Problem Set 1 Micro Theory, S. Wang Question 1.1. Show that “f(λx)=λf(x), ? x ∈ R n + , λ > 1”implies“f(λx)= λf(x), ? x ∈R n + , λ > 0.” Question 1.2. Use a Lagrange function to solve c(w 1 ,w 2 ,y) for the following problem: ? ? ? ? ? ? ? c(w 1 ,w 2 ,y) ≡ min x 1 ,x 2 w 1 x 1 + w 2 x 2 s.t. x ρ 1 + x ρ 2 = y ρ Question 1.3. Use a graph to solve the cost function for the following problem: ? ? ? ? ? ? ? c(w 1 ,w 2 ,y) ≡ min x 1 ,x 2 w 1 x 1 + w 2 x 2 s.t. y = ax 1 + bx 2 Question 1.4. Find the cost function for the following problem: c(w 1 ,w 2 ,y) ≡ min x 1 ,x 2 w 1 x 1 + w 2 x 2 s.t. y =min{ax 1 ,bx 2 } Question 1.5. Consider the factor demand system: x 1 (w 1 ,w 2 ,y)= null b 11 + b 12 null w 2 w 1 null1 2 null y, x 2 (w 1 ,w 2 ,y)= null b 22 + b 21 null w 1 w 2 null1 2 null y where b 11 ,b 12 ,b 21 ,b 22 > 0 are parameters. Find the condition(s) on the parameters so that this demand system is consistent with cost minimizing behavior. What is the cost function then corresponding to the above factor demand system? 1 Question 1.6. The Ace Transformation Company can produce guns (y 1 ), or butter (y 2 ), or both; using labor (x), as the sole input to the production process. Feasible production is represented by a production possibility set with a frontier x = null y 2 1 + y 2 2 . (a) Write the production function on the implicit form G(y 1 ,y 2 ,y 3 )=0. Does G satisfy Assumptions 2.1 and 2.2? (b) Suppose that the company faces the following union demands. In the next year it must purchase exactly ˉx units of labor at a wage rate w; or no labor will be supplied in the next year. If the company knows that it can sell unlimited quantities of guns and butter at prices p 1 and p 2 respectively, and chooses to maximize next year’s profits, what is its optimal production plan? Question 1.7. A consumer has a utility function u(x 1 ,x 2 )=? 1 x 1 ? 1 x 2 . (a) Compute the ordinary demand functions. (b) Show that the indirect utility function is ?( √ p 1 + √ p 2 ) 2 /I. (c) Compute the expenditure function. (d) Compute the compensated demand functions. Question 1.8. A consumer has expenditure function e(p 1 ,p 2 ,u)=p 1/4 1 p b 2 u. What is the value of b? Question 1.9. Suppose the consumer’s utility function is homogeneous of degree 1. Show that the consumer’s demand functions have constant income elasticity equals 1. Question 1.10. What axiom is violated by (0,0.75; 100,0.25) null [0, 0.5; (0,0.5; 100,0.5), 0.5] ? 2 Question 1.11. For the insurance problem: max (1?p)u(I 1 )+pu(I 2 ) s.t. (1?π)I 1 + πI 2 = w?πl where l>0 is the loss, p ∈ (0, 1) is the probability of the bad event, π ∈ (0, 1) is the price of insurance, w is initial wealth, I 1 = w?πq, and I 2 = w?l +(1?π)q. (a) If the insurance market is not competitive and the insurance company makes a posi- tive expected profit: πq?pq > 0, will the consumer demand full-insurance (q ? = l), under-insurance (q ? <l), or over-insurance (q ? >l)? Show your answer. (b) Show the above solution on a diagram. Question 1.12. There are two consumers A and B with utility functions and endow- ments: u A (x 1 A ,x 2 A )=alnx 1 A +(1?a)lnx 2 A ,W A =(0,1) u B (x 1 B ,x 2 B )=min(x 1 B ,x 2 B ),W B =(1,0) Calculate the equilibrium price(s) and allocation(s). Question 1.13. Consider a two-consumer, two-good economy. Both consumers have thesameCobb-Douglasutilityfunctions: u i (x 1 i ,x 2 i )=lnx 1 i +lnx 2 i ,i=1,2. There is one unit of each good available. Calculate the set of Pareto e?cient allocations and illustrate it in an Edgeworth box. Question 1.14. Consider an economy with two firms and two consumers. Denote g as the number of guns, b as the amount of butter, and x as the amount of oil. The utility functions for consumers are u 1 (g,b)=g 0.4 b 0.6 ,u 2 (g,b)=10+0.5lng +0.5lnb. Each consumer initially owns 10 units of oil: ˉx 1 =ˉx 2 =10. Consumer 1 owns firm 1 which has production function g =2x; consumer 2 owns firm 2 which has production function b =3x. Find the competitive equilibrium. 3 Answer Set 1 Answer 1.1. For any y ∈R n + and 0 <t<1, let x ≡ ty and λ ≡ 1 t . We then have f(y)=f(λx)=λf(x)= 1 t f(ty). Therefore, tf(y)=f(ty), ? t>0,y∈ R n + , where the equality for t ≥ 1 is already given. Answer 1.2. See Varian (2nd ed.) p.31-33, or Varian (3rd ed.) p.55-56. Answer 1.3. From Figure 1.2, we see that the minimum point is ( y a , 0) or (0, y b ) depending on the ratio of w 1 w 2 . Therefore, the cost is w 1 a y or w 2 b y. That is, c(w 1 ,w 2 ,y)=min null w 1 a y, w 2 b y null . x x 1 2 . y_ a cxwxw =+ 2211 ybxax =+ 21 Isoquant Figure 1.2. Cost Minimization with Linear Technology Answer 1.4. Since the production is not di?erentiable, we cannot use FOC to solve the problem. One way to do is to use a graph. 4 x x 1 2 ax =bx 1 2 y_ b y_ a y=f(x) Figure 1.3. Cost Minimization with Leontief Technology From Figure 1.3, we see that the minimum point is ( y a , y b ). Therefore, the cost function is: c(w 1 ,w 2 ,y)= null w 1 a + w 2 b null y. Answer 1.5. If the demand system is a solution of a cost minimization problem, then it must satisfy the properties listed in Proposition 1.6. Property (1) in the proposition is obviously satisfied. Property (2) requires symmetric cross-price e?ects, that is, ?x 1 ?w 2 = ?x 2 ?w 1 or 1 2 b 12 y √ w 1 w 2 = 1 2 b 21 y √ w 1 w 2 . Therefore, b 12 = b 21 . With b 12 = b 21 , the substitution matrix is ? ? ? ?x 1 ? w 1 ?x 1 ? w 2 ?x 2 ? w 1 ?x 2 ? w 2 ? ? ? = b 12 ? ? ? ? 1 2 w ? 3 2 1 w 1 2 2 y 1 2 w ? 1 2 1 w ? 1 2 2 y 1 2 w ? 1 2 1 w ? 1 2 2 y ? 1 2 w 1 2 1 w ? 3 2 2 y ? ? ? . We have ?x 1 ? w 1 < 0, and null null null null null null null ?x 1 ? w 1 ?x 1 ? w 2 ?x 2 ? w 1 ?x 2 ? w 2 null null null null null null null = b 2 12 null null null null null null null ? 1 2 w ? 3 2 1 w 1 2 2 y 1 2 w ? 1 2 1 w ? 1 2 2 y 1 2 w ? 1 2 1 w ? 1 2 2 y ? 1 2 w 1 2 1 w ? 3 2 2 y null null null null null null null = 1 4 b 2 12 y 2 null w ?1 2 w ?1 1 ?w ?1 1 w ?1 2 null =0. Thus, the substitution matrix is negative semi-definite. Finally, property (4) is implied by the fact that the substitution matrix is negative semi-definite. Therefore, to be consistent with cost minimization, we need and only need condition: b 12 = b 21 . 5 Let b ≡ b 12 = b 21 . Then the cost function is c(w 1 ,w 2 ,y)=w 1 x 1 + w 2 x 2 =[w 1 b 11 + w 2 b 22 +2b √ w 1 w 2 ]y. Answer 1.6. (a) The production set is defined by x ≥ null y 2 1 + y 2 2 which means that if the firm wants to produce (y 1 ,y 2 ) it needs at least x amount of labor. Since the labor x is an input, it should be negative in the definition of implicit production function. This means that we can choose y 3 = ?x and define G(y 1 ,y 2 ,y 3 ) ≡ null y 2 1 + y 2 2 +y 3 with Y = {(y 1 ,y 2 ,y 3 ) | y 1 > 0,y 2 > 0,y 3 ≤ 0}. The production process is then defined by G(y 1 ,y 2 ,y 3 ) ≤ 0 for y ∈ Y. We first have G y 1 = y 1 null y 2 1 + y 2 2 > 0,G y 2 = y 2 null y 2 1 + y 2 2 > 0,G y 3 =1> 0, thus Assumption 2.1 is satisfied. The 2nd order conditions are null null null null null null null null null null null 0 G y 1 G y 2 G y 1 G y 1 y 1 G y 1 y 2 G y 2 G y 2 y 1 G y 2 y 2 null null null null null null null null null null null = null null null null null null null null null null null null 0 y 1 √ y 2 1 +y 2 2 y 2 √ y 2 1 +y 2 2 y 1 √ y 2 1 +y 2 2 y 2 2 ( y 2 1 +y 2 2 ) 3/2 ? y 1 y 2 ( y 2 1 +y 2 2 ) 3/2 y 2 √ y 2 1 +y 2 2 ? y 1 y 2 ( y 2 1 +y 2 2 ) 3/2 y 2 1 ( y 2 1 +y 2 2 ) 3/2 null null null null null null null null null null null null = ? 1 null y 2 1 + y 2 2 < 0 and null null null null null null null null null null null null null null null 0 G y 1 G y 2 G y 3 G y 1 G y 1 y 1 G y 1 y 2 G y 1 y 3 G y 2 G y 2 y 1 G y 2 y 2 G y 2 y 3 G y 3 G y 3 y 1 G y 3 y 2 G y 3 y 3 null null null null null null null null null null null null null null null = null null null null null null null null null null null null null null null 0 G y 1 G y 2 1 G y 1 G y 1 y 1 G y 1 y 2 0 G y 2 G y 2 y 1 G y 2 y 2 0 10 00 null null null null null null null null null null null null null null null = ? null null null null null null null G y 1 y 1 G y 1 y 2 G y 2 y 1 G y 2 y 2 null null null null null null null =0. Therefore, Assumptions 2.2 is satisfied. (b) The problem is π =maxp 1 y 1 + p 2 y 2 ?wˉx s.t. y 2 1 + y 2 2 =ˉx 2 6 The solution are: y 1 = p 1 ˉx null p 2 1 + p 2 2 ,y 2 = p 2 ˉx null p 2 1 + p 2 2 . π =ˉx null null p 2 1 + p 2 2 ?w null . Therefore the supplies are: (y 1 ,y 2 )= ? ? ? ? ? ? null p 1 ˉx √ p 2 1 +p 2 2 , p 2 ˉx √ p 2 1 +p 2 2 null if w ≤ null p 2 1 + p 2 2 (0, 0) if w> null p 2 1 + p 2 2 . Answer 1.7. (a) [5]. The consumer’s problem is ? ? ? ? ? v(p, I)=max ? 1 x 1 ? 1 x 2 s.t. p 1 x 1 + p 2 x 2 = I Let L ≡? 1 x 1 ? 1 x 2 + λ(I ?p 1 x 1 ?p 2 x 2 ). The FOC’s 1 x 2 1 = λp 1 , 1 x 2 2 = λp 2 imply x 1 = null p 2 p 1 x 2 . Substituting this into the budget constraint will immediately give us x ? 2 = I p 2 + √ p 1 p 2 . By symmetry, we also have x ? 1 = I p 1 + √ p 1 p 2 . (b) Substituting the consumer’s demands into the utility function will give us v(p, I)=? p 1 + √ p 1 p 2 I ? p 2 + √ p 1 p 2 I = ? p 1 + p 2 +2 √ p 1 p 2 I = ? ( √ p 1 + √ p 2 ) 2 I . (c) Let u = v(p, e), i.e. u = ? ( √ p 1 + √ p 2 ) 2 e which immediately gives us the expenditure function: e(p, u)=? ( √ p 1 + √ p 2 ) 2 u . 7 (d) Substituting e(p, u) for I intheconsumer’sdemandfunctionsweget ˉx 1 (p, u)= e(p, u) p 1 + √ p 1 p 2 = ? ( √ p 1 + √ p 2 ) 2 (p 1 + √ p 1 p 2 )u = ? √ p 1 + √ p 2 u √ p 1 = ? 1 u null 1+ √ p 2 √ p 1 null . By symmetry, ˉx 2 (p, u)=? 1 u null 1+ √ p 1 √ p 2 null . Answer 1.8. Since e(p, u) is linearly homogeneous in p, b = 3 4 . Answer 1.9. We can easily show that v(p, λI)=λv(p, I), ?λ > 0, given that fact that u(λx)=λu(x), ? λ > 0. Then, v p i (p, I) is linearly homogeneous in I, and v I (p, I) is homogeneous of degree 0 in I. By Roy’s identity, we then have x ? i (p, λI)=? v p i (p, λI) v I (p, λI) = ? λv p i (p, I) v I (p, I) = λx ? i (p, I). Taking the derivative w.r.t. λ , we then have I ?x ? i (p,λI) ?I = x ? i (p, I). Setting λ =1, we then have I x ? i ?x ? i (p, I) ?I =1. Answer 1.10. If RCLA were not violated, then [0, 0.5; (0,0.5; 100,0.5), 0.5] ～ (0,0.75; 100,0.25) which would immediately imply a contradiction. Therefore, RCLA must has been vio- lated. Answer 1.11. (a) [5]. At the optimal point (I ? 1 ,I ? 2 ), (1?p)u 0 (I ? 1 ) pu 0 (I ? 2 ) = 1?π π . The expected profitisπq?pq =(π?p)q>0. Then, π >p. Thus, 1?π π < 1?p p . Then, u 0 (I ? 1 ) <u 0 (I ? 2 ) or I ? 1 >I ? 2 , i.e., w?πq ? >w?l+(1?π)q ? . It implies q ? <l, that is, we have under-insurance. 8 (b) [5]. When π = p, in Example 3.4, we have shown that the solution must be on the 45 ? line. When π >p,thebudgetlineisflatter, and the tangent point must be below the 45 ? line. That is, the individual is under-insured. I I 1 2 w w-l slope= slope= 1-p p 45 o . . 1-π π . Figure 5.1. Insurance in a non-competitive market Answer 1.12. Individual A’s utility function is equivalent to u A (x 1 A ,x 2 A )= (x 1 A ) a (x 2 A ) 1?a . Let p = p 1 and p 2 =1. Then the income is I A = p · 0+1· 1=1, and the demands are: x 1 A = aI A p = a p ,x 2 A = (1?a)I A 1 =1?a. For individual B, by its utility function, we know that the demands must satisfy x 1 B = x 2 B . Then by budget constraint px 1 B + x 2 B = I B = p · 1+1· 0=p, the demands are: x 1 B = x 2 B = I B 1+p = p 1+p . In equilibrium, the total supply of good 1 must be equal to the total demand for good 1: a p + p 1+p =1. Therefore, p ? = a 1?a and the allocation is (x 1 A ) ? =(x 2 A ) ? =1?a, (x 1 B ) ? =(x 2 B ) ? = a. 9 Answer 1.13. By Proposition 1.27, the following equation defines the set of P.O. points: 1/x 1 1 1/x 2 1 = 1/x 1 2 1/x 2 2 or x 2 1 x 1 1 = x 2 2 x 1 2 . Feasibility requires x 1 1 + x 1 2 =1 and x 2 1 + x 2 2 =1. Let x ≡ x 1 1 and y ≡ x 2 1 . Then above two equations imply y x = 1?y 1?x =? y = x. Therefore, the set of P.O. allocations = null [(x, y), (1?x, 1?y)] | x = y, x≥ 0 null . This set is the diagonal line in the following diagram. 1 2 y=x P.O. x y Figure 4.4. P.O. Allocations Answer 1.14. Denote g =guns,x=oil,b= butter, price of guns P g , price of butter P b , price of oil P x =1 (we can arbitrarily choose one of prices. We can do that because of the homogeneity of demand functions). The two consumers are: consumer 1: u 1 (g,b)=g 0.4 b 0.6 ,g=2x, ˉx 1 =10. consumer 2: u 2 (g,b)=g 0.5 b 0.5 ,g=3x, ˉx 2 =10. Firm 1’s problem: π 1 ≡ max x P g g ?x =max x (2P g ?1)x. 10 It implies x 1 is indeterminate,P g = 1 2 , π 1 =0. Note that the only possible equilibrium is when P g = 1 2 . Zero-profitargumentisnot accurate here. Firm 2’s problem: π 2 ≡ max x P b b?x =max x (3P b ?1)x. It implies x 2 is indeterminate,P g = 1 3 , π 2 =0. Consumer 1’s problem: max g,b u 1 (g,b)=g 0.4 b 0.6 s.t. P g g + P b b =ˉx 1 + π 1 Its solution is g 1 = 0.4ˉx 1 P g =8,b 1 = 0.6ˉx 1 P b =18. Consumer 2’s problem: max g,b u 2 (g,b)=g 0.5 b 0.5 s.t. P g g + P b b =ˉx 2 + π 2 The solution is g 2 = 0.5ˉx 2 P g =10,b 2 = 0.5ˉx 2 P b =15. Market clearing conditions: x 1 + x 2 =ˉx 1 +ˉx 2 ,g 1 + g 2 =2x 1 ,b 1 + b 2 =3x 2 . Because of Walras Law, we only need two of these three conditions to determine the equilibrium. They imply that x ? 1 =9 and x ? 2 =11. Therefore, the equilibrium is: x ? 1 =9,x ? 2 =11,g ? 1 =8,g ? 2 =10,b ? 1 =18,b ? 2 =15,P ? g = 1 2 ,P ? b = 1 3 . End 11