Question

Problem 3

An econometrician has statistically estimated the following Marshallian demand functions for a good ?: x^m(p_x,I)=0.5*(I/p_x) and  y^m(p_y,I)=0.5*(I/p_y), In addition, she was able to derive the following indirect utility function consistent with her statistical estimations: ?(p_x,p_y,I)=0.5*I*p_x^-0.5*p_y^-0.5

Now she claims that the Slutsky equation does not hold for her functions and asks you to check this:

a) Compute the expenditure function from the information given.

b) Compute the compensated (Hicksian) demand curve for good ?.

c) Use the results from part b) and the given Marshallian demand function for good ? to show that the Slutsky equation does hold also in this case.

Answer 1

(a) v= V = 0.5 1 Pre-0.5 py-0.5 0.5 III . Preo. 5 Pyo.5 - au SPX J pý = . Expenditure fu E = 20 JPx Joy DE JPx = xh = sul py 25px xha v JPG J Px = JPx JPY

(6) Now suppose Px fer Px sutsky equation 0x 8 px = Oxh 8px PE(IE) = SE + IC SE = axb & Px xh = wspy Sess - VSP Governor)

Now put v= 2 x SEE - (1 ksoy) [ 55 Jes Pr] 2 - I u Px. Px EN DESI OPS IE = = -0.25 1 __ P 3 ope -xox 8I

x = 0.51 8.5 ох OI Px Px - IE= - 4 - 0.25 I TE = -0.25I - P 0.25 I P² -0.51 - OX of Thus, slubsky equation holds true.