The Ohio State University Department of Economics Econ. 805 Winter 00 Prof. James Peck Homework Set Questions and Answers. Consider the following pure exchange economy with two consumers and two goods. Consumer 's utility function and endowments are given by: u (x, x ) = a log(x ) + log(x ) ω = (,). Consumer 's utility function and endowments are given by: u (x, x ) = log(x ) + a log(x ) ω = (,). The parameters, a and a, are both positive. (a) Compute the aggregate excess demand function for this economy, and verify by direct computation that Walras Law holds. (b) Define a competitive equilibrium for this economy. (c) Calculate the competitive equilibrium price and allocation. (d) For what values of a and a will consumer be a net purchaser of good at the competitive equilibrium? ANSWER: (a) Solving consumer 's utility maximization problem, we simultaneously solve the marginal rate of substitution condition, a x / x = p, and the budget equation holding with equality, p x +x = p +, where prices are normalized so that the price of good is p and the price of good is. We get x = a (p+)/[p(a +)] and x = (p+)/(a +). Solving consumer 's utility maximization problem, we get the first order conditions: x / (a x ) = p and p x + x = p +. Solving for consumer 's demand functions, we get: x = (p+)/[p(+a )] and x = a (p+)/(+a ). Therefore, the excess demand function is: z(p) = ( a (p+)/[p(a +)] + (p+)/[p(+a )] -, (p+)/(a +) + a (p+)/(+a ) - ). To verify that Walras Law holds, we must calculate pz (p) + z (p), which equals a (p+)/(a +) + (p+)/(+a ) - p+ (p+)/(a +) + a (p+)/(+a ) - = (p+) + (p+) - p - = 0. It checks. (b) A competitive equilibrium is a (normalized) price vector (p,) and an allocation (x, x, x, x ) such that:
(i) (x, x ) solves max a log(x ) + log(x ) s.t. p x +x! p + x " 0 and (x, x ) solves max log(x ) + a log(x ) s.t. p x + x! p + x " 0. (ii) x + x! x + x!. Note: because both utility functions are strictly monotonic, budget and market clearing inequalities must hold with equality. (c) To calculate the competitive equilibrium price and allocation, we look for a price where z(p) = 0. Looking at market, we must solve: (p+)/(a +) + a (p+)/(+a ) - = 0. This can be rewritten as: (+a )(p+) + (a +)a (p+) = (a +)(+a ). Therefore, p( + a + a + a a ) + (+a + a + a a ) = (a +)(+a ). p( + a + a a ) = + a + a a, which implies the equilibrium price, p* is p* = ( + a + a a ) / ( + a + a a ). Simplifying the equilibrium allocation is tedious, but you can simply express the answer as: x = a (p*+)/[p*(a +)], x = (p*+)/(a +), x = (p*+)/[p*(+a )], and x = a (p*+)/(+a ), where p* = ( + a + a a ) / ( + a + a a ). (d) Consumer will be a net purchaser of good when x >. a (p*+)/[p*(a +)] > can be rewritten as a (p*+) > p*(a +), or a p* + a > a p* + p*. Thus, the condition is: p* < a /. Plugging in the value of p*, we have ( + a + a a ) / ( + a + a a ) < a /.
Cross-multiplying: 4 + 6 a + a a < a + a a + (a ) a, or (a ) a + a a - 4 a - 4 > 0. Factoring the above expression, we have: (a +)[a a - 4] > 0. Since a is positive, the condition for consumer to be a net purchaser of good is: a a > 4.. Construct an example of an economy (a specification of the consumers, the utility functions, and the endowments) that does not have a competitive equilibrium. Carefully and clearly explain your answer. Which assumptions from our existence theorem are violated in your example? ANSWER: The easiest answers involve dropping the assumption of strict quasiconcavity, and drawing a carefully labeled Edgeworth box. Alternatively, you could specify functional forms. For example, suppose there is just one consumer and goods. u(x,x ) = (x ) + (x ) and ω = (,). At any price vector, this consumer will spend all of his/her income on one commodity or the other, whichever is relatively cheaper. Normalizing p = p / p, we have excess demand for good when p <, excess demand for good when p >, and excess demand for one of the goods (whichever the consumer chooses) when p =.. Assume that we have a pure exchange economy with n consumers and k goods, in which endowments are strictly positive for each consumer and each commodity. Assume also that all utility functions satisfy strict quasiconcavity, strict monotonicity, and continuity. (A) (B) Show that if the economy has more than one competitive equilibrium allocation, then the initial endowments cannot be Pareto optimal. Dropping the assumption of strict quasiconcavity while maintaining the other assumptions, give an example of an economy that has more than one competitive equilibrium allocation, but where the initial endowment is Pareto optimal. A carefully drawn and labeled Edgeworth Box is good enough. ANSWER: (A) Suppose not. Then the initial endowments are PO and there is at least one other distinct PO allocation, x* (the other CE allocation FFTWE). In the utility maximization problems determined by (p*,x*), each consumer can afford his/her * endowment. Therefore, ui( xi ) ui( ω i) for all i. But since endowments are PO, the inequality cannot be strict for any consumer, or else x* would Pareto ** * dominate ω. But now the allocation, xi = ( xi +ωi)/ must provide at least as much utility to each consumer as x* or ω, and strictly higher utility for any * consumer where xandω are distinct. Thus, x** Pareto dominates x* and ω. # i i
(B) Suppose there are two consumers and two goods. Let each consumer have the utility function, ui( xi) = xi + xi and let each consumer have the endowment (/,/). The initial endowment is PO, and any allocation on the budget line defined by a price ratio of is a CE allocation. The indifference curves are all straight lines with a slope of -. 4. This question concerns a pure exchange economy with K commodities and n consumers, where all utility functions are strictly monotonic, strictly quasi-concave, and continuous. For each of the following statements, if the statement is true, then prove it. If the statement is false, then provide a counterexample. A carefully drawn, labeled, and explained Edgeworth Box diagram is enough for a counterexample. (A) If x* and x** are strongly Pareto optimal, then u ( x ) u ( x ) for all i. i i i i (B) If x* and x** are strongly Pareto optimal, then u ( x ) u ( x ) for some i. i i i i (C) If u ( x ) = u ( x ) for all i, and if x* $ x**, then x* cannot be strongly Pareto optimal. i i i i ANSWER: (A) This statement is false. Starting with just about any two Pareto optimal points will be a counterexample. For example, two consumers, two goods, Cobb-Douglas utility functions, and an aggregate endowment of unit of each good. The contract curve is the diagonal of the Edgeworth box, so let x* be given by x * = (/4, /4) and x * = (/4, /4), and let x** be given by x ** = (/4, /4) and x ** = (/4, /4). The statement is false, since consumer strictly prefers x ** to x *. (B) This statement is true. If not, then every consumer strictly prefers x**, but then x** would Pareto dominate x*, contradicting the Pareto optimality of x*. (C) This statement is true. If ui( xi ) = ui( xi ), then the two bundles are on the same indifference curve. Consider instead the allocation, x%, in which everyone receives the midpoint of the line segment connecting x* and x**, x% = (x* + x**)/. Since x* and x** are feasible, then x% is feasible. If x i * = x i **, then obviously this consumer receives the same utility under x i * and x i %, since it is the same bundle. For all consumers i such that x i * $ x i **, and we know there is at least one such consumer, then consumer i strictly prefers x i %, because the line segment cuts through the upper contour set, by strict quasi-concavity. Therefore, x% Pareto dominates x*, so x* cannot be Pareto optimal.
5. Consider the following pure exchange economy with consumers and two commodities. Consumer has the endowment vector (,) and the utility function u ( x, x ) = log( x ) + log( x ). Consumer has the endowment vector (,0) and the utility function u ( x, x ) = log( x ) + log( x ). Consumer has the endowment vector (0,) and the utility function u ( x, x ) = log( x ) + log( x ). (A) Define a competitive equilibrium for this economy. (B) Calculate the competitive equilibrium price and allocation. ANSWER: (A) A competitive equilibrium is a price vector, (p, p ), and an allocation, ( x*, x*, x*, x*, x*, x* ), such that: ) x * solves max log( x ) + log( x ) Subject to px + px = p + p (Equality due to monotonicity), x " 0, ) x * solves max log( x) + log( x ) Subject to px + px = p (Equality due to monotonicity), x " 0, ) x * solves max log( x) + log( x ) Subject to px + px = p (Equality due to monotonicity), x " 0, 4) market clearing: x + x + x =, x + x + x =. (B) To make life easier, we will normalize the price of good to be, so the price vector is (p,). We first derive the demand functions for the three consumers. The two relevant equations are the budget equation and the marginal rate of substitution equation (MRS = p). For consumer, we have:
x x p = and px + x = p +. Solving for the demands, we have: x = ( p+ )/ p and x = ( p+ )/. For consumer, we have: x x = p and px x p + = Solving for the demands, we have: x = / and x = p/. For consumer, we have: x x = p and px x + = Solving for the demands, we have: x = / p and x = /. Now we pick one of the market clearing equations to solve for p. Good is easier, so we have: (p+)/ + p/ + / =. Solving for p, we get p =. (We could have guessed that, because of the symmetry between goods and, but it is better to solve it.) Now we plug p = to get the final allocation: ( x*, x*, x*, x*, x*, x* ) = (,, /, /, /, /). 6. Consider the following pure exchange economy with 00 consumers and two commodities. For i =,..., 00, consumer i has the utility function i i i i i u ( x, x ) = log( x ) + log( x ). For i =,..., 00, consumer i has the endowment vector (,). For i = 0,..., 00, consumer i has the endowment vector (,). (A) Define a competitive equilibrium for this economy. (B) Calculate the competitive equilibrium price and allocation. ANSWER: (A) A competitive equilibrium is a price vector, (p,p ), and an allocation {(x i,x i ) for i =,... 00} satisfying: () for i =,..., 00, the nonnegative vector (x i,x i ) solves: max log(x i ) + log(x i ) Subject to p x i + p x i! p + p.
() for i = 0,..., 00, the nonnegative vector (x i,x i ) solves: max log(x i ) + log(x i ) Subject to p x i + p x i! p + p. () markets clear 00 00 x i= i i= i 400 and x 500. (B) Solving for the C.E., compute demand functions by solving the budget equation and the equation found by equating marginal rates of substitution to the price ratio, p. (Normalize p to be p, and p to equal.) For i =,..., 00, we have: x i / x i = p and p x i + x i = p +. Solving, we have: x i = (p+)/p and x i = (p+)/. For i = 0,..., 00, we have: x i / x i = p and p x i + x i = p +. Solving, we have: x i = (p+)/p and x i = (p+)/. To find the equilibrium price, use market clearing for one of the markets. Using market, and noticing that there are 00 consumers of the first type and 00 consumers of the second type, we have: 00[ (p+)/ ] + 00[ (p+)/ ] = 500. Solving, we find that p = 5/4. Plugging a price of 5/4 into the demand functions, we find the allocation: (x i,x i ) = (7/5, 7/4) for i =,..., 00 and (x i,x i ) = (/0, /8) for i = 0,..., 00.