Economics 2010a Fall 2003 Edward L. Glaeser Lecture 10 10. More on Production a. Derived Demand—Marshall’s Laws b. Long Run/Short Run, LeChatelier, Dynamics c. Aggregating Supply d. Theory of the Firm, the Holdup Problem e. Agency Issues f. Application: The Coase Theorem Marshall-Hicks laws of Derived Demand (1) The demand for a good is more elastic the more readily substitutes can be obtained. (2) The more important the good, the more elastic the derived demand (Hicks’ addition– if substitutes are readily available). (3) The demand for an input is higher, the more elastic is the supply of other inputs. (4) The more elastic the demand for the final good– the more elastic is the demand for the input. All of these statements are supposed to be about Wj Zj /Zj /Wj Some of these comparative statics we know what to do with: (1) and (2). To get (3) and (4), we need some new ingredients. (1) The demand for a good is more elastic the more readily substitutes can be obtained. In the limit, this is obvious– if there exists a perfect substitute, then the derived demand elasticity is infinite. Start with the FOC for an input j P /f/Z j = Wj. Differentiation gives us: /f/Z j /Zj /Wj = 1 P ?>ifij /f /Zi /Zi /Wj Or using the first order condition and manipulating: Wj Zj /Zj /Wj = 1 Zj ?>ifij WiZi WjZj Wj Zi /Zi /Wj Unrigorously– just looking at the equation gives you the unimportance result (i.e., when z is small you expect this expression to be bigger) and the substitutes result (when the other goods are able to adjust a lot– you expect a bigger demand elasticity). Let’s make this rigorous with two good K and L, where we look at labor demand, the price of labor is W, the price of capital is R. Just plugging into the formula gives us: W L /L /W = 1 L ? RK WL W K /K /W It is more useful to also use the two first order conditions and note that fLL /L/W + PfKL /K/W = 1 and PfKL /L/W + PfKK /K/W = 0. Manipulating these equations yields: /L /W = fKK PfLLfKK ? PfKL2 < 0 and /K /W = fKL PfLLfKK ? PfKL2 This still isn’t all that helpful– let’s try a separable production function: f K,L = aKJ + bLK W L /L /W = W L 1 PbK K? 1 LK?2 = PbKLK?1 LPbK K? 1 LK?2 = 1 K? 1 Well– that isn’t all that interesting. It certainly tells us that the unimportance result can be general (Hicks’ point). How about Cobb-Douglas: f K,L = KJLK In the CRS case, you can’t solve for scale, only for factor proportions. Use cost minimization to solve: minK,L,V RK + WL +V KJL1?J ? Q This gives us: JRK = 1 ?J WL or, using the Q constraint, K = Q W 1?J RJ 1?J And L = Q RJW 1?J J or WL /L/W = ?J That’s a little bit better– in this case, the elasticity of demand for labor is equal to one minus labor’s share in the production function. Of course, we can’t say anything about the degree of substitutability. How about the more interesting CES production function f K,L = aKa + bLa 1a Again, as long as this is CRS, we can only do cost minimization not profit maximization. This gives us R = VaKa?1 aKa + bLa 1a ?1 W = VbLa?1 aKa + bLa 1a ?1 or K = L aWbR 11?a , from which L = a 11?a b ?a1?a WR a1?a + b ? 1a Q K = a + b 11?a a ?a1?a RW a1?a ? 1a Q Holding quantity constant– the elasticity of labor with respect to wages equals: L = a 11?a b ?a1?a WR a1?a + b ? 1a Q /L /W = 1 1?a a 1 1?a b ?a 1?a R ?a 1?a W 2a?1 1?a a 1 1?a b ?a 1?a WR a 1?a + b ? 1+aa Q and W L /L /W = 1 1 ?a 1 a ?11?a b 11?a R a1?a W ?a1?a + 1 This is increasing with a, decreasing with b, increasing with W and decreasing with R. If the terms in the denominator don’t move too much with a, then as a rises, the elasticity rises– this is Marshall’s substitutability point. To get law # 3: we need to allow R to be a function of K, and then we get /L /W = f W R 1 W ? 1 R /R /K /K /W Q where f WR = a 11?a b ?a1?a 1?a W R a 1?a a 11?a b ?a1?a WR a1?a + b 1+aa This a little artificial, since we are holding overall output constant. But nonetheless, from this it should be clear that as /R/K gets bigger, the value of /L/W gets smaller. That is Marshall’s third law. To get law number 4, just go to a single input case: maxL Pf L ? WL, which yields Pfv L = WL Now differentate this totally with respect to W, allowing P to change as well dP dQ dQ dW + Pf vv L /L /W = L + W /L /W Using the fact that dQdW = fv L /L/W , we get /L /W = f L ? Pf L W fvv L + f L ? QP dPdQ Higher demand elasticities make the denominator smaller, because QP dPdQ is one divided by the demand elasticity. What does this mean? Dynamic Supply Issues: The first thing that is useful to keep in mind is that everything is essentially dynamic, and that everything that we are doing is a static approximation to that. Perhaps the most extreme version of this is when we try to say something about long run/short run distinctions using the basic model. The essence of the LeChatelier/Samuelson principle is that the long run response to a price change is larger than the short run response to the same change. Long run/short run distinction is handled by just assuming that some inputs are fixed in the short run. Typically in the K, L formulation– this means assuming that capital is fixed and labor is flexible. To prove the LeChatelier principle in the two input world (K and L). Note that the long run comparative statics are: /L /W = fKK PfLLfKK?PfKL2 < 0 and /K /W = fKL PfLLfKK?PfKL2 /Q /W = fLfKK+fKfKL PfLLfKK?PfKL2 In respose to a price increase we have that PfLL /L/W + PfKL /K/W = ?fL and PfKL /L/W + PfKK /K/W = ?fK Which gives us: /L /P = ?fLfKK+fKfKL PfLLfKK?PfKL2 and /K /W = ?fKfLL+fLfKL PfLLfKK?PfKL2 /Q /P = ?fL2fKK ? fK2 fLL + 2fLfKfKL PfLLfKK ? PfKL2 = 1Pf LLfKK ? PfKL2 ? fK 2 ?fLL ? fL 2 ?fKK 2 + 2fLfK fKL + 2 ?fLLfKK > 0 The short run supply responses are: just come from differentiating: PfL L,K = W This yields: /L/W = 1Pf LL < 0 and /Q /W = fL PfLL < 0 /L /P = ?fL PfLL and /Q /P = ?fL2 PfLL To show that the long response is bigger for wages show that: 1 PfLL > fKK PfLLfKK ? PfKL2 or fLLfKK ? fKL 2 >fKKfLL Well that’s got to be true. For the long run quantity response to be bigger than the short run response: fL PfLL > fLfKK + fKfKL PfLLfKK ? PfKL2 or rearranging ? fLfKL2 > fKfLLfKL If fKL < 0 this is clearly false– so the long run supply elasticity is smaller than the short run (because you can use more capital and the goods are substitutes). If fKL > 0 then the condition becomes: WfKL > ?RfLL which might or might not hold. In the case of price shocks to get a bigger long run price elasticity it must be that: ?fLfKK + fKfKL PfLLfKK ? PfKL2 > ?fL PfLL or RfKLfLL > WfKL2 which again is obviously false if the two goods are subtitutes, and possibly true if not. For quantities we have ?fL2fKK ? fK2 fLL + 2fLfKfKL PfLLfKK ? PfKL2 > ?fL2 PfLL or fL2fKK + fK2 fLL ? 2fLfKfKL fLL > fL2 fLLfKK ? fKL2 or fK2 fLL2 ? 2fLfKfKLfLL + fL2fKL2 = fLfKL ? fKfLL 2 > 0 So we now that: (1) supply elasticities with respect to price are bigger in the long run and (2) input demand elasticities with respect to input price are bigger in the long run. A little bit of extra dynamics: The depletable resource problem. You are a firm with a depletable resource that costs you nothing to extract. You do not have any market power. What should we expect prices to look like? Firms maximize: Xt=0K e?rtp t x t dt subject to the constraint X ? Xt=0T x t dt. This produces first order condition: e?rtp t = V, which is to say that for regions where the firm is consistently extracting resources, it must be that rp t = pv t Similarly, we can ask about some form of irreversible investment where it costs K and than you receive P t in perpetuity. In the case where P t is deterministic this can be written: maxT X t=T K e?rtP t dt ? e?rTK This yields first order condition: P T = rK You invest when the price is equal to the interest rate times the cost of capital. A few caveats: (1) this only makes sense if price is rising, if price is falling and P 0 > rK, you want to invest immediately as long as Xt=0K e?rtP t dt > K; (2) you also need to check that Xt=TK e?rtP t dt ? e?rTK to figure out if investment makes any sense at all. Alternatively, use the notation: V T = Xt=TK e?r t?T P t dt In that case, choose T to maximize: e?rT V T ? K which yields: Vv T = r V T ? K , which is the same condition. Now assume that V is stochastic, specifically a Geometric Brownian motion: dV = JVdt +aVdz where dz is the increment of a Weiner process which means: Az = Ot t where Ot is normally distributed random variable with mean zero and standard deviation one. Let F V,t denote the value of the opportunity to invest: F V,t = max V ? K, 1 + rdt ?1E F V + dV,t + dt |V The goal is to figure out the optimal stopping rule– i.e. the point at which it makes sense to invest. This is somewhat similar to the marriage market example discussed in workshop. Here are the ingredients that you need: (1) Ito’s Lemma: Given a simple brownian motion of the form: dx = a x,t dt + b x,t dz where dz is the increment of a Weiner process, and a function F x,t – then dF = /F/t + a x,t /F/x + 12 b2 /2F/x2 dt + b x,t /F/x dz In the case of a geometric brownian motion with drift– a V,t = JV and b V,t = aV , and in a case where F is not itself a function of t– just of V (or x) then Ito’s lemma implies: dF = JV /F/V + 12 a2V2 /2F/V2 dt + /F/x aVdz At the point of investment it must be that: F VD,t = VD ? K This is just saying that when it is appropriate to invest, the value of the option is just the value of investment. Moreover, at that point the smooth pasting condition applies– the derivatives of the two sides need to be equal as well. FV VD,t = 1 One more thing to know– in the region where you don’t invest: rFdt = E dF The expected gain in value must equal the interest rate times the time period. Using Ito’s lemma– this means that: rF = JVFv V + 12 a2V2Fvv V The general solution to this second order differential equation is F V = A1VB1 + A2VB2 Each of these must solve: r = J? 12 a2 Bi + 12 a2Bi2 where B1 > 0 and B2 < 0 because we know that the value of this thing doesn’t go to infinity as V goes to zero. Or Bi = 12 ? Ja2 – 1a2 Ja2 ? 12 2 + 2ra2 We can exclude the root where B2 < 0 So B1 = 12 ? Ja2 + 1a2 Ja2 ? 12 2 + 2ra2 Thus we know that A1VDB1 = VD ? K And A1B1VDB1?1 = 1 which together give you: VD = B1B 1?1 K or VD = K 1 2 ? J a2 + J a2 ? 12 2+ 2r a2 ? 12 ? J a2 + J a2 ? 12 2+ 2r a2 = K a2 2 ?J+ a2 2 ?J 2+2ra2 ? a22 ?J+ a22 ?J 2+2ra2 And this we can work with– consider the case where J = 0, VD = 1+ 8r a2 +1 1+ 8r a2 ?1 K which is obviously decreasing with the interest rate and increasing with the variance of the shock. As the variance increases, the option value from waiting gets more valuable and you have a higher cutoff value. In the case where variance is zero– this becomes: VD = K Aggregation of supply: We are often as interested in whether supply curves shift up or demand curves ship down. In an individual supply curve– we know that P = Cv Q is a good starting point for supply curves slope and hence a supply curve slopes up if and only if C . – the cost function is convex. From MWG Proposition 5.C.2, we know that C . is convex if and only if f . is concave. This is one way to get an upward sloping supply curve. If f . displays constant returns to scale (which gives us zero profits– because CRS implies >ZifZi = 0 = >ZiWi/P ) The firm level equations don’t do much for us. In general, homogeneous firms tend to mean that supply is perfectly elastic. There are two ways out of this: (1) assume that there are limits to some input and allow the price of that input to be determined endogenously– so it increases as the quantity supplied increases, (2) decreasing returns to scale and limited entry, (3) heterogeneity across firms. As an example consider a firm that is Cobb-Douglas with N inputs + there is a fixed amount of the first input. The f.o.c. for the all inputs are JiPQ = WiZi and total output equals, which means that the price must equal: < WiJi Ji As asserted earlier– this is indepenent of quantity. If the total industry level of Z1 is fixed at Z, then J1PQ Z = W1 then price equals: Q Z J1 1?J1 < i>2 Wi Ji Ji 1?J1 or the quantity can be written: P 1?J1 J1 Z< i>2 Wi Ji ?Ji J1 In this case the supply elasticity equals (quantity with respect to price equals) 1?J1J1 which is determined by the share of output produced in the limited commodity. Obviously, if we have a limited number of firms, we can have concave production, convex costs, and upward sloping demand– firms then have positive provides and dQ dP = N Cvv Q where N is the number of firms and C vv Q is the second derivative of the cost function. Aggregating with heterogeneity. An easy case is one in which firms can only produce a fixed number of units if they enter Qi for each firm at marginal cost Ci and pay fixed costs Fi for entering. Thus a firm only enters when P ? Ci Qi ? Fi ? 0 Order firms by i so that P ? C i Q i ? F i is strictly increasing with i, where i is characterized by distribution function G i . If we assume a dense enough distribution of firms, we know that there exists a marginal i* such that P ? C iD Q iD ? F iD = 0 Then total supply equals: Xi>iD Q i dG i And the impact of price on quantity equals is ?/iD/P g iD Q iD = Q2 iD g iD P ? C iD Qv iD ? Cv iD Q iD ? Fv iD This expression is certainly positive and depends (1) the density of marginal new entrants, (2) the output of new entrants (squared– because it matters both for the response to price and for the importance of new firms) and (3) the extent to which there is heterogeneity across firms (more heterogeneity– less entry). Simplify and expand the problem at once. All entrant firms face a cost curve C Q which is identical. Thus each firm in the market sets P = Cv Q , and we define QD P with this equation. The entry condition gives us PQD ? C QD = F iD However, each firm has a different fixed cost of operating (this is not irreversible investment– just an cost of being in business). Total supply equals: 1 ? G iD P QD P Differentiation yields that the impact of price on supply equals: 1 ? G iD P Cvv QD P ? g i D /iD /P Q D P = 1 ? G iD P Cvv QD P ? QD2 P g iD Fv iD This combine the intensive and extensive margins of supply. Social multipliers in the supply side: Now assume a fixed number of firms and assume that C = C Q,NQ . where Q is the average supply in the industry. Any one firm sets: P = C1 Q,NQ and the individual level price elasticity is /Q/P = 1C 11 Aggregate elasticity is: /Q /P = 1 ? C12Q /N/P C11 + C12N /Q/P Again, the important effects come from the cross partials which will cause a positive socail multiplier when positive and a negative otherwise. The theory of the firm– hold-up problems– ex post appropriation. Ex post appropriation is the most usual way to make forms of theft socially costly. Assume that you it costs you cQ2 to produce Q units of a good. Assume further that with some probability ^ those goods are taken from you. You end up investing to the point where ^P = 2cQ This shows up big time in trying to understand cross-national differences in GDP. Rule of law is really critical– and this simple model probably captures the bulk of the rule of law problem. Assume that a supplier needs to produce goods before a fixed contract can be specified. If you prefer think about it as building a factory at a cost K, which can then produce N input goods at a cost c. The goods are valued by the input user at an amount V. As long as V > c trade should take place– but at what price. If only this one firm can use the goods, let’s say that the total price is equal to aV + 1 ?a c > c If N aV + 1 ?a c > K then everything is fine and the factory gets built. But if N aV + 1 ?a c < K then the factory won’t get built even if it is in everyone’s interest for it to be built. Ex post appropriation makes it impossible to get the investment occurring. One solution is for the buying firm to provide for the input supplier. This will be more attractive when the customer firm has more bargaining power relative to the supplier, or when the ratio of fixed costs to marginal costs is quite high. This model gives no reason why you wouldn’t consolidate but it’s a start– the contracting approach to the theory of the firm. Agency Problems– You get one page First– people generally don’t like to assume that firms maximize profits anymore. Generally – we need something about incentive problems within the firm. The general set-up of the problem is that there is a principal, we is: (1) designing a contract for the agent and (2) trying to maximize profits. There is an agent who is maximizing utility subject to the contract. Thus let the contract be a function of (1) profits ^ and (2) some other external variable (observable) – x or w ^,x Profits themselves are a function of the agent’s action, a, the observable external variable, x, and an unobservable external variable y or ^ a,x,y The principal maximizes subject to two constraints: (1) Incentive compatibility– the agent chooses her/his action to maximize the expectation of U w ^ a,x,y ,x ,a which produces an optimal a, denoted aD which is itself a function of the contract. The agent can choose a either before or after the states of the world are revealed. I will focus on the ex ante decision-making. (2) Individual rationality– the agents utility, given this optimal action, is sufficiently high that it exceeds the outside reservation utility, i.e. E U w ^ aD,x,y ,x ,aD ? U Subject to these constraints the principal maximizes the expectation of : ^ aD,x,y ,x ? w ^ aD,x,y ,x ,aD This is particularly nice economics because it involves two sided optimization– the agent is maximizing subject to the constraint that the principal is maximizing. A few simple results: (1) If the agent has utility that is quasi-linear with respect to wealth (i.e. risk neutral), and faces no credit constraints, the optimal contract is of the form ^ aD,x,y + k. The IR constraint then becomes: E w ^ aD,x,y ,x ,aD + v aD = U So the principals maximization problem can be rewritten: E ^ aD,x,y ? V aD + U This is maximized when a* is chosen to maximize E ^ aD,x,y ? V aD Either risk aversion or credit constraints can cause this to collapse. Most typically the literature has a trade off between incentives and risk-aversion. In that case, you could imagine that var(x)=0. ^ a,y = a + y and U w,a = w ? .5bw2 ? ca = w ? .5bw2 ? .5b var w ? ca The optimal linear contract: w ^ = J+K^ This induces an effort level of a = K 1?bJ ?cbK2 and individual rationality implies that: w ? .5bw2 ? .5b var w ? ca = U or J? .5bJ2 + K 1?bJ ?c 2bK2 ? .5K2b var y = U Now the agent maximizes: 1 ?K K 1?bJ ?cbK2 ?J subject to the IR constraint. And this is doable. The Coase Theorem If property rights are well defined, and if contracting/enforcement costs are negligible, then (1) externalities will be fully internalized (i.e. we will be on the pareto frontier), and (2) in the absence of income effects, the outcome will be independent of the assignment of property rights. Obvious example– firm 1’s costs are Q12 + cQ1Q2 Firm 2’s costs are Q22 Both prices are one. If they make decisions totally independently, firm two sets quantity equal to .5, firm one sets quantity equal to .5-.25c Joint maximixation would imply Q2 = Q1 = 12+c You can get this by taxing the second firm by c2 + c per unit of output. If you have assigned the second firm, the right to do whatever he wants, the first firm can offer a contract F Q2 that is declining in Q2 The only requirement of this contract is that firm 2 cannot be worse off with the contract than without it (i.e. incentive compatibility). This implies that .25 = F Q2 + Q2 ? Q22 If enforcement is not a problem, firm one can set F Q2 = .25 ? Q2 + Q22 for some desired Q2 and zero otherwise. This makes the contract incentive compatible. Now firm 1 chooses Q1 and Q2 to maximize: Q1 ? Q12 ? cQ1Q2 ? F Q2 which gives us: Q1 = .5 ? .5cQ2 and cQ2 = ?Fv Q2 Differentiating F Q2 = .75 ? Q2 + Q22 gives us ?Fv Q2 = ?1 + 2Q2 or Q2 = 12+c The overall contract needs to pay firm 2– the difference in profits. If person 1 has the property right– i.e. the ability to stop person two from producing at all, then the right thing occurs as well. The only way that different property rights could matter is if the size of the transfer caused the desire to avoid the pollution to matter. The big missing thing here is enforcement. Essentially, either Pigovian taxes or the Coase theorem relies critically on the ability to punish an actor who pollutes. Transaction costs are almost surely overrated relative to the ability to punish cheaters. Take an even simpler case– firm 2’s action brings it benefit B and imposes cost C on firm 1. As long as C > B, the action should not occur. Firm two either needs to be taxed some quantity above B or bribed some quantity above B for foregoing the action. But once the action has occurred, firm 2 is going to work hard to avoid paying any penalties. This type of framework can help us to understand: (1) quantity restrictions rather than Pigovian taxes because they are easier to enforce, (2) government ownership if the private firm is effectively too powerful to be disciplined, (3) regulatory agencies that are incentivized more strongly than judges, or that have special expertise. The broad point is that enforcement really matters.