ECON3010 Problem Set 3 and Solutions (2014 ) PDF

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Problem Set 3 on Edgeworth box, inter-temporal choices and Cobb-douglas utility equilibrium with detailed Solutions. ...

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Econ 231 Intermediate Microeconomic Theory

Alejandro Molnar Fall 2014

Problem Set 3 Solutions Due in class on: Friday Oct 3 Instructions: Please turn in your problem set using a cover sheet with your name on it. The cover sheet could be the printout of the problem set or a blank page. To provide some incentives not to forget this, including the cover sheet scores 10 points. Points on this problem set: 160

Cover Sheet. 10 points Q1. Edgeworth box with perfect complements. 20 points Aaron likes to consume exactly 2 cokes for every 1 burger, and Yang likes exactly 1 coke for every 1 burger. They only consume these combinations and do not want any extra coke or burger without the other. Currently Aaron has 5 cokes and 5 burgers, and Yang has 7 cokes and 3 burgers. They have no more money to buy cokes or burgers but can trade with each other. With Aaron at the lower left corner and Yang at the upper right corner, cokes on the horizontal axis and burgers on the vertical axis: A) Draw the Edgeworth box. Show the endowment point and draw indifference curves for both that go though the endowment point. Aaron’s endowment point is (5, 5) and Yang’s is (7, 3). There are 7 + 5 = 12 cokes and 5 + 3 = 8 burgers in total. Aaron consumes exactly 2 cokes (c) for every 1 burger (b). His preference could be represented by U A (c, b) = min[c, 2b], and his indifference curves are shown in green lines on graph Q1(A). Yang consumes exactly 1 coke (c) for every 1 burger (b). His preference could be represented by U Y (c, b) = min[c, b] and his indifference curves are shown in blue lines on graph Q1(A). The endowment point is the red dot on the graph.

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Graph Q1 (A)  Y coke  7

Yang

 A burger  5

 Y burger  3 Endowment

Aaron

 A coke  5

B) Can you suggest an allocation that would make one of the consumers better of f without making the other worse off ? If so, provide an example. There is more than one answer to this question. For example, Aaron gives Yang 1 burger and Yang gives Aaron 3 cokes. At the endowment, Aaron can consume 2.5 of his preferred combinations at (5, 5) for a utility of 5, and Yang can consume 3 of his combinations at (7, 3) for a utility of 3. After the trade, Aaron consumes 4 combinations at (8, 4) for a utility of 8, and Yang consumes 4 combinations at (4, 4) for a utility of 4. They are both better of f with this trade. In fact, any point inside the box that is bounded by both of their indifference curves going through the endowment point would be a Pareto improvement (with the exception of the opposite corner of the box to the endowment, where both would have the same utility as at the endowment). C) Is the allocation with (4, 1) to Aaron, and the rest to Yang Pareto efficient? Is the allocation with (4, 3) to Aaron, and the rest to Yang Pareto efficient? Why or why not? Yes, (4, 1) to Aaron, and the rest to Yang is Pareto efficient. When Aaron is at (4, 1), Yang is at (8, 7) since there are 12 cokes and 8 burgers in total. Aaron consumes 1 combination and Yang consumes 7 combinations. It is Pareto efficient since taking a little bit of burger from Yang to give to Aaron (so that Aaron would increase is number of combinations beyond 1) would come at the expense of Yang’s utility. We cannot take burgers from Aaron because this would harm Aaron. We could remove some coke from Aaron without harming 2

Aaron’s utility, but this would not provide additional utility to Yang. We could remove up to one coke from Yang, but this would not increase Aaron’s utility. Since there are no Pareto improvements to be made, (4, 1) to Aaron and the rest to Yang is Pareto efficient. No, (4, 3) to Aaron, and the rest to Yang is NOT Pareto efficient. When Aaron is at (4, 3), Yang is at (8, 5) since there are 12 cokes and 8 burgers in total. Aaron consumes 4/2 = 2 combinations and Yang consumes 5 combinations. To show this is not Pareto efficient, all we have to do is provide an example of a Pareto improvement. Two possible examples are: (1) Yang can give Aaron 3 cokes for free, making Aaron better off; or (2) Aaron can give Yang 1 burger for free, making Yang better off. Neither agent is worse of f after the trades in these two examples.

Q2. General equilibrium with Cobb-Douglas utility. 50 points Consider an economy with 2 individuals: Aaron and Yang, and 2 goods: x and y. There is no production in this economy. Aaron has 18 units of x and 4 units of y; Yang has 3 units of x and 6 units of y. Aaron’s utility function over x and y is U A (x, y) = xy, Yang’s preferences are represented by U Y (x, y) = xy 2 . With Aaron at the lower axis and Yang at the upper axis, cokes on the horizontal axis and burgers on the vertical axis: A) Draw an Edgeworth box for this economy and show the endowment point on the graph. What are the utilities for both agents in autarky? Aaron’s endowment point is (18, 4) and Yang’s is (3, 6). There are 18 + 3 = 21 units of good x and 4 + 6 = 10 units of good y in total. The Edgeworth box is drawn below, and the endowment point is the red dot. The utility of Aaron in autarky (no trade) is U A (18, 4) = 18 · 4 = 72, and the utility of Yang in autarky is U Y (3, 6) = 3 · 62 = 108.

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Graph Q2 (A)  Yx  3

Yang

Endowment

Ay4

 Yy  6

Aaron

 A x  18

B) Provide an example of a mutually beneficial trade, and find the utilities for both agents with this trade. There is more than one answer to this question. For example, Aaron can trade with Yang to get 1 unit of y for 2 units of x. Aaron’s new bundle is (18 − 2, 4 + 1) = (16, 5) and his new utility is U A (x, y) = 16·5 = 80 > 72. Yang’s new bundle is (3+2, 6−1) = (5, 5) and his new utility is U Y (x, y) = xy 2 = 5·52 = 125 > 108. This trade makes both Aaron and Yang better off. Note: Any allocation choice in the interior of the Pareto-improving set you find in part (C) will be a mutually beneficial trade. Also, if you solved for the general equilibrium allocations in question (E), this will be a Pareto-efficient allocation (as you might guess from the First Welfare Theorem), and therefore a Pareto-improvement over the initial endowment. C) Draw the set of allocations that are Pareto-improving over the endowment. On graph Q2(C), the endowment point is in red. Green curves represent Aaron’s indifference curves, and blue curves represent Yang’s indifference curves. The lens - shaped shaded area A represents the set of Pareto-improving allocations (bounded by indifference curves through the endowment). A movement from the endowment point to any allocation in A would make at least one individual better off, and no individuals worse off.

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Graph Q2 (C)  Yx  3 U Y ( x, y )  x. y

Yang 2

A

 Ay  4

 Yy  6 U A ( x, y )  x. y

Aaron

 A x  18

D) Draw the contract curve (all Pareto optimal points). Indicate which Pareto optimal trades would be blocked by A or Y, as well as the core of this economy. On graph Q2(D), the endowment point is in red. Green curves represent Aaron’s indifference curves, and blue curves represent Yang’s indifference curves. The black lines represent the contract curve, with the solid part of the line as core and the dashed lines as trades blocked by Aaron and Yang. Remember the core is the part of the contract curve that goes through the set of Pareto-improving allocations A. Note: For the definition of contract curve, please take a look at the slides. You can also take a look at Professor Molnar’s handwritten note (from the exercise he went over in class) on how to find the mathematical expression for the contract curve. The approach makes use of the property that for convex indifference curves the contract curve is defined by the points on the Edgeworth box where the slopes of both agent’s indifference curves are equal.

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Graph Q2 (D) Yang

 Yx  3

Core

U Y ( x, y )  x. y A

 Ay  4 U A ( x, y )  x. y

2

 Yy  6 Trades blocked

Aaron

 A x  18

E) Find the general equilibrium prices and quantities consumed by each agent in this economy. Let the prices for x and y be px and py . Aaron’s endowment of x and y are wx A = 18 and wy A = 4. Yang’s endowment of x and y are wx Y = 3 and wy Y = 6. Solve for the equilibrium levels of consumption for both Aaron and Yang. Aaron’s total wealth is mA = 18px +4py . His budget set is px xA + py yA = mA = 18px + 4py . Aaron maximizes his utility U A (xA , yA ) = xA .yA subject to the budget constraint. That is, max U A (x, y) = xA .yA such that px .xA + py .yA = mA = 18px + 4py . p M RS By the method of marginal rate of substitution, M RSxy = xyAA = pxy . Therefore, mA px .xA = py .yA = 2 = 9px + 2py at the solution. This implies Aaron’s optimal demands are: py ∗ = 9+2 xA px px ∗ yA = 9 + 2 py

Note: you could also have used other methods to find the demand. One method is substitution by re-writing xA in terms of yA from the budget constraint, or vice versa, and then maximizing the utility function in that variable by taking the first order condition. Another method you could use is to recall that for a Cobb-Douglas preference like this one, we’ve already found that the general

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formula for demand of good 1 is given by: x1 (p1 , p2 , m) =

α m α + β p1

and then you can adapt this to the problem at hand, being careful to identify correctly what the income and prices are. Now consider Yang’s problem. His total wealth is mY = 3px + 6py . His budget set is px xY +py yY = mY = 3px +6py . Yang maximizes his utility U Y (xY , yY ) = xY .yY 2 subject to the budget constraint. That is, max U Y (x, y) = xY yY2 such that px xY + py yY = mY = 3px + 6py . M RS

By the method of marginal rate of substitution, M RSxy = 2 xy Y = ppyx . (Or you Y can also use first order conditions with method of substitution by re-writing xA in yA from the budget constraint, or vice versa). Therefore, 2px xA = py yA and px xY + py yY = mY . 2px xY = py yY = 2/3mY at the solution. This implies Yang’s optimal demands are: px xY = 1/3mY = px + 2py ⇒ xY∗ = 1 + 2py /px py yY = 2/3mY = 2px + 4py ⇒ y∗Y = 2px /py + 4 General equilibrium is defined as the set of prices and quantities where the market clears. That is, the total amount of consumption on a good x or y equals the sum of endowment. x∗A + x∗Y = wA∗ + w ∗Y = 18 + 3 = 21 and ∗ y∗A + yY∗ = wA + wY∗ = 4 + 6 = 10. Plugging in the optimal demand functions of both agents we found above, we get p p p p the following two equations: 9 + 2 pyx + 1 + 2 pyx = 21 and 9pyx + 2 + 2 pyx + 4 = 10. Solving either market clearing condition, we find that the equilibrium price ratio = 2.75. is pp21 = 11 4 Now plug the price ratio back into the demand functions for both agents to find the equilibrium quantities demand. 9 ∗ +2 = 58 = 2.75 For Aaron, x∗A = 9+2.(2.75) = 14.5, and yA . So Aaron consumes 11 58 14.5 units of x and 11 units of y. Note: Aaron’s utility in this equilibrium is ∗ y ∗ = (29/2)(58/11) = 76.45 > 72, so Aaron gains from trade. U A (x, y) = xA A 52 2 + 4 = 11 For Yang, xY∗ = 1 + 2.(2.75) = 6.5, and yY∗ = 2.75 . So Yang consumes 52 6.5 units of x and 11 units of y. Note: Yang’s utility in this equilibrium is 2 U Y (x, y) = x∗Y yY∗ = (13/2)(52/11)2 = 145.26 > 108, so Yang gains from trade.

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Q3. Intertemporal choice determines the interest rate. 30 points Consider an economy with 2 agents: Old and Young, and 2 periods: today (period 1) and future (period 2). The old agent (O) earns an income of 100k today and no income in the future after retiring. His utility function is U O (c1 ; c2 ) = c1 c0.9 2 , where c 1 is consumption today and c2 is consumption in the future. The young agent (Y ) has no income today and earns an income of 100k in future after graduating. His utility function is U Y (c1 ; c2 ) = ln(c1 ) + 0.9ln(c2 ), where c1 and c2 are as above. The agents can borrow and lend to each other at an interest rate r. For simplicity, we assume the prices of consumption are normalized, i.e. p1 = p2 = 1. A) Write the budget sets for the two agents. The old agent O has an endowment of (100, 0) and the young agent Y has an endowment of (0, 100). If O wants to consume in period 2, he must lend some money to Y in the period 1 to get a payment in period 2. If Y wants to consume in the period 1, he has to borrow some from O . For the old agent O, his consumption in period 1 is cO 1 and his saving (or lending) . to Y in period 1 is 100 − cO 1 In period 2, O gets (100 − cO 1 ) · (1 + r) as repayment from Y . O’s consumption in period 2 is cO2 = (100 − c1O )(1 + r), determined by his lending in period 1 and the interest rate r. Therefore, the old agent’s budget set is: c2O = (100 − c1O)(1 + r), or (1 + r)c1O + c2O = 100(1 + r) + 0 For the young agent Y , his consumption in period 1 is cY1 that he borrows from O. In period 2, he earns an income of 100k but has to repay the amount he previously borrowed, cY1 (1 + r), to O. Thus his consumption in period 2 is cY2 = 100−(c1Y )(1+ r), determined by his borrowing in period 1 and the interest rate r. Therefore, the young agent’s budget set is: c2Y = 100 − (cY1 )(1 + r), or (1 + r)cY1 + c2Y = 0 + 100 In both cases, the budget sets for the agents described above are just particular cases of the general budget set for intertemporal choice seen in the lectures, which just says that the present value of consumption must equal the present

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value of income: c1 +

c2 m2 = m1 + 1+r 1+r

You could also express this in terms of future consumption, which is just multiplying both sides of the budget set by (1 + r): (1 + r)c1 + c2 = (1 + r)m1 + m2 In this problem, just substitute in the incomes m1 and m2 into this equation for each agent to find their respective budget constraints. B) Determine both consumers’ demand functions (for saving, or lending). The old agent maximizes his utility U O (c1 ; c2 ) = c1 c20.9 subject to his budget constraint (1 + r)c1O + c2O = 100(1 + r). Solving the budget constraint in terms of C2O and substituting in to the utility function, we can rewrite his optimization problem as: max c1O · [(100 − c1O )(1 + r)]0.9 By taking the derivative with respect to cO 1 and setting it equal to zero (the first order condition for maximization), we get: −0.1 + [(100 − c1O)(1 + r)]0.9 = 0 −0.9c1O(1 + r)[(100 − cO 1 )(1 + r)]

⇒0.9c1O(1 + r)[(100 − cO1 )(1 + r)]−0.1 = [(100 − c1O)(1 + r)]0.9 ⇒0.9c1O(1 + r) = (100 − cO 1 )(1 + r) ⇒0.9c1O = 100 − cO 1 ⇒1.9c1O = 100 Thus, the old agent’s optimal demands for consumption in both periods are given by: c1O =

100 1.9

c2O = 0.9(1 + r)c1O =

90(1 + r ) 1.9

90 O So his saving/lending amount is sO 1 = 100 − c1 = 1.9 ≈ 47.368

Now the young agent maximizes his utility U Y (c1 ; c2 ) = ln(c1 ) + 0.9ln(c2 ) subject to his budget constraint (1 + r)c1Y + c2Y = 100.

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Solving the budget constraint in terms of cY2 and substituting in to the utility function, we can rewrite his optimization problem as: max ln(c1Y ) + 0.9ln[100 − c1Y (1 + r )] By taking the derivative with respect to cY1 and setting it equal to zero (the first order condition for maximization), we get: 1 −(1 + r) +0.9 =0 c1Y 100 − cY1 (1 + r) 0.9(1 + r ) 1 ⇒ Y = 100 − cY1 (1 + r) c1 ⇒0.9(1 + r)cY1 = 100 − cY1 (1 + r) ⇒1.9cY1 (1 + r) = 100 100 ⇒cY1 = 1.9(1 + r )

Thus, the young agent’s optimal demands for consumption in both periods are given by: c1Y =

100 (1.9)(1 + r )

c2Y = 0.9(1 + r)c1Y =

90 ≈ 47.368 1.9

Note: Since a utility function is unique up to a positive, monotonic transformation, the utility function of the young agent can alternatively be written as U Y (c1 , c2 ) = exp[ln(c1 ) + 0.9ln(c2 )] = c1 · c20.9, which is a Cobb-Douglas function equivalent to the old agent’s utility function. Or, similarly, we could log-transform the old agent’s utility function to be the same as the young agent’s utility function as follows: U O (c1 , c2 ) = ln(c1 c20.9) = ln(c1 ) + 0.9ln(c2 ). With either transformation the agents have equivalent utility functions, but since they have different endowments their optimal demand functions are still different. Note: alternatively, you could have solved this problem by recalling the usual demand function for Cobb-Douglas preferences (both Young and Old have CobbDouglas preference after the appropriate transformations). What you do then is write down the demand function using the appropriate “prices” and income. The income is straightforward, for example in the present value formulation of the budget set it’s just the present value of the agent’s income. For the prices, you could recognize from looking at the budget line that the “price” for consumption in period 1 is just 1 (this is what it means to write the present value formulation of the budget line - that we’ve expressed everything in terms 1 . of period 1 consumption), and the “price” for period 2 consumption is 1+r Note: Both agents O and Y make their optimal decisions to maximize utilities given budget constraints. Their optimal levels of consumption could potentially 10

depend on the interest rate r, or be constant. In this question, 0.9 in the utility functions stands for the time discount factor. The problem states that p1 = p2 , meaning that nominal prices for consumption are normalized and we’re not concerning us with the issue of inflation. The “price” ratio is determined by r, as r is the opportunity cost of present consumption in terms of future consumption. If the interest rate turns out to be positive (as it is in the real world), this means that given the same number for income in the current or future period, “current wealth” is more valuable than “future wealth”. C) Determine the competitive equilibrium interest rate r and allocations for each agent’s consumption in each period. What is the amount saved or lent by each agent from period 1 to period 2? We already found the optimal demand functions for both agents. In a general equilibrium, we must also satisfy the market clearing condition. That is, in either period, the sum of the two agents’ consumption equals the sum of their endowment wealth. So we must have c1O + cY1 = w1O + w1Y = 100 + 0 = 100, and Y O Y cO 2 + c2 = w2 + w2 = 0 + 100 = 100. (Hint: Think of it as an Edgeworth box example) Y By the results from part B), we can plug in the optimal demands cO 1 , c 1 to solve 100 ∗ = 100 ∗ = 0.11. for r as follows: 100 = 100 ⇒ 1 + r + ≈ 1.11 ⇒ r (1.9)(1+r) 1.9 90 Thus, the equilibrium interest rate is 11.11%, or 0.11. O Plugging in r ∗ , the amount saved or lent is sO 1 = 100 − c1 = 90/(1.9) = 47.37, and the optimal demands of both agents in both periods is: ∗

cO 1 = 100/1.9 = 52.63,

∗

cO 2 = 90.(1 + r)/1.9 = 52.63

cY1 = 100/1.9(1 + r) = 47.37, ∗

cY2

∗

= 90/1.9 = 47.37

So the equilibrium condition can be summarized as follows: The old agent O consumes 52.63k in both periods. The young agent Y consumes 47.37k in both periods. In period 1, O lends 47.37k to Y . In period 2, Y repays 52.63k to O . The equilibrium interest rate is r ∗ = 0.11. Note that the equilibrium interest rate also satisfies this condition: 1 1 = = 0.9 1 + r∗ 1 + 0.11 And 0.9 was the discount factor that both agent’s preferences have on future consumption. The reason that we found a positive interest rate, with this particular value, is that agents have a preference for present consumption over future consumption. It is this preference that determines the interest rate. D) (Not really harder, but optional and will not be graded) Compare M RS O and M RS Y at the equilibrium allocation. Is the equilibrium allocation Pareto efficient and why? (Think of the M RS condition that must hold) Check the M RS condition. 11

M RS O = M RS Y =