Question

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Consider a pure exchange economy with two goods, wine (x) and cheese (y) and two con- sumers, A and B. Let cheese be the numeraire good with price of $1. Consumer A's utility function is UA(x, y) = xy and B's utility function is UB(x, y) = min [x, y). A has an initial allocation of 10 x and no y, and B has an initial allocation of 10 units of y and no x. (a) Put wine x on the horizontal axis and cheese y on the vertical axis. Measure goods for consumer A from the lower left and goods for consumer B from the upper right. Solve for the contract curve of Pareto efficient allocations in this economy and show this on your graph. (b) Mark the initial allocation with the letter W. Draw the indifference curves for each person through this point. Calculate utility at this allocation for both consumers. Is the initial resource allocation consistent with Pareto efficiency? Explain. Find the competitive equilibrium prices and consumption for each type of consumer. Derive A's and B's demand functions (Marshallian). Calculate the equilibrium price of wine assuming price of cheese is $1. Using demand for wine, show that Walras' Law holds. Show the budget constraint and indifference curves at the equilibrium. Label the equilibrium E. Show that E is on the contract curve. Suppose instead, you wished to obtain the equilibrium E1, where utility of A is equal to UB = 7. Find corresponding price of wine pi that must hold in this instance, and show the set of possible new endowments (which will be a line) that satisfies the Second Theorem of Welfare Economics, given pi.

Answer 1

Two goods Wine cx). - Cheese Cy) Two consumers: A & B cheese be the numeraire good i.e. py-1 eltility functions o UA (x, y) = ny ] UB(x, y)= min [x, y Initial Allocation A = Clo, o) B-10,10) neit wine (s) on houizontal axis and cheese cy) on vertical asis. We make an edguwouth bon Meing the ginen information cele - contract come TB-yB 5 TA -Jeg W initial endowment O 5 Wine (x) 10 wome of this economy will be to consumice Biibecause he/she. contract according

is the rigid agent so x = y. so, price - Gross den I (c) A 10 A&R the of on (b) the is initial shown endowment by pourt W o pout W geureth Price of cheese, fy - 1 will be Marlent fou wire, good a in equilibrium when Total demand = Total Supply x + 2B - 10 using the demand functionthet equations to mt man 10. wie 2 px Pact py 100+ 0(1) + OCPr) + 108 2 10- bextl slove + 10 =10 Plotting by B as o as ICA the cont lps tos P1+1 2x = het how at puit pa= 7

(c) so, price of wine = px= 1. Gross demand DCA100) + 0(1) termin وفي XA = 5 units yote ya , 0017.10(1) YA = 5 units m 2(1) [ Total Supply - xA ] x 8 = 10-5 = 5 units y = 10-5 = 5 units [Total Supply - ne] Plotting the demand of A as (5,5) and B as (5,5), we decaw the IC as ICA and ICB passing therough the contract were.