6. (First-Best Allocation) Consider an exchange economy with two consumers (1 an

Question

6. (First-Best Allocation) Consider an exchange economy with two consumers (1 and 2) and two /4 goods (A and B). The preference of both consumers are given as ul(ZIA 2:18) = Z143 and the initial endowments for consumer 2 is w2 = (w2A.w2B) = (5,3) good A and 5 units of good B before any economic activity. u2 is similarly defined.) ment amount for both goods (a) u2(a 2A,T2B) = x 12 to The initial endowments for consumer l is = (w1A, tule) = (3,5) 3/4 1/4 (Note: u:1 = (wlA,ww) = (3,5) means that the mighty God gines consanner l trith 3 units of Hint: t is possible to have many solutions, given the symmetric preference and total endow- Compute Suppose the social welfare is given by W(r) ul(TIA+a: 1 B) + t42(#21,#2B). the First-Best allocation for is economy, which maximizes W() subject to physical feasibility constraint. What is the uity for cach consumer? Hint: transforming the constrained optimization to an unconstrained one will greatly simplify the calculation. Suppose the endlowmelt distribution Is now tul-(WIA: tul B)-(3, 1) and u/2 = (WyA,WyB) = (5,7). Does it change the first-best allocation? (b) (c) Draw the Edgeworth Box for this economy. Draw four indifference curves for cach consumer. Highlight the Pareto efficient set. Mark the First-Best allocation as point A in the graph. Is it Pareto efficient?

Explanation / Answer

a) For ease of typing, we refer the goods as x and y instead of A and B. The two consumers are 1 and 2.

The utility functions and endowments are as below: U1=x1^(1/4)y1^(3/4); U2=x2^(3/4)y2^(1/4) The endowments for consumers 1 and 2 are (3,5) and (5,3) respectively.

The social welfare function is given by W=U1(x,y)+U2(x,y) To find the first best allocation we must take partial derivatives of the social welfare function with respect to x and y. So, we get-

dW/dx= 1/4(y/x)^3/4+3/4(y/x)^1/4 and dW/dy=3/4(x/y)^1/4+1/4(x/y)^3/4, Now we equate dW/dx with dW/dy.

The utility is given by substituting the initial endowments into the utility functions, that is, U1=3^(1/4)5^(3/4) and U2=5^(3/4)3^(1/4)

b) Given that endowments change we plug in the new set of endowment values into the utility functions. So we get, U1=3^(1/4)1^(3/4) and U2=5^(3/4)7^(1/4)

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