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ECA5101 Microeconomic Analysis I Semester 1 AY Homework 2 — Solution Consumer Theory, Exchange Economy, and Externality Due at the beginning of lecture 6. Question 1 (Quasi-Linear Utility Function) Consider the utility function ! . For simplicity, assume we always have interior solutions. a) Derive the demand functions for x and y respectively. The marginal utilities are ! and ! . The tangency condition is thus ! . Thus the demand for x is ! . The budget line is ! , substituting the demand for x into the budget line, we have the demand for y as ! . b) Derive the compensated demand functions for x and y respectively. Compare the compensated demand function of x to the demand function of x. What can you conclude? Solving the expenditure minimization problem, the compensated demand function for x is ! and the compensated demand function for y is ! . The compensated demand function for x is the same as the demand function for x. This is because the demand for x is independent of income. Thus there is no income effect with respect to x when the price of x changes. If there is no income effect, the demand and the compensated demand are the same. c) Suppose the price of x increases from a to b. Show that the compensating variation and the change in consumer surplus are the same. As the compensated demand curve and the demand curve are the same, we have ! . d) Derive the income and substitution effects for x and y respectively using the Slutsky equation. For x, the substitution effect is ! . The income effect is ! . For y, the substitution effect is ! . The income effect is ! . U(x, y) = x + y MUx = 1 2 x MUy = 1 1 2 x = Px Py x = P2y 4P2x Pxx + Pyy = I y = I Py − Py 4Px xc = P2y 4P2x yc = U − Py 2Px CV = ∫ b a P2y 4P2x dPx = ∫ b a P2y 4P2x dPx = ΔCS ∂xc ∂Px = − P2y 2P3x −x ∂x ∂I = 0 ∂yc ∂Py = − 1 2Px −y ∂y ∂I = 1 4Px − I P2y !1 National University of Singapore Department of Economics ECA5101 Microeconomic Analysis I Semester 1 AY 2021/2022 Question 2 (Pareto Efficiency vs. Maximizing Total Utility) Consider an exchange economy with two consumers, A and B, and two goods, x and y. Suppose the indifference curves of both consumers are well behaved (downward sloping, smooth and convex). Let ! be the utility function of A and ! be the utility function of B. Suppose the total endowment of x is ! and the total endowment of y is ! in this economy. An alternative way of understanding Pareto efficiency is as follows. For an allocation to be Pareto efficient, consumer A must be maximizing his/her utility given any utility level ! of consumer B. Otherwise it is possible to have a Pareto improvement. a) Let’s start from writing down consumer A’s utility maximization problem — A maximizes utility subject to the constraint that B receives a utility of ! . For simplicity, use ! to represent the utility function of consumer B. The utility maximization problem is ! s.t. ! b) Solve the problem in part a) using the Lagrange multiplier method. Show that the first- order conditions give us ! . The Lagrangian function is ! . The first two first-order conditions are ! ! Combining the two equations, we get ! . c) Suppose we maximize the sum of the utilities of the two consumers instead. Write down the maximization problem and derive the first-order conditions. Is the solution to this problem also a solution to the problem in part a)? How about the reverse? The maximization problem is ! The first-order conditions are ! ! Anything that satisfies the above two conditions will also satisfy ! . Thus, the solution to this problem is also a solution to the problem in part a). In other words, the allocations where the total utility is maximized are Pareto efficient. However, the reverse is not true. The allocations satisfying ! do not necessarily satisfy UA(xA, yA) UB(xB, yB) x¯ y¯ U¯ U¯ UB(x¯ − xA, y¯ − yA) max x A,yA UA(xA, yA) UB(x¯ − xA, y¯ − yA) = U¯ MRSAx,y = MRSBx,y Λ(xA, yA, λ) = UA(xA, yA) + λ(UB(x¯ − xA, y¯ − yA) − U¯ ) ∂Λ ∂xA = MUAx − λMUBx = 0 ∂Λ ∂yA = MUAy − λMUBy = 0 MRSAx,y = MRSBx,y max x A,yA UA(xA, yA) + UB(x¯ − xA, y¯ − yA) MUAx −MUBx = 0 MUAy −MUBy = 0 MRSAx,y = MRSBx,y MRSAx,y = MRSBx,y !2 National University of Singapore Department of Economics ECA5101 Microeconomic Analysis I Semester 1 AY 2021/2022 ! and ! . Thus, a Pareto efficient allocation does not necessarily maximize the total utility of the two consumers. Question 3 (Exchange Model for Perfect Complements and Perfect Substitutes) Consumer A has utility function ! and consumer B has utility function ! . Each consumer is endowed with is 1 unit of x and 1 unit of y. a) Draw an Edgeworth box with x on the horizontal axis and y on the vertical axis. Measure goods for consumer A by the distance from the lower left corner of the box. Label the endowment allocation. Draw the indifference curve that passes through the endowment allocation for each consumer. See graph below. The Edgeworth box is a square and the endowment allocation is in the center of the box. The green line is the indifference curve for consumer A that passes through he endowment allocation. The red line is the indifference curve for consumer B that passed through the endowment allocation. b) Is the endowment allocation Pareto efficient? The endowment allocation is not Pareto efficient. For example, if we move to point A in the graph below, consumer A’s utility will not change but consumer B’s utility will be higher. c) Derive the equation of the contract curve in terms of xB and yB. Draw the contract curve in your graph. As we can see from the graph above, the Pareto efficient allocation must be at the kinks of consumer B’s indifference curves. As long as we are not at the kink point, we can always MUAx = MUBx MUAy = MUBy U(xA, yA) = xA + yA U(xB, yB) = min(2xB, yB) !3 National University of Singapore Department of Economics ECA5101 Microeconomic Analysis I Semester 1 AY 2021/2022 move along the indifference curve for consumer A (so that consumer A’s utility does not change) to a kink point where consumer B gets higher utility. Thus the contract curve is ! . Graphically, it is the purple line shown below. d) Set x as a numeraire. Find the equilibrium price of y and the equilibrium allocation. (Hint: for consumer A to consume at a Pareto efficient allocation, what should the prices be?) By the First Welfare Theorem, the equilibrium allocation will be a point on the contract curve. First, given the endowment, the point xA=1, yA=0 will never be the optimal basket for consumer A because consumer A gets a higher utility at the endowment allocation. Thus the point xA=1, yA=0 will not be the equilibrium allocation. By the same logic, the origin of consumer B will not be the equilibrium allocation. Note that consumer A views the two goods as perfect substitutes. Except for the point xA=1, yA=0 and the origin of consumer B, all the other points on the contract curve are in the interior of the Edgeworth box. For consumer A to maximize utility at an interior point, the prices should be such that the per dollar marginal utilities of two goods are the same. Since the marginal utility for each good is 1, and since the price of x is set to be $1, this would imply that the price of y is $1. Thus the equilibrium price of y is $1. Consumer B’s budget line given these prices is ! . To maximize utility, consumer B will consume at the kink point ! . Thus ! and ! . The market clearing conditions are ! and ! . Thus the equilibrium allocation is ! . This is point A in the graph of part b). Question 4 (Externality and Transaction Cost) Tintin and Lily are neighbors. Lily hates dogs, but Tintin loves them. Tintin currently keeps a dog, Snowy. Snowy loves to bark at Lily, and this scares Lily. Suppose that Tintin receives a utility gain equivalent to $500 from owning Snowy, and Lily suffers a utility loss equivalent to $800 from having Snowy living next door. a) What is the efficient outcome? The efficient outcome is not to keep Snowy because the total cost of having Snowy is $800, which is higher than the total benefit, $500. yB = 2xB xB + yB = 2 yB = 2xB xB = 2 3 yB = 4 3 xA + xB = 2 yA + yB = 2 (xA = 4 3 , yA = 2 3 , xB = 2 3 , yB = 4 3 ) !4 National University of Singapore Department of Economics ECA5101 Microeconomic Analysis I Semester 1 AY 2021/2022 b) Suppose the government passes a new law that stipulates that one is allowed to keep pets at home only if there are no complaints from one’s neighbors. Suppose that transaction cost is 0, what will the free market outcome be? To keep Snowy, Tintin has to pay Lily at least $800, which he will not do since the utility gain for him is only $500. Thus Tintin will not keep Snowy. c) Suppose instead of requiring the neighbor’s consent, the new law stipulates that one is allowed to keep pets at home without the consent of one’s neighbors. Suppose that transaction cost is 0, what will the free market outcome be? Now Lily is willing to pay Tintin anything between $500 and $800 for Tintin not to keep Snowy, and Tintin will agree. The market outcome is not to keep Snowy. d) Suppose Tintin and Lily speaks different languages and do not understand each other. To negotiate, they need a translation software. Suppose the cost of the software is $100. What is the efficient outcome? Reconsider part c), would the free market outcome be efficient? The $100 is a transaction cost. The efficient outcome is unaffected by the transaction cost, so it is still not to keep Snowy. For part c), the free market outcome is still efficient. Suppose Tintin pays for the software, then Lily is willing to pay anything between $600 to $800 for Tintin not to keep Snowy and Tintin will agree. If Lily pays for the software, then she is willing to pay Tintin anything between $500 to $700 to not to keep Snowy and Tintin will agree. So the outcome is still efficient. The key is that the transaction cost ($100) is lower than the gain from trade ($300). Hence the efficient outcome is still achieved in the market. Note that the efficient outcome does not depend on whether there is property right, who owns the property right, or whether there is transaction cost. As long as the cost of having Snowy is higher than the gain from keeping Snowy, the efficient outcome is not to keep Snowy. What the efficient outcome is and whether the efficient outcome can be achieved in the market are two separate questions. Even if the cost of translator is $1000, the efficient outcome is still not to keep Snowy. However, in this case because the transaction cost is higher than the gain from trade, the efficient outcome will not be achieved in the market. For example, suppose Tintin has the property right and Tintin pays for the software. For Tintin not to keep Snowy, Lily has to pay Tintin as least $500+$1000=$1500, which she will not pay since it is higher than the loss of having Snowy ($800). Thus the market outcome will be to keep Snowy and this is inefficient. In other words, when the transaction cost is $1000, market does not allocate resources efficiently.
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