Problem set 1 Micro Theory S. Wang Question1.1. Show that“f(X)=f(x),Vx∈R,A>1” implies“f(A)= Af(x),Vx∈R,A>0.” estion 1.2. Use a Lagrange function to solve c(w1, w2, y) for the following problem c(w1, w2, y)= min 1-1+w2.C2 x1,x2 Question 1.3. Use a graph to solve the cost function for the following problem y)≡ Inin w1. I1+2x t +bC2 Question 1. 4. Find the cost function for the following problem c(1,2,y)≡minn11+22 t mIn Question 1.5. Consider the factor demand system (2,2,y)=|b1+b2( n2(,m2,y)=|h+a2 where b11, b12, b21, b22>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?
Problem Set 1 Micro Theory, S. Wang Question 1.1. Show that “ f(λx) = λf(x), ∀ x ∈ Rn +, λ > 1 ” implies “ f(λx) = λf(x), ∀ x ∈ Rn +, λ > 0. ” Question 1.2. Use a Lagrange function to solve c(w1, w2, y) for the following problem: ⎧ ⎪⎪⎨ ⎪⎪⎩ c(w1, w2, y) ≡ min x1, x2 w1x1 + w2x2 s.t. xρ 1 + xρ 2 = yρ Question 1.3. Use a graph to solve the cost function for the following problem: ⎧ ⎪⎪⎨ ⎪⎪⎩ c(w1, w2, y) ≡ min x1, x2 w1x1 + w2x2 s.t. y = ax1 + bx2 Question 1.4. Find the cost function for the following problem: c(w1, w2, y) ≡ min x1, x2 w1x1 + w2x2 s.t. y = min {ax1, bx2} Question 1.5. Consider the factor demand system: x1(w1, w2, y) = % b11 + b12 w2 w1 1 2 & y, x2(w1, w2, y) = % b22 + b21 w1 w2 1 2 & y where b11, b12, b21, b22 > 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(y1), or butter (32), or both; using labor(a), as the sole input to the production process. Feasible production is represented by a production possibility set with a frontier x= vgi+32 (a) Write the production function on the implicit form G(y1, y2, 33)=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 i 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 p1 and p2 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(a1, 2)=-+- (a) Compute the ordinary demand functions (b) Show that the indirect utility function is -(VPi+VP2)2/I (c)Compute the expenditure function (d) Compute the compensated demand functions Question 1.8. A consumer has expenditure function e(p1, P2, u)=Pip2u. 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)>[0,0.5;(0,0.5;100,0.5),0.5
Question 1.6. The Ace Transformation Company can produce guns ( y1 ), or butter ( y2 ), 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 = sy2 1 + y2 2. (a) Write the production function on the implicit form G(y1, y2, y3)=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 x¯ 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 p1 and p2 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(x1, x2) = − 1 x1 − 1 x2 . (a) Compute the ordinary demand functions. (b) Show that the indirect utility function is −( √p1 + √p2)2/I. (c) Compute the expenditure function. (d) Compute the compensated demand functions. Question 1.8. A consumer has expenditure function e(p1, p2, u) = p 1/4 1 pb 2u. 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) " [0, 0.5; (0, 0.5; 100, 0.5), 0.5] ? 2
Question 1.11. For the insurance problem: max(1-p)u(Ii)+pu(I2) t.(1-)l1+l2 where I>0 is the loss, p E(0, 1)is the probability of the bad event, T E(O, 1) is the price of insurance, w is initial wealth, I1=W-T, and I2=w-1+(1-q (a) If the insurance market is not competitive and the insurance company makes a posi- tive expected profit: Tq-pq>0, will the consumer demand full-insurance('=l) under-insurance(q"0)? 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 ua(rA, ra)=aIn A+(1-a)In a, VA=(0 uB(aB, B)=min(=B, 3), Calculate the equilibrium price(s) and allocation(s Question 1. 13. Consider a two-consumer, two-good economy. Both consumers have the same Cobb-Douglas utility functions u;(ai 2)=In. +In There is one unit of each good available. Calculate the set of Pareto efficient allocations and illustrate it in an edgeworth box. Question 1. 14. Consider an economy with two firms and two consumers. Denote g the number of guns, b as the amount of butter, and a as the amount of oil. The utilit functions for consumers are u1(,b)=94b06, u2(g,b)=10+0.5lng+0.5lnb Each consumer initially owns 10 units of oil: i1=i2= 10. Consumer 1 owns firm 1 which has production function g=2 r; consumer 2 owns firm 2 which has production function b=3 r. Find the competitive equilibrium 3
Question 1.11. For the insurance problem: max (1 − p)u(I1) + pu(I2) s.t. (1 − π)I1 + πI2 = 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, I1 = w − πq, and I2 = w − l + (1 − π)q. (a) If the insurance market is not competitive and the insurance company makes a positive expected profit: πq − pq > 0, will the consumer demand full-insurance (q∗ = l), under-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 endowments: uA(x1 A, x2 A) = a ln x1 A + (1 − a) ln x2 A, WA = (0, 1) uB(x1 B, x2 B) = min(x1 B, x2 B), WB = (1, 0) Calculate the equilibrium price(s) and allocation(s). Question 1.13. Consider a two-consumer, two-good economy. Both consumers have the same Cobb-Douglas utility functions: ui(x1 i , x2 i) = ln x1 i + ln x2 i , i = 1, 2. There is one unit of each good available. Calculate the set of Pareto efficient 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 u1(g, b) = g0.4 b 0.6 , u2(g, b) = 10 + 0.5 ln g + 0.5 ln b. Each consumer initially owns 10 units of oil: x¯1 = ¯x2 = 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.l. For any yE R+ and 00,yER+, where the equality for t> 1 is already oIven Answer 1. 2. See Varian(2nd ed. )p 31-33, or Varian(3rd ed. )p55-56 Answer 1. 3. From Figure 1.2, we see that the minimum point is(4, 0)or(0, 6) depending on the ratio of 3. Therefore, the cost is y or by. That is c(w1, w2, 9)=min wn.w ax,+ bx,=y Isoquant Wirtw2x2= Figure 1. 2. Cost Minimization with Linear Technology Answer 1.4. Since the production is not differentiable, we cannot use FoC to solve the problem. One way to do is to use a graph 4
Answer Set 1 Answer 1.1. For any y ∈ Rn + and 0 0, y ∈ Rn +, 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 w1 w2 . Therefore, the cost is w1 a y or w2 b y. That is, c(w1, w2, y) = min qw1 a y, w2 b y r . x x 1 2 . _y a w x + w x = c 1 1 2 2 ax + bx = y 1 2 Isoquant Figure 1.2. Cost Minimization with Linear Technology Answer 1.4. Since the production is not differentiable, we cannot use FOC to solve the problem. One way to do is to use a graph. 4
Figure 1.3. Cost Minimization with Leontief Technology From Figure 1.3, we see that the minimum point is(4, 4). Therefore, the cost function 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 effects, that is 2 '12 Therefore, b12= b21. With 012= b21, the substitution matrix is 5w12w2y w1 a w a w2 (3u12u22y-2uiu2'y/ We have <0 b2 =1m(1-听)=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: b12= b21 5
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(w1, w2, y) = w1 a + w2 b 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 effects, that is, ∂x1 ∂w2 = ∂x2 ∂w1 or 1 2 b12 y √w1w2 = 1 2 b21 y √w1w2 . Therefore, b12 = b21. With b12 = b21, the substitution matrix is ⎛ ⎜⎝ ∂x1 ∂ w1 ∂x1 ∂ w2 ∂x2 ∂ w1 ∂x2 ∂ w2 ⎞ ⎟⎠ = b12 ⎛ ⎜⎝ −1 2w− 3 2 1 w 1 2 2 y 1 2w−1 2 1 w− 1 2 2 y 1 2w− 1 2 1 w−1 2 2 y −1 2w 1 2 1 w− 3 2 2 y ⎞ ⎟⎠ . We have ∂x1 ∂ w1 < 0, and ∂x1 ∂ w1 ∂x1 ∂ w2 ∂x2 ∂ w1 ∂x2 ∂ w2 = b2 12 −1 2w−3 2 1 w 1 2 2 y 1 2w− 1 2 1 w−1 2 2 y 1 2w− 1 2 1 w− 1 2 2 y −1 2w 1 2 1 w− 3 2 2 y = 1 4 b2 12y2 w−1 2 w−1 1 − w−1 1 w−1 2 = 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: b12 = b21. 5
Let b= b12= b21. Then the cost function is c(w1, w2, y)=W111+w2. c2=1b11 +w2b22+2byw12ly Answer 1.6 (a) The production set is defined by ≥V+ which means that if the firm wants to produce(/, g2) it needs at least a amount of labor. Since the labor is an input, it should be negative in the definition of implicit production function. This means that we can choose 33=- and define G(,,)=V听++ with Y={(m,m,)>0,m>0,功≤0 The production process is then defined by G(y1, y2, y3)0, yi t y2 thus Assumption 2. 1 is satisfied. The 2nd order conditions are yiy1y19192 Gy2 Gy2yn gg2y2 + y+2 0 Gu Gy2 gys 0 Gy Gya 1 yI Cy1y1 Uy1y2 Cy193 y1 Uy191 Cy1y2 9191y192 0. GGGG GoaL G Gya Guay gaya grays Therefore, Assumptions 2.2 is satisfi (b)The problem is 丌=maxp1h+P2y-a t.2+v2=x2 6
Let b ≡ b12 = b21. Then the cost function is c(w1, w2, y) = w1x1 + w2x2 = [w1b11 + w2b22 + 2b √w1w2]y. Answer 1.6. (a) The production set is defined by x ≥ t y2 1 + y2 2 which means that if the firm wants to produce (y1, y2) 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 y3 = −x and define G(y1, y2, y3) ≡ t y2 1 + y2 2+y3 with Y = {(y1, y2, y3) | y1 > 0, y2 > 0, y3 ≤ 0}. The production process is then defined by G(y1, y2, y3) ≤ 0 for y ∈ Y. We first have Gy1 = s y1 y2 1 + y2 2 > 0, Gy2 = s y2 y2 1 + y2 2 > 0, Gy3 = 1 > 0, thus Assumption 2.1 is satisfied. The 2nd order conditions are 0 Gy1 Gy2 Gy1 Gy1y1 Gy1y2 Gy2 Gy2y1 Gy2y2 = 0 √ y1 y2 1+y2 2 √ y2 y2 1+y2 2 √ y1 y2 1+y2 2 y2 2 (y2 1+y2 2) 3/2 − y1y2 (y2 1+y2 2) 3/2 √ y2 y2 1+y2 2 − y1y2 (y2 1+y2 2) 3/2 y2 1 (y2 1+y2 2) 3/2 = − 1 sy2 1 + y2 2 < 0 and 0 Gy1 Gy2 Gy3 Gy1 Gy1y1 Gy1y2 Gy1y3 Gy2 Gy2y1 Gy2y2 Gy2y3 Gy3 Gy3y1 Gy3y2 Gy3y3 = 0 Gy1 Gy2 1 Gy1 Gy1y1 Gy1y2 0 Gy2 Gy2y1 Gy2y2 0 1 0 00 = − Gy1y1 Gy1y2 Gy2y1 Gy2y2 = 0. Therefore, Assumptions 2.2 is satisfied. (b) The problem is π = max p1y1 + p2y2 − wx¯ s.t. y2 1 + y2 2 = ¯x2 6
The solution are: P1. p2. 所+n Pi +p2 丌=(12+n2- Therefore the supplies are 0-() +西2 Answer 1.7 (a)5. The consumer's problem is v(p, I)=max -- st.p11+p2x2=1 Let L=---++A(I-P1-1-p2 C2). The FOC's a2 imply T1=v22 x2. Substituting this into the budget constraint will immediately 2+√P1P2 By symmetry, we also have I n1=p1+√P2 (b)Substituting the consumer's demands into the utility function will give us v(p, I)=-P1+vp1p2_ P2+VP1P2 p1+p+2√1P 匝+VP2)2 (c) Let u=v(p, e),i.e (V+√P)2 which immediately gives us the expenditure function e(,a)=(+V)2
The solution are: y1 = p1x¯ sp2 1 + p2 2 , y2 = p2x¯ sp2 1 + p2 2 . π = ¯x t p2 1 + p2 2 − w . Therefore the supplies are: (y1, y2) = ⎧ ⎪⎪⎨ ⎪⎪⎩ √ p1x¯ p2 1+p2 2 , √ p2x¯ p2 1+p2 2 if w ≤ sp2 1 + p2 2 (0, 0) if w > sp2 1 + p2 2. Answer 1.7. (a) [5]. The consumer’s problem is ⎧ ⎪⎨ ⎪⎩ v(p, I) = max − 1 x1 − 1 x2 s.t. p1x1 + p2x2 = I Let L ≡ − 1 x1 − 1 x2 + λ(I − p1x1 − p2x2). The FOC’s 1 x2 1 = λp1, 1 x2 2 = λp2 imply x1 = tp2 p1 x2. Substituting this into the budget constraint will immediately give us x∗ 2 = I p2 + √p1p2 . By symmetry, we also have x∗ 1 = I p1 + √p1p2 . (b) Substituting the consumer’s demands into the utility function will give us v(p, I) = −p1 + √p1p2 I − p2 + √p1p2 I = −p1 + p2 + 2√p1p2 I = −( √p1 + √p2)2 I . (c) Let u = v(p, e), i.e. u = −( √p1 + √p2)2 e which immediately gives us the expenditure function: e(p, u) = −( √p1 + √p2)2 u . 7
(d) Substituting e(p, u) for I in the consumer's demand functions we get 五(.0)=-n)=-所+②=-+V=1 +√P1 PIP2)u /p1 y symmetry 2(P,u) Answer 1.8. Since e(p, a)is linearly homogeneous in P, b=4 Answer 1.9. We can easily show that u(p, AD)=Au(p, D),vA>0, given that fact that u(Ar)=Au(),vA>0. Then, Up (p, I) is linearly homogeneous in I, and ur(p, I)is homogeneous of degree 0 in I. By Roys identity, we then have a(p, AD) Upi (p, AD) A(p, I (p,A)(D,D) Taking the derivative w.r. t. A, we then have ax(p, AT) etting a=1. we then have I ar*(p r a1 Answer 1.10. If RCLA were not violated. then 0,0.5;(0,0.5;10,0.5),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, 15) p/(2) The expected profit is Tq-pq=(T-p)q>0.Then, T>p. Thus Then, u(1I, i.e., w-T">w-L+(1-T)a*. It implies q *<l, that is, we have under-insurance 8
(d) Substituting e(p, u) for I in the consumer’s demand functions we get x¯1(p, u) = e(p, u) p1 + √p1p2 = − ( √p1 + √p2)2 (p1 + √p1p2)u = − √p1 + √p2 u √p1 = −1 u 1 + √p2 √p1 . By symmetry, x¯2(p, u) = −1 u 1 + √p1 √p2 . 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, vpi (p, I) is linearly homogeneous in I, and vI (p, I) is homogeneous of degree 0 in I. By Roy’s identity, we then have x∗ i(p, λI) = −vpi (p, λI) vI (p, λI) = −λvpi (p, I) vI (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 violated. Answer 1.11. (a) [5]. At the optimal point (I∗ 1 , I∗ 2 ), (1 − p)u0 (I∗ 1 ) pu0 (I∗ 2 ) = 1 − π π . The expected profit is πq − pq = (π − p)q > 0. Then, π > p. Thus, 1 − π π 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 T=P, in Example 3.4, we have shown that the solution must be on the 45 line. When T>p, the budget line is Hatter, and the tangent point must be below the 45 line. That is the individual is under-insured slope=t slope P gure 5.1. Insurance in a non-competitive market Answer 1.12. Individual A's utility function is equivalent to ua(aA, xa) (=A)(al-a. Let p= P1 and P2= 1. Then the income is IA=P0+1.1=1 and the demands are (1-a) For individual B, by its utility function, we know that the demands must satisfy aB=x3 Then by budget constraint pzB +aa=IB=p 1+1.0=p, the demands are IB P 1+p1+p In equilibrium, the total supply of good 1 must be equal to the total demand for good 1 p I+p Therefore, p and the allocation is 9
(b) [5]. When π = p, in Example 3.4, we have shown that the solution must be on the 45◦ line. When π > p, the budget line is flatter, 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 45o . . 1-π π . Figure 5.1. Insurance in a non-competitive market Answer 1.12. Individual A’s utility function is equivalent to uA(x1 A, x2 A) = (x1 A)a(x2 A)1−a. Let p = p1 and p2 = 1. Then the income is IA = p · 0+1 · 1=1, and the demands are: x1 A = aIA p = a p , x2 A = (1 − a)IA 1 = 1 − a. For individual B, by its utility function, we know that the demands must satisfy x1 B = x2 B. Then by budget constraint px1 B + x2 B = IB = p · 1+1 · 0 = p, the demands are: x1 B = x2 B = IB 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 (x1 A) ∗ = (x2 A) ∗ = 1 − a, (x1 B) ∗ = (x2 B) ∗ = a. 9
Answer 1.13. By Proposition 1.27, the following equation defines the set of P O. points Or Feasibility requires r2+a2 Letr≡ I and y≡r. Then above two equations imply y Therefore the set of P O allocations=f[(, y),(1-a,1-y)llx=y,x20 This set is the diagonal line in the following diagram Figure 4.4. P.O. Allocations Answer 1. 14. Denote g= guns, a=oil,b= butter, price of guns Pg, price of butter Po, price of oil Pr=l (we can arbitrarily choose one of prices. We can do that because of the homogeneity of demand functions). The two consumers are consumer1:u1(g,b)=904606,g=2x,1=10. 2(G,b)=905605,g=3x,2=10 Firm ls problem: 丌1≡maxP9-x=max(2P-1)x
Answer 1.13. By Proposition 1.27, the following equation defines the set of P.O. points: 1/x1 1 1/x2 1 = 1/x1 2 1/x2 2 or x2 1 x1 1 = x2 2 x1 2 . Feasibility requires x1 1 + x1 2 = 1 and x2 1 + x2 2 = 1. Let x ≡ x1 1 and y ≡ x2 1. Then above two equations imply y x = 1 − y 1 − x =⇒ y = x. Therefore, the set of P.O. allocations = [(x, y), (1 − x, 1 − y)] | x = y, x ≥ 0 . 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 Pg, price of butter Pb, price of oil Px = 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: u1(g, b) = g0.4b0.6, g = 2x, x¯1 = 10. consumer 2: u2(g, b) = g0.5b0.5, g = 3x, x¯2 = 10. Firm 1’s problem: π1 ≡ maxx Pg g − x = maxx (2Pg − 1)x. 10