EC 720 - Math for Economists Samson Alva Department of Economics Boston College October 4 2011 1. Profit Maximization Problem Set 2 Solutions (a) The Lagrangian for this problem is L(y k l λ) = py rk wl λ(y k a l b ). (b) The NDCQ is satisfied since the constraint function g(k l) k a l b y has no maximum i.e. Dg 0 for any (k l). According to the Kuhn-Tucker theorem the values y k l that solve the firm s problem together with the associated value λ for the multiplier must satisfy the first-order conditions the constraint the nonnegativity condition L 1 (y k l λ ) = p λ = 0 L 2 (y k l λ ) = r + aλ (k ) a 1 (l ) b = 0 L 3 (y k l λ ) = w + bλ (k ) a (l ) b 1 = 0 L 4 (y k l λ ) = (k ) a (l ) b y 0 λ 0 the complementary slackness condition λ [(y k ) a (l ) b ] = 0. (c) The first-order condition for y reveals that λ > 0 whenever p > 0. Assuming that this is the case the complementary slackness condition requires the constraint to hold as an equality. Hence the first-order conditions can be used together with the binding constraint to solve for y k l λ in terms of the model s parameters a b p r w: y = k = ( (a ( (a r ) a ( b w r ) 1 b ( b w 1 ) b p a+b ) 1 1 a b ) b p) 1 1 a b
l = ( (a r ) a ( b w λ = p. ) 1 a p) 1 1 a b (d) The solutions from above imply that: i. The optimal y k l all rise when the output price p rises holding all other parameters fixed. ii. The optimal y k l all fall when the rental rate for capital r rises holding all other parameters fixed. iii. The optimal y k l all fall when the wage rate w rises holding all other parameters fixed. iv. The optimal y k l all remain unchanged when p r w all double at the same time. 2. Cobb-Douglas Utility Maximization (a) The Lagrangian for the consumer s problem is L(c 1 c 2 λ) = c a 1c 1 a 2 λ( c 1 + c 2 I). (b) The NDCQ is satisfied since we have a non-degenerate linear constraint. According to the Kuhn-Tucker theorem the values c 1 c 2 that solve the consumer s problem together with the associated value λ for the multiplier must satisfy the first-order conditions the constraint the nonnegativity condition L 1 (c 1 c 2 λ ) = a(c 1) a 1 (c 2) 1 a λ = 0 L 2 (c 1 c 2 λ ) = (1 a)(c 1) a (c 2) a λ = 0 L 3 (c 1 c 2 λ ) = I c 1 c 2 0 λ 0 the complementary slackness condition λ ( c 1 + c 2 I) = 0. (c) The first-order conditions for c 1 c 2 reveal that λ > 0 if the prices are both strictly positive a condition that must hold for the problem to have a well-defined solution in the first place. Assuming that this is the case the complementary slackness conditions requires the constraint to hold as an equality. Hence the first-order order conditions can be used together with the binding 2
constraint to solve for c 1 c 2 λ in terms of the model s parameters I a: c 1 = ai c 2 = (1 a)i ( ) a ( ) 1 a a 1 a λ =. (d) The solutions for c 1 c 2 shown above imply that the consumer spends the fraction a of his or her income on good 1 the remaining fraction 1 a of his or her income on good 2. Hence one could estimate a directly from the observation on the fraction of income spent on good 1. (e) The Lagrangian for the consumer s problem is now L(c 1 c 2 λ) = a ln(c 1 ) + (1 a) ln(c 2 ) λ( c 1 + c 2 I). Once again we have that NDCQ is satisfied. According to the Kuhn-Tucker theorem the values c 1 c 2 that solve the consumer s problem together with the associated value λ for the multiplier must satisfy the first-order conditions the constraint the nonnegativity condition L 1 (c 1 c 2 λ ) = a c 1 λ = 0 L 2 (c 1 c 2 λ ) = 1 a c 2 λ = 0 L 3 (c 1 c 2 λ ) = I c 1 c 2 0 λ 0 the complementary slackness condition λ ( c 1 + c 2 I) = 0. As before the first-order conditions for c 1 c 2 reveal that λ > 0 if the prices are both strictly positive again a condition that must hold if the problem is to have a well-defined solution in the first place. Assuming that this is the case the complementary slackness conditions requires the constraint to hold as an equality. Hence the first-order order conditions can be used together with the binding constraint to solve for c 1 c 2 λ in terms of the model s parameters I a: c 1 = ai 3
c 2 = (1 a)i λ = 1 I. The solutions for c 1 c 2 shown above continue to imply that the consumer spends the fraction a of his or her income on good 1 the remaining fraction 1 a of his or her income on good 2. This is because the two utility functions from this problem the previous one represent the same underlying preference ordering. Hence one could again estimate a directly from the observation on the fraction of income spent on good 1. 3. Utility Maximization - Second-Order Conditions (a) We know from above that c 2 = c 1 = ai (1 a)i λ = 1 I. (b) Following the pattern shown in the theorem the bordered Hessian matrix H is 0 H = a/c 2 1 0 0 (1 a)/c 2 2 therefore H π (1 a) = 1 + ap2 2 > 0. c 2 2 c 2 1 Also the permuted bordered Hessian matrix H π is 0 H π = (1 a)/c 2 2 0 0 a/c 2 1 therefore H π = ap2 2 + (1 a)p2 1 c 2 1 c 2 2 so our solutions satisfy the SONC. (c) To check the SOSC we need H to have a strictly positive determinant which we know from part b) is true. Thus our solution satisfies the SOSC. 4. The Constraint Qualification 4 > 0
(a) Since the utility function is strictly increasing it will always be optimal for the consumer to spend all of his or her income on the single good. Hence c = I/p. (b) With the Lagrangian defined as the Kuhn-Tucker conditions are L(c λ) = U(c) λ(pc I) L 1 (c λ ) = U (c ) λ p = 0 L 2 (c λ ) = I pc 0 λ 0 λ (I pc ) = 0. Since U (c) > 0 for all values of c the first-order condition requires λ > 0 which then implies via the complementary slackness condition that c = I/p. Substituting this result back into the first-order condition yields the solution λ = U (I/P ). (c) With the Lagrangian defined as the Kuhn-Tucker conditions are L(c λ) = U(c) λ(pc I) 3 L 1 (c λ ) = U (c ) 3pλ (pc I) 2 = 0 L 2 (c λ ) = (I pc ) 3 0 λ 0 λ (I pc ) 3 = 0. But substituting the value of c = I/p found above into the new first-order condition yields U (I/P ) 3pλ (I I) 2 = U (I/P ) = 0 which cannot hold if U (c) > 0 for all values of c. What has gone wrong is that with the constraint written in an unnecessarily complicated way the constraint qualification which requires that fails to hold. 5. Perfect Substitutes 3(I pc ) 2 0 5
(a) The solution is for the consumer to spend all of his or her income on the good with the lowest price: c 1 = I/ c 2 = 0 if < c 1 = 0 c 2 = I/ if >. If = then any values of c 1 c 2 satisfying the budget constraint with equality will be optimal. (b) According to the Kuhn-Tucker theorem the values of c 1 c 2 that solve the consumer s problem together with the associated values of λ µ 1 µ 2 must satisfy the first-order conditions L 1 (c 1 c 2 λ µ 1 µ 2) = 1 λ + µ 1 = 0 the constraints L 2 (c 1 c 2 λ µ 1 µ 2) = 1 λ + µ 2 = 0 L 3 (c 1 c 2 λ µ 1 µ 2) = I c 1 c 2 0 L 4 (c 1 c 2 λ µ 1 µ 2) = c 1 0 L 5 (c 1 c 2 λ µ 1 µ 2) = c 2 0 the nonnegativity conditions λ 0 µ 1 0 µ 2 0 the complementary slackness conditions λ (I c 1 c 2) = 0 µ 1c 1 = 0 µ 2c 2 = 0. The first-order conditions the nonnegativity conditions for µ 1 µ 2 imply that λ > 0 so that the budget constraint must always bind. Hence there are three possibilities to consider. The first possibility is that µ 1 > 0 µ 2 = 0. In this case the complementary slackness condition requires that c 1 = 0 the binding budget constraint implies that c 2 = I/. The first-order conditions require that λ = 1/ µ 1 = λ 1 = / 1. But this last condition is consistent with µ 1 > 0 only if >. The second possibility is that µ 1 = 0 µ 2 > 0. In this case reasoning analogous to that above implies that c 1 = I/ c 2 = 0 λ = 1/ µ 2 = / 1. But this last condition is consistent with µ 2 > 0 only if > p1. The final possibility is that µ 1 = µ 2 = 0. The first-order conditions imply that this case can only occur when = so that 6
λ = 1/ = 1/ any values of c 1 c 2 satisfying the budget constraint with equality will work. These solutions serve to confirm the guesses from part (a) above. 6. Elasticities of Dem (a) Differentiate both sides of equation (5) with respect to one of the prices p j to obtain c 1( p 3 I) p j + c 2( p 3 I) p j + p 3 c 3( p 3 I) p j + c j( p 3 I) = 0. Now use the definition of the price elasticities ε ij to rewrite this expression as [c 1( p 3 I)/p j ]ε 1j + [c 2( p 3 I)/p j ]ε 2j +p 3 [c 3( p 3 I)/p j ]ε 3j + c j( p 3 I) = 0. Finally multiply through by p j divide through by I rearrange to obtain s 1 ε 1j + s 2 ε 2j + s 3 ε 3j = s j. (b) Differentiate both sides of equation (5) with respect to I to obtain c 1( p 3 I) I + c 2( p 3 I) I + p 3 c 3( p 3 I) I = 1. Now use the definition of the income elasticities η i to rewrite this expression as [c 1( p 3 I)/I]η 1 + [c 2( p 3 I)/I]η 2 + p 3 [c 3( p 3 I)/I]η 3 = 1 or more simply s 1 η 1 + s 2 η 2 + s 3 η 3 = 1. (c) Differentiate both sides of (6) with respect to r to obtain c i (r r rp 3 ri) +p 3 c i (r r rp 3 ri) p 3 + c i (r r rp 3 ri) + I c i (r r rp 3 ri) p I = 0. Now divide through by c i (r r rp 3 ri) set r = 1 to obtain ε i1 + ε i2 + ε i3 + η i = 0. 7