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"<I, or over-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), 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 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 0<t<l, let a=ty and A=f. We then have f(y=f( r)=Af(a)=f(ty) Therefore, tf(y)=f(ty),Vt>0,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 <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 ∈ 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