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.

(A) Find the aggregate endowment (or total supply) of good x.

(B) Find Mr. 1’s demand for good x in terms of px. Your answer should be a formula in terms of the letter px.

(C) Similarly, write down both Mr. 2 and Mr. 3’s demand for good x

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)

A)

Total endowment of x (total supply of x ) = 10+5+20

= 35

B)

X_A = .25M₁/P_x by using the x= (a/a+b)(M/P_x)

= M₁/4P_x

C)

X_B = .5M₂/P_x

= M₂/2P_x

X_c = .75M₃/P_x

= 3M₃/4p_x

THANK YOU

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