Indirect utility function explained
In economics, a consumer's indirect utility function
gives the consumer's maximal attainable
utility when faced with a vector
of goods prices and an amount of
income
. It reflects both the consumer's preferences and market conditions.
This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices. A consumer's indirect utility
can be computed from his or her utility function
defined over vectors
of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector
by solving the
utility maximization problem, and second, computing the utility
the consumer derives from that bundle. The resulting indirect utility function is
The indirect utility function is:
- Continuous on Rn+ × R+ where n is the number of goods;
- Decreasing in prices;
- Strictly increasing in income;
- Homogenous with degree zero in prices and income; if prices and income are all multiplied by a given constant the same bundle of consumption represents a maximum, so optimal utility does not change;
- quasi-convex in (p,w).
Moreover, Roy's identity states that if v(p,w) is differentiable at
and
, then
- | \partialv(p0,w0)/(\partialpi) |
\partialv(p0,w0)/\partialw |
=xi(p0,w0),
i=1,...,n.
Indirect utility and expenditure
The indirect utility function is the inverse of the expenditure function when the prices are kept constant. I.e, for every price vector
and utility level
:
Example
Let's say the utility function is the Cobb-Douglas function
which has the Marshallian demand functions
[1] x1(p1,p2)=
{\rmand} x2(p1,p2)=
,
where
is the consumer's income. The indirect utility function
is found by replacing the quantities in the utility function with the demand functions thus:
v(p1,p2,w)=
=
0.6
0.4=\left(
\right)0.6\left(
\right)0.4=(0.60.6*.4.4)w0.6+0.4
=K
w,
where
Note that the utility function shows the utility for whatever quantities its arguments hold, even if they are not optimal for the consumer and do not solve his utility maximization problem. The indirect utility function, in contrast, assumes that the consumer has derived his demand functions optimally for given prices and income.
See also
Further reading
- Book: Cornes, Richard . Duality and Modern Economics . New York . Cambridge University Press . 1992 . 0-521-33601-5 . Individual Consumer Behavior: Direct and Indirect Utility Functions . 31–62 .
- Book: Jehle, G. A. . Geoffrey A. Jehle . Reny . P. J. . Philip J. Reny . 2011 . Advanced Microeconomic Theory . Third . Harlow . Prentice Hall . 978-0-273-73191-7 . 28–33 .
- Book: Luenberger, David G. . David Luenberger . Microeconomic Theory . New York . McGraw-Hill . 1995 . 0-07-049313-8 . 103–107 .
- Book: Mas-Colell, Andreu . Andreu Mas-Colell . Michael D. . Whinston . Michael Whinston . Jerry R. . Green. Jerry Green (economist). 1995 . Microeconomic Theory . New York . Oxford University Press . 56–57 . 0-19-507340-1 .
- Book: Nicholson, Walter . Microeconomic Theory: Basic Principles and Extensions . Hinsdale . Dryden Press . Second . 1978 . 0-03-020831-9 . 57–59 .
Notes and References
- Book: Varian, H. . 1992 . Microeconomic Analysis . registration . 3rd . New York . W. W. Norton ., pp. 111, has the general formula.