Some Duality Results


INDIRECT UTILITY FUNCTION

The indirect utility function is defined as the maximum utility that can be attained given money income and goods prices.

u*(p1,p2,M) = max U(x1,x2) s.t. p1 x1 + p2 x2 = M

Properties of the indirect utility function:

The first two properties are obvious. To see why u* is quasi-convex, consider the following diagram.

Let x10 and x20 be the optimal demand bundle associate with prices (p10, p20) and income M. Let U(x10, x20)=u. Therefore u* at the point (p10, p20) equals u. Consider some other point on the straight line. When goods prices fall on this straight line, the original goods bundle is still affordable. Thus the consumer can attain at least a utility of at least u. But typically, the optimal choice of goods will allow a utility level higher than u. If we draw an "indifference curve" on the price space, therefore, such an indifference curve will lie above the straight line. Prices to the north-east of the indifference curve are worse than prices on the indifference curve. From the diagram, we conclude that the lower contour sets are convex. This is precisely the definition of quasi-convexity.

The fourth property, Roy's identity, is a consequence of the envelope theorem. Since the Lagrangian of the problem is

U(x1,x2)+ λ(M - p1 x1 - p2 x2)
We have ∂u*/∂p1 = - λ* x1* and ∂u*/∂M = λ*. Dividing one equation by the other yields Roy's identity.

EXPENDITURE FUNCTION

The expenditure function is defined as the minimum expenditure required to attain a utility level u, given goods prices. That is,

C*(p1, p2, u) = min p1 x1 + p2 x2 s.t. U(x1, x2)=u
You can see that the expenditure function is formally equivalent to the cost function introduced in producer theory. All their mathematical properties are the same, so you can refer back to the earlier notes.

It is also clear that you can derive the cost function from the indirect utility function, and vice versa. If the minimum cost of achieving utility level u is M, then the maximum utility from income M is u. The cost function and indirect utility function are therefore inverses of each other. We have

C*( p1, p2, u*(p1, p2, M) ) ≡ M
and
u*( p1, p2, C*(p1, p2, u) ) ≡ u

DIRECT AND INDIRECT UTILITY FUNCTIONS

Given an indirect utility function u*(p1, p2, M), we can find the (direct) utility function U(x1, x2) by this relationship:

U(x1, x2) = minp1, p2 u*(p1, p2, M) s.t. p1 x1 + p2 x2 = M

To see why this is true. Let x=(x1, x2) be the demanded bundle when prices are p=(p1, p2) and income is M. Then, by definition, u*(p, M) = U(x). Let p' be any other price vector that satisfies p'x=M. Since x is still affordable, we must have

u*(p',M) ≥ U(x) for all p' x = M
So U(x) is indeed the solution to the minimization problem.

SLUTSKY EQUATION

Start off with a Marshallian demand x1= x1*( p1, p2, M). Let utility at this demand bundle be u. When p1 changes, holding M constant, the level of utility will change. Suppose, now, when p1 changes, M is also changed to keep utility constant at u, then we get a Hicksian demand curve, x1= x1~ (p1, p2, u). To keep utility constant at u, money income has to be equal to C*(p1, p2, u). Therefore we may write

x1*( p1, p2, C*(p1, p2,u) ) ≡ x1~ ( p1, p2, u )

Differentiate both sides with respect to p1,

∂x1*/ ∂p1 + (∂x1*/∂M) (∂C*/∂p1) = ∂x1~ /∂p1
Note that ∂C*/∂p1 = x1~ = x1*. Therefore
∂x1*/ ∂p1 + x1*( ∂x1*/∂M) = ∂x1~/ ∂p1
Since we already know that the right hand side is negative, we conclude that the left hand side is also negative. Alternatively, we can write
∂x1*/ ∂p1 = ∂x1~/ ∂p1 - x1* (∂x1*/∂M)
The first term on the right hand side is always negative, and we call it the substitution effect. The second term is the income effect, and it may be either positive or negative.


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