Question

Two six-year-old twins, Lisa (L) and Malcolm (M), have the following initial endowments of Cookies and Apples: Let CL, AL denote Lisa's consumption and CM, AM denotes Malcolm's consumption of Cookies and Apples. Lisa and Malcolm have the following utility functions: a) Calculate a competitive equilibrium when Lisa and Malcolm decide to trade with each other. b) Draw an Edgeworth diagram with Lisa in the lower-left corner and Malcolm in the upper-right corner, where consumption of Cookies is measured on the horizontal axis and consumption of Apples is measured on the vertical axis. Identify the initial endowments and draw the indifference curves of Lisa and Malcolm consistent with these endowments. c) Show graphically the Pareto dominating space given the initial endowments and show that the competitive equilibrium is Pareto-efficient. d) Explain the contract curve concept, and depict it graphically for Lisa and Malcolm. Explore how different positions on the contract curve can be achieved by means of redistribution of endowments. (Choose a different endowment point and calculate a new equilibrium. Try to find the equation for the contract curve) e) What are the relations among the different parts of this question and the First- and Second Welfare Theorems?

Answer 1

Endowment for Lisa : 5 cookies and 5 apples

Endowment for Malcolm : 5 cookies and 5 apples

So, the total endowment/availability in economy of Apples = 5+5 = 10, and of Cookies = 5+5 = 10 (edgeworth box dimensions)

Utility functions : for Lisa U(C,A) = C*A (so, comparing with Cobb-Douglas transformation, a = 1, b = 1) and for Malcolm U(C,A) = C + 2A (substitutes case)

a) Finding Competitive equilibrium :

Given the utility functions of Lisa and Malcolm, we can find their demand functions using price ratio and Marginal rate of substitution(MRS), using the tangency condition (MRS = Pc/Pa), Pc is price of cookie and Pa is price of an apple. Normalizing the price of apples, i.e, Pa = 1, so price ratio = Pc/Pa = Pc/1 = Pc

MRS = MUc/MUa

MUc = $\partial U/\partial C$ and MUa = $\partial U/\partial A$

Then, for Lisa, MUc = $\partial U/\partial C$ = A ; and MUa = $\partial U/\partial A$ = C, and MRS = MUc/MUa = A/C

So, at tangency, A/C = Pc. Also, with the given prices and endowments,

income of Lisa, ML = Pc*C + Pa*A = 5*Pc + 5

This is a Cobb-Douglas transformation, so we know the demand function is as follows:

C = (a/(a+b))*ML/Pc = (1/(1+1))*(5*Pc + 5)/Pc = (5*Pc + 5)/2Pc

And, A = (b/(a+b))*ML/Pa = (1/(1+1))*(5*Pc + 5)/1 = (5*Pc + 5)/2

For Malcolm, MUc = $\partial U/\partial C$ = 1 and MUa = $\partial U/\partial A$ = 2, and MRS = MUc/MUa = 1/2

In case of substitutes, we know demand may lie on corners as well. Income of Malcolm would be same as Lisa (due to same endowments and prices), so income of Malcolm, MC = 5*Pc + 5

If MRS > price ratio, i.e, if 1/2 > Pc, Malcolm will consume only C, so, C = MC/Pc = (5*Pc + 5)/Pc, A=0

If 1/2 < Pc, Malcolm will consume only A, so, A = MC/Pa = (5*Pc + 5)/1 = 5*Pc+5, C = 0

If 1/2 = Pc, Malcolm will consume both C and A

Considering all the cases obtained above:

Case 1 : If Pc < 1/2, For Malcolm, A = 0, so all apples are consumed by Lisa

Thus, for Lisa, A = 10 i.e, (5*Pc + 5)/2 = 10 (from demand function of Lisa)

On solving we get, Pc = (2*10)/5 - 1 = 3, but 3 > 1/2, so this case is rejected.

Case 2: If Pc > 1/2, For Malcolm, C = 0, so all Cookies are consumed by Lisa

Thus, for Lisa, C = 10 i.e, (5*Pc + 5)/2Pc = 10 (again from demand function of Lisa)

On solving we get, Pc = 1/3 < 1/2, so this case is rejected. (as we assumed Pc > 1/2)

Case 3 : If Pc = 1/2,

For Lisa, using her demand functions: C = (5*Pc + 5)/2Pc = (5*(1/2) + 5)/2*(1/2) = 2.5 + 5 = 7.5 cookies

And apples, A = (5*Pc + 5)/2 = (5*(1/2) + 5)/2 = 3.75 apples

Then, remaining Cookies, 10 - 7.5 = 2.5 cookies is demanded by Malcolm

and remaining apples, 10 - 3.75 = 6.25 apples is demanded by Malcolm

And competitive equilibrium is Pc = 1/2 and Pa = 1, when they trade.

b) Following is the diagram for endowment and the respective and consistent Indifference curves passing throught them:

c)  

Pareto dominating space is the shaded region in the above diagram. Notice that at the competitive equilibrium quantity, Malcolm's Indifference curve is same as before, while Lisa's indifference curve has shifted upwards (generating higher utility for her). So, one person is better off, while other is not worse off (so it is a Pareto improvement move).

At the competitive equilibrium, there is no pareto dominating space, i.e, the two ICs (IC of Malcolm, and new IC (IC2) of Lisa) are tangent to each other, so no movement or allocation of goods can now take place such that one is better off without hurting the other one or without worsening off the other one's utility, making the competitive equilibrium Pareto efficient.