Government Policies--Consumer and Producer Surplus ? Review ? Consumer surplus is the total benefit or value that consumers receive beyond what they pay for the good. ? Producer surplus is the total benefit or revenue that producers receive beyond what it cost to produce a good. Consumer and Producer Surplus Producer Surplus Between 0 and Q 0 producers receive a net gain from selling each product-- producer surplus. Consumer Surplus Quantity 0 Price S D 5 Q 0 Consumer C 10 7 Consumer BConsumer A Between 0 and Q 0 consumers A and B receive a net gain from buying the product-- consumer surplus 29 Evaluating the Gains and Losses from Government Policies--Consumer and Producer Surplus ? To determine the welfare effect(福利效 应) of a governmental policy we can measure the gain or loss in consumer and producer surplus. ? Welfare Effects ? Gains and losses caused by government intervention in the market. 30 Change in Consumer and Producer Surplus from Price Controls The loss to producers is the sum of rectangle A and triangle C. Triangle B and C together measure the deadweight loss. B A C The gain to consumers is the difference between the rectangle A and the triangle B. Deadweight Loss: B+C Quantity Price S D P 0 Q 0 P max Q 1 Q 2 Suppose the government imposes a price ceiling（最高限价）P max which is below the market-clearing price P 0 . E F 31 Change in Consumer and Producer Surplus from Price Controls ? Observations: ? The total loss is equal to area B + C. ? The total change in surplus = (A - B) + (-A - C) = -B - C ? The deadweight loss(社会无谓损失) is the inefficiency of the price controls or the loss of the producer surplus exceeds the gain from consumer surplus. 32 Change in Consumer and Producer Surplus from Price Controls ? Observation ? Consumers can experience a net loss in consumer surplus when the demand is sufficiently inelastic 33 Effect of Price Controls When Demand Is Inelastic B A P max C Q 1 If demand is sufficiently inelastic, triangle B can be larger than rectangle A and the consumer suffers a net loss from price controls. Example Oil price controls and gasoline shortages in 1979 S D Quantity Price P 0 Q 2 34 Price Controls and Natural Gas Shortages ? 1975 Price controls created a shortage of natural gas. ? What was the deadweight loss? 35 Application:Import Quotas（进 口配额）and Tariffs（关税） ? Many countries use import quotas and tariffs to keep the domestic price of a product above world levels 36 Import Tariff or Quota That Eliminates Imports Q S Q D P W Imports A BC By eliminating imports, the price is increased to P O . The gain to producer is area A. The loss to consumers A + B + C, so the deadweight loss is B + C. Quantity Price How high would a tariff have to be to get the same result? D P 0 Q 0 S In a free market, the domestic price equals the world price P W . 37 D CB Q S Q D Q’ S Q’ D A P* P w Quantity Price D S ? The increase in price can be achieved by a quota or a tariff. ? Area A is again the gain to domestic producers. ? The loss to consumers is A + B + C + D. Import Tariff or Quota (general case) 38 Import Tariff or Quota (general case) ? If a tariff is used the government gains D, so the net domestic product loss is B + C. ? If a quota is used instead, rectangle D becomes part of the profits of foreign producers, and the net domestic loss is B + C + D. D CB Q S Q D Q’ S Q’ D A P* P w Quantity D S Price 39 The End 40 Last Revised: Nov. 12, 2005