Jill buys two goods: X and Y and has the following utility function: U(X,Y) = 2X

Question

Jill buys two goods: X and Y and has the following utility function: U(X,Y) = 2X0.25Y0.75  

a. Find the equation for her generalized demand curve for X*(Px,Py,I)           

b. Find the equation for her generalized demand curve for Y*(Py,Px,I)         

c. Are the above generalized demand functions homogeneous of degree 0? (Show your work)

d. Is X a normal good?     

e. Is X a gross substitute or complement to Y?  

f. Find the compensated demand curve for Xc(Px, Py, U)              

g. Find the compensated demand curve for Yc (Py, Px, U)          

h. Are the compensated demand functions homogenous of degree 0 in Px and Py if utility is held constant?        

i. Find the expenditure function: E(Px, Py, U)          

j. Use Shephard’s Lemma to verify that your compensated demand functions Xc and Yc are correct.  

k. Find the indirect utility function: V(Px,Py,I)

Explanation / Answer

U = 2X0.25Y0.75

Budget constraint is:

I = X. Px + Y. Py

(a)

MRS = MUx / MUy where

MUx = DU / DX = 2 x 0.25 x (Y / X)0.75

MUy = dU / dY = 2 x 0.75 x (X / Y)0.25

So, MRS = (1 / 3) x (Y / X)

Under optimality, MRS = Px / Py

(1 / 3) x (Y / X) = Px / Py

So,

Y = 3X (Px / Py) and

X = (Y / 3) (Py / Px)

Substituting in budget line:

I = X. Px + Y. Py,

I = X. Px + Py. 3X (Px / Py)

I = X. Px + 3X. Px = 4X. Px

So,

X = I / (4 Px) [Demand function for X]

(b) Again, from budget constraint,

I = X. Px + Y. Py

I = (Y / 3) (Py / Px). Px + Y. Py

I = Y. Py. (1/3) + Y. Py

I = (4 / 3) Y. Py

Y = (I x 3) / (4 Py) [Demand function for Y]

(c)

X = I / (4 Px)

Y = (I x 3) / (4 Py)

A function f(x) is homogenous of degree 0 if,

f(kx) = f(x)

Both the demand functions here are independent of X or &. Hence, increasing X or Y will leave the functions unchanged.

So, both demand functions are homogenous of degree 0.

(d) From demand function of X, if Income (I) increases, quantity demanded of X also increases, Therefore, X is a normal good.

Note: Out of 11 questions, first 4 are answered.

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