Consider an individual\'s indirect utility function where p represents the price
ID: 1122855 • Letter: C
Question
Consider an individual's indirect utility function where p represents the price-vector and y represents the individual's wealth (a) State Roy's identity and use it to show that the ordinary (Marshallian) By demand function for good 1 is given by xi(p.y) -Comment on this expression [25 marks] (b) Derive the uncompensated terms ap1 ay Comment on these expressions. [25 marks] (c) From the indirect utility function, derive the individual's expenditure function e(p, u)-WP21-), where u is utility, and derive the Hicksian (compensated) demand function for good 1, x1(p, u) [25 marks] (i) Continuing from (c), derive the substitution term a p1 (ii) Use the Slutsky equation to derive the substitution term(0) from ap1 your results at (b).Explanation / Answer
a. Roy’s identity states that if the indirect utility function v(p,y) is differential at (p0,y 0) and v(p0,y 0)/ y 0 then xi (p0,y 0) = - [v(p0,y 0)/ pi] / [v(p0,y 0)/ y] ; i = 1,......,n and (p0,y 0) are price and income vectors.
Using above definition, x1 (p,y ) = - [v(p,y)/p1] / [v(p,y )/y]
Therefore, x1 (p,y )= y/p1
b. x1 (p,y)/ p1 = - y/p12
x1 (p,y)/ p2 = 0
x1 (p,y)/ y = /p1
c. Using the relation v(p, e(p,u)) = u we can derive the expenditure function from indirect utility function.
v(p, e(p,u)) = e(p,u) p1- p2-(1-)
v(p, e(p,u)) = u
e(p,u) p1- p2-(1-) = u
e(p,u) = u p1 p2(1-)
The hicksian demand function can be derived using the relation between marshallian and hicksian demand function,
x1h (p,u) = x1(p, e(p,u))
= e(p,u)/p1
= u p1 p2(1-)/p1
= u p1-1 p2(1-)
d. i. x1h (p,u)/ p1= (-1) u p1-2 p2(1-)
ii.The slutsky equation is given as,
x1 (p,y)/p1 = x1h (p,u)/ p1 - x1 (p,y )[ x1 (p,y)/y]
- y/p12 = x1h (p,u)/ p1 - /p1*/p1
= p1-2
= u p1-2 p2(1-)