Question

Edgeworth box with quasi linearity There are two economic agents, A and B. Utility functions are the following: (a) The endowment for the economy is (z,y) = (10,20). Find the set of all Pareto efficient allocation such that xA > 0, zB > 0,VA > 0,yB > 0. 0 (i.e. yi E (-00,00) for i E {A, B). The endowment for the (b) Assume xA 0 and FB economy is (T,) (10,20. Find teset of a Pareto efficient allocation (c) Assume rA 2 0,xp0,yA 0,y20. The endowment for the economy is (r,) (10,20). Find the set of all Pareto efficient allocation. (d) Pick a Pareto efficient allocation in part (c) such that y 0. Find the Price taking equilibrium supporting the allocation

Answer 1

a. At the pareto efficient allocations, the marginal rate of substitution of both consumers is equal:

XB 4xA XB 2VXA

Also,

XA+ XB 10- X10-XA

Plugging the value in the equation above, we get:

10 XA= 4XA XA = 2

This is the condition for Pareto Optimality.

PE-(XA YA XA 2, yA E (0, 20]

b. In this case, the set of Pareto Optimal allocations will be the same, the only difference would be the range of values y can take:

PE-(XA YA XA 2, yA E (0, 00)

c. The set of Pareto Optimal allocations is

PE-(XA YA XA 2, yAE[0,20]

d. Let's take the allocation ((2,20),(8,0)) and let's assume the price of Y is 1

Consumer 1's problem:

max XA+ YA s.t. XAp+ y = 2P+20

At equilibrium, MRS = Price Ratio:

pXA 4p 2/XPXA

Substituting the value of x in the budget constraint:

8p +80p 1 р у - 402 +У 3D 2р + 20 4p

Hence, consumer 1's demand function is

1 8p2+80p 1 (XA, YA)- ( = 4p2 4p

Consumer 2's problem:

$max \ 2\sqrt{x_B}+y_B \ s.t. \ x_Bp+y=8p$

At equilibrium, MRS = Price Ratio

pXB XB

Substituting the value of x in the budget constraint:

$p\frac{1}{p^2}+y=8p \rightarrow y=\frac{8p^2-1}{p}$

Consumer 2's demand function is:

$(x_B,y_B)=(\frac{1}{p^2},\frac{8p^2-1}{p})$

Total demand is equal to total endowment,

$\frac{1}{p^2}+\frac{1}{4p^2}=10 \rightarrow 1=8p^2 \rightarrow p=\frac{1}{2\sqrt2}$

Hence, the competitive equilibrium prices are:

$(p_X,p_Y)=(\frac{1}{2\sqrt2},1)$