Unformatted text preview: 1 ECO 204 2008‐2009 Ajaz Hussain HW 4 Solutions Question 1 Ajax has utility function U = Q11/5Q24/5. a) What are α and β in U = Q1αQ2β? Answer: α = 1/5 and β = 4/5. b) Does Ajax’s utility function U = Q11/5Q24/5 represent “more is better” preferences over Q1 and Q2? Show Ajax has diminishing marginal utility. Answer: Yes. There are two ways of seeing this. Note that as either Q1 and/or Q2 increase, utility increases. More formally, the marginal utility of goods 1 and 2 ‐‐ for positive amounts of goods 1 and 2‐‐ are positive: MU1 = dU/dQ1 = (1/5)(Q2/Q1)4/5 MU2 = dU/dQ2 = (4/5)(Q1/Q2)1/5 Observe the utility function has diminishing marginal utility: as Q1 rises, MU1 increases, but at a diminishing rate. The same is true for MU2: as Q2 rises, MU2 increases, but at a diminishing rate. c) Why does the utility function U = (1/5) ln(Q1) + (4/5)ln(Q2) also represent Ajax’s preferences over Q1 and Q2? ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 2 Answer: Observe that: (1/5) ln(Q1) + (4/5)ln(Q2) = ln(Q11/5Q24/5 ) Now if A > B, then ln(A) > ln(B). Thus if bundle A‐‐ consisting of goods 1 and good 2‐‐ is preferred to bundle B‐‐ consisting of goods 1 and 2, then both U = (1/5) ln(Q1) + (4/5)ln(Q2) and U = Q11/5Q24/5 will assign a higher value to A than B. As an example, suppose bundle A has Q1 = 2 and Q2 = 2 while bundle B has Q1 = 1 and Q2 = 1. By more is better, A is preferred to B. In fact, both utility functions give this result: U = Q11/5Q24/5 → U(A) = (2)1/5(2)4/5 > (1)1/5(1)4/5 = U(B) and: U = (1/5) ln(Q1) + (4/5)ln(Q2) → U(A) = (1/5)ln(2) + (4/5)ln(2) > (1/5) ln(1) + (4/5)ln(1) = U(B) Note: As an exercise, you should reproduce the results below on percentage of income spent on goods 1 and 2 and price, income and cross‐price elasticities using the transformation of the utility function U = (1/5) ln(Q1) + (4/5)ln(Q2). d) Given an arbitrary level of utility, what is the equation of Ajax’s indifference curve over Q1 and Q2? Answer: If Ajax has utility function U = Q11/5Q24/5 then: U = Q11/5Q24/5 → Q24/5 = U/Q11/5 → Q2 = U5/4/Q11/4 In a Q1, Q2 plot this will have the form of an “imperfect substitutes” indifference curve. ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 3 e) Suppose all good are sold at uniform prices. Assume Ajax is a price taker. Show that Ajax will always spend 20% of his income Y on good 1 and the remainder on good 2. Answer: From lecture 5, recall expenditure on good 1 is {α/(α + β)} = 1/5 while that on good 2 is {β/(α + β)} = 4/5. There are several ways to show this formally. Here is one: since these are imperfect substitute indifference curves‐‐ given uniform prices‐‐ at the optimal choice, the budget line will be tangent to the indifference curve. Thus for Ajax’s UMP: MU1/MU2 = P1/P2 → {(1/5)(Q2/Q1)4/5}/{(4/5)(Q1/Q2)1/5} = P1/P2 → (1/4)(Q2/Q1) = P1/P2 → (1/4)(P2Q2) = P1Q1 This says, the expenditure on good 1 is 25% of the expenditure on good 2. Now, because these are more is better preferences, we know Ajax will spend his entire income. Thus: P1Q1 + P2Q2 = Y Substituting (1/4)(P2Q2) = P1Q1 above, we have: (1/4)(P2Q2) + P2Q2 = Y → (5/4)(P2Q2) = Y → P2Q2 = (4/5)Y Ajax spends 80% of his income on good 1. Therefore, he spends 20% on good 1 (review lecture 5): P1Q1 = (1/5)Y Two other ways to solve this question would have been: ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 4 By direct substitution: use the budget constraint to express Q2 in terms of Q1, substitute in U = Q11/5Q24/5 and maximize U with respect to Q2 which you can substitute back in the budget constraint to show that 80% of Y is spent on good 2. • By Lagrangean: set up the problem as: L = Q11/5Q24/5 ‐ λ(P1Q1 + P2Q2 ‐ Y) This becomes a 3 variable problem in Q1, Q2 and λ. Solve for these and substitute Q1 into the budget constraint to show that 80% of Y is spent on good 2. f) Derive Ajax’s demand equation for good 1. Answer: The demand equation is Q1 = f(P1). From above, we have: P1Q1 = (1/5)Y • → Q1 = (1/5)Y/P1 → Q1 = (1/5) Y P1‐1 With Y ceteris paribus this is the demand equation for good 1. From lecture 4, you should recognize that this is a constant elasticity demand equation. Note: As an exercise, you should derive good 2’s demand equation. g) What is Ajax’s price elasticity of good 1? Answer: From above, we had: Q1 = (1/5) Y P1‐1 Thus, the price elasticity is ‐1. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d ln Q)/(d ln P1) or (iii) (dQ/dP1)( P1/Q). Since Ajax always spends 20% of his income on good 1, it’s not surprising that the price elasticity is ‐1. ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 5 Note: As an exercise, you should derive good 2’s price elasticity. h) What is Ajax’s income elasticity of good 1? Answer: From above, we had: Q1 = (1/5) Y P1‐1 Thus, the income elasticity is 1. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d ln Q)/(d ln Y) or (iii) (dQ/dY)(Y/Q). Note: As an exercise, you should derive good 2’s income elasticity. i) What is Ajax’s cross‐price elasticity for good 1? Answer: From above, we had: Q1 = (1/5) Y P1‐1 Notice this can be written as: Q1 = (1/5) Y P1‐1 P20 Thus, the cross‐price elasticity of good 1 (with respect to good 2) is 0. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d ln Q)/(d ln P2) or (iii) (dQ/dP2)(P2/Q). Note: As an exercise, you should derive good 2’s cross‐price elasticity. Question 2 Jenn has utility function U = Q11/5. a) What are α and β in U = Q1αQ2β? Answer: α = 1/5 and β = 0. Right away, from the Cobb‐Douglas utility function above, you should recognize that Jenn will spend all her income on good 1 and none on good 2 (from lecture 5, recall expenditure on good 1 is {α/(α + β)} = 1 while that on good 2 is ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 6 {β/(α + β)} = 0). b) Does Jenn’s utility function U = Q11/5 represent “more is better” preferences over Q1 and Q2? Answer: Yes. There are two ways of seeing this. Note that as Q1 increases, utility increases. More formally, the marginal utility of good 1‐ ‐ for positive amounts of goods 1 ‐‐ is positive: MU1 = dU/dQ1 = (1/5)(1/Q1)4/5 Observe the utility function has diminishing marginal utility: as Q1 rises, MU1 increases, but at a diminishing rate. c) Why does the utility function U = (1/5) ln(Q1) also represent Jenn’s preferences over Q1 and Q2? Answer: Observe that: (1/5) ln(Q1) = ln Q11/5 Now if A > B, then ln(A) > ln(B). Thus if bundle A‐‐ consisting of goods 1 and good 2‐‐ is preferred to bundle B‐‐ consisting of goods 1 and 2, then both U = (1/5) ln(Q1) and U = Q11/5 will assign a higher value to A than B. As an example, suppose bundle A has Q1 = 2 while bundle B has Q1 = 1. By more is better, A is preferred to B. In fact, both utility functions give this result: U = Q11/5 → U(A) = (2)1/5 > (1)1/5 = U(B) and: U = (1/5) ln(Q1) → U(A) = (1/5)ln(2) > (1/5) ln(1) = U(B) ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 7 Note: As an exercise, you should reproduce the results below on percentage of income spent on goods 1 income and cross‐price elasticities using the transformation of the utility function U = (1/5) ln(Q1). d) Given an arbitrary level of utility, what is the equation of Jenn’s indifference curve over Q1 and Q2? Answer: If Jenn has utility function U = Q11/5 then: U = Q11/5 → Q11/5 = U → Q1 = U5 In an Q1, Q2 plot this will have the form of “vertical” indifference curves. e) Suppose all goods are sold at uniform prices. Assume Jenn is a price taker.. Show that Jenn will always spend 100% of his income Y on good 1 and the nothing on good 2. Answer: From lecture 5, recall expenditure on good 1 is {α/(α + β)} = 1 while that on good 2 is {β/(α + β)} = 0. Observe that you cannot show this formally by appealing to a tangency argument‐‐ since these are vertical indifference curves‐‐ given uniform prices‐‐ at the optimal choice, the budget line will be not be tangent to the indifference curve. Thus for Jenn’s UMP: MU1/MU2 ≠ P1/P2 Thus we have to appeal to results from graphical analysis‐‐ see Figure 1. There, observe that Jenn will spend her entire income on good 1. Since Jenn has more is better preferences, she will spend her entire income and thus: P1Q1 = Y → Q1 = Y/P1 ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 8 Figure 1 Another way to show this would’ve been by setting up the Lagrangean. Set up the problem as: L = Q11/5 ‐ λ(P1Q1 ‐ Y) This becomes a 2 variable problem in Q1 and λ. The first order condition with respect to λ yields Q1 = Y/P1. This approach has the added advantage of also yielding the Langrane multiplier λ which (recall from MATH 133) gives the value (in utility terms) of raising income by a dollar. f) Derive Jenn’s demand equation for good 1. Answer: We had: Q1 = Y/P1 This is the demand equation. Again, this is a constant elasticity demand function. g) What is Jenn’s price elasticity of good 1? Answer: From above, we had: ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 9 Q1 = Y P1‐1 Thus, the price elasticity is ‐1. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d ln Q)/(d ln P1) or (iii) (dQ/dP1)( P1/Q). Since Jenn always spends 100% of his income on good 1, it’s not surprising that the price elasticity is ‐1. h) What is Jenn’s income elasticity of good 1? Answer: From above, we had: Q1 = Y P1‐1 Thus, the income elasticity is 1. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d ln Q)/(d ln Y) or (iii) (dQ/dY)(Y/Q). g) What is Jenn’s cross‐price elasticity for good 1? Answer: From above, we had: Q1 = Y P1‐1 Notice this can be written as: Q1 = Y P1‐1 P20 Thus, the cross‐price elasticity of good 1 (with respect to good 2) is 0. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d ln Q)/(d ln P2) or (iii) (dQ/dP2)(P2/Q). Question 3 G‐Indart has utility function U = min(2Q1 , 4Q2) a) What are α and β in U = min(αQ1,βQ2)? Answer: α = 2, β = 4. ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 10 b) Does G‐Indart’s utility function U = min(2Q1 , 4Q2) represent “more is better” preferences over Q1 and Q2? Answer: No. There are many ways to see this. First, note that graphically, G‐Indart’s indifference curves look like: Figure 2 Any two bundles on the horizontal or vertical portion has the same utility, despite the fact that one bundle may have more the other bundle. For example, suppose bundle A has: (Q1, Q2) = (1, 2) and bundle B has (Q1, Q2) = (1, 1). Note how bundle A has “more” than bundle B. Observe: U(A) = min(2(1) , 4(2) ) = min(2,8) = 2 and: U(B) = min(2(1) , 4(1) ) = min(2,4) = 2 Thus, both bundles have the same utility despite the fact that bundle A is “more” than bundle B. ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 11 c) Suppose G‐Indart purchases goods 1 and 2 for (uniform) prices P1 and P2. Assume G‐ Indart is a price taker. Derive an expression for G‐Indart’s expenditure on goods 1 and 2. Answer: From lecture 5, recall that we cannot use the tangency argument: MU1/MU2 = P1/P2 to solve G‐Indart’s UMP. This is because for complements, the ratio MU1/MU2 is not defined at the optimal choice owing to the “kink”. Observe in Figure 2 that G‐Indart spends his entire income and that the optimal choice must be on the line Q2 = (1/2)Q1. Substitute this in the budget constraint: P1Q1 + P2Q2 = Y → P1Q1 + P2 (1/2)Q1 = Y → Q1(P1 + P2/2) = Y → Q1(2P1 + P2)/2 = Y → Q1 = 2Y/(2P1 + P2) This is the demand equation for good 1. To obtain an expression for the expenditure on good 1, simply do: P1Q1 = 2YP1/(2P1 + P2) d) Derive G‐Indart’s demand equation for good 1. Answer: From above, this is: Q1 = 2Y/(2P1 + P2) Note that unlike the Cobb‐Douglas UMP in questions 1 and 2, demand for good 1 here depends on the price of goods 1 and 2. This is obvious since G‐Indart prefers the two ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 12 goods as complements and so the price of both goods will determine his demand for good 1 (and good 2). Even though the question does not ask for it, you can derive good 2’s demand equation by noting that since: Q2 = (1/2)Q1 → Q2 = (1/2) {2Y/(2P1 + P2)} = Y/(2P1 + P2) e) What is G‐Indart’s price elasticity of good 1? Answer: It’s best to use the formula E = (dQ/dP)(P/Q) for this question. First re‐write the demand function as: Q1 = 2Y(2P1 + P2)‐1 Now: dQ1/dP1 = ‐4Y/(2P1 + P2)2 Thus: E = (dQ1/dP1)(P1/Q1) = ‐4(Y/(2P1 + P2)2 )(P1/Q1) → E = ‐4(Y/(2P1 + P2)2 )(P1/{2Y/(2P1 + P2)}) → E = ‐ 2P1/(2P1 + P2) Which is not a constant elasticity. As an exercise, compute the cross‐price elasticity: (dQ1/dP2)(P2/Q1). f) What is G‐Indart’s income elasticity of good 1? Answer: Again, start with: Q1 = 2Y/(2P1 + P2) Use the formula E = (dQ/dY)(Y/Q): ECO 204, 2008‐2009. Ajaz Hussain. Department of Economics, University of Toronto 13 Now: dQ1/dY = 2/(2P1 + P2) Thus: E = (dQ1/dY)(Y/Q1) = (2/(2P1 + P2)) (Y/Q1) → E = (2/(2P1 + P2)) (Y/{2Y/(2P1 + P2)}) → E = 1 Which is a constant elasticity. ECO 204, 2008‐2009. 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