Question

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) = 2x+y and B's utility function is UB(x; y) = xy. A's initial allocation is 10 units of x and 0 units of y. B's initial allocation is 0 units of x and 30 units of y.

(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. 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.

(b) Solve for the contract curve of Pareto efficient allocations in this economy and show this on your graph.

(c) 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 indierence curves at the equilibrium. Label the

equilibrium E. Show that E is on the contract curve.

(d) Suppose instead, you wished to obtain the equilibrium E1, where consumption of good

x by A rises to x1

A = 5, but is also on the contract curve. Find corresponding price of

wine p1 that must hold in this instance, and show the set of possible new endowments

(there will be a line that shows the complete set) that would satisfy the Second Theorem

of Welfare Economics, given p1.

Answer 1

Two Goods Wine (x) and cheese (4) Pn . let cheese is numeraire good UA (",y)= anty Up (wy) = ny A's endowment - (10,0) + WA 's endowment = (0,30) + W8 10 30 LCR A Y A ICB Paredo Parcelona Nicient А т. paint Initial Endowment 30 . w 1 C Initial endowment will not be pareton efficient. - All Pareto-ficient points lying on up = 9 will be pareto efficient pito

30,10 14 Ta Q Contract Come so, all the points lying on Me = 9 will be pareto-effia -cient, 3 10 (c.) for competitive equilibrium, set of ç fareto: Efficient CE are Using this we know that will lie on ug ay line, mo

Now, budget line should nous through endowment bundle la point e shown on the above diagram will be DE 2 ut 4 m - ię budget line will be 2n+y = m * Pu et ln 22 J & Py = 1 7. will to will Pn 22 T clear the market (Verd, मेक As 20 Income = 1082 a B's Income a 3001 - 30 * Budget line facing the A will be . 2nty = 20 e Budget line gereing the & will be ent y = 30 How, we con chind the simply marimize dd ya the utility to - P.TO

le for - Gusulmer-A, Man UA (hy) sit en ty = 20 ię zuty sit luty 220 Ic Intyo Co IC and Budget line are overlapping, 4 Consumer will demard enogthing S on the line satisfying enty = 2 for Gusumer - B, Man UB (my) Sit 2n+9230 • P.To

Man my sit anty - 30 at tergential noint vre R. slope of B.L = slope of Il HUN 오 Hoy & quay 1. so, putting in bil, - aut anago + qu? 30 7.5 y = 15
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