Question

Consider a pure exchange economy with 2 goods and 3 agents. Good y is the numeraire, so we set py = 1. • Mr. 1 is Cobb-Douglas of type λ = 0.25 and is endowed with (10, 4). • Mr. 2 is Cobb-Douglas of type λ = 0.5 and is endowed with (5, 9). • Mr. 3 is Cobb-Douglas of type λ = 0.75 and is endowed with (20, 20). In the following steps, you will find the equilibrium of this model economy.

(D) Write down the aggregate demand for good x in the economy. Again, your answer should be in terms of the letter px.

(E) Solve for the magic, equilibrium price p ∗ x that makes the supply of good x equal the demand of good x.

(F) Calculate how many units of good x each of the three agents will receive in equilibrium. Check: the three numbers should add up to 35.

(G) Write down all three agents’ demands for good y in terms of px.

Answer 1

SOLUTION:-

We have P_y = 1

U_A(x , y) = x^.25y^.75

U_B(x,y) = x^.5y^.5

U_C(x,y) = x^.75y^.25

e_A = (10,4)

e_B = (5,9)

e_C = (20,20)

the general rule for demand function of x and y when we have cobb douglas utility function U = x^ay^b

x = (a/a+b)(M/P_x)

y = (a/a+b)(M/P_y)

D)

aggregate demand of X = XA+XB+XC

= M₁/4P_x + M₂/2P_x +3M₃/4P_x

E)

at equilibrium total demand of x = total supply of x

M₁ = 10P_x + 4 since P_y = 1

M₂ = 5P_x + 9

M₃ = 20P_x +20

so (M₁ +2M₂ +3M₃)/4P_x = 35

now put the values of M1 M2 M3

[ (10P_x + 4 + 2 (5P_x + 9) = 3 (20P_x + 20)]/4P_x = 35

after some algebra

80P_x + 82 = 140P_x

60P_x = 82

P_x = 41/30 this equilibrium price

F)

XA = M₁/4P_x

= (10P_x +4)/4P_x put P_x = 41/30

= 265/82

XB = M₂/2P_x

= (5P_x + 9)/2P_x

= 475/82

Xc = 3M₃/4P_x

= 3( 20P_x + 20)/4P_x

= 2130/82

check = 265/82+475/82+2130/82

= 35

hence verify

G) Similarly

YA = ( b/a+b)(M1/P_y)

= .75(10P_x + 4) since Py = 1

YB = .5M2/Py

= .5(5P_x + 9)

Yc = .25M3/Py

= .25( 20P_x + 20)