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Unit-7: Recent Developments in Demand Theory

To solve the problem of utility maximization, we write as— Notes

Max U (X)

Subject to ∑ pi X ≤ Y …(1)
i i
Where X = Utility bundle of i products
i
U = The utility from utility bundle
P = Price of i products
i
Y = Total income of consumer
Suppose that λ = P /Y and now the problem of utility maximization can be written as follows—
i i
Max U (X)
Subject to ∑ λ X ≤ 1 …(2)

i i i
Where λ is normalized price.
i
In this form, there are two sets of problem of utility maximization— (i) Consumption units with price
and (ii) Normalized price with λ = λ ,……, λ price.
i n
The optimum demand function can be given as follows—

X = D (λ) i = 1, …… n …(3)
i i
The maximum utility level can be obtained by substitution of equation (3) into equation (1). Further, this
optimum utility bundle depends on vector of prices which shows in equation (3). The indirect utility
function is derived from it.
V (λ) = U (d), (λ) …… d (λ) …(4)
n
V is called indirect utility function because it depends upon income level and a set of normalized
prices λ.

Properties of Indirect Utility Function

Following are the characteristics of Indirect Utility Function:
1. If U is continuous, then V is also continuous to all positive sets of λ.
2. U is not increased because if the income decreases or price increases, then it does not maximize the
utility function. It is correct if U increases in utility bundle ith.
3. U does not decrease if ith is normalized price however U is increasing in ith utility bundle.
4. If there is a corner solution means X = 0 then the utility of consumer not changes if P increases. For
t
example, if the price of Maruti Zen is increased then it does not affect on maximum consumer’s
utility levels.

Graphical Presentation

The indirect utility function is drawn by indirect indifference curves. Suppose that only two consumer
goods are 1 and 2 whose normalized prices are λ and λ which are on vertical and horizontal axis as
1
2
shown on Fig. 7.3. An indirect indifference curve IIC shows the combinations of normalized prices on
2
λ which is untouched to maximum utility level. If the consumer is not satisfied from any of the product
2

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