Finance:Indirect utility function

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In economics, a consumer's indirect utility function [math]v(p, w)[/math] gives the consumer's maximal attainable utility when faced with a vector [math]p[/math] of goods prices and an amount of income [math]w[/math]. 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 [math]v(p, w)[/math] can be computed from his or her utility function [math]u(x),[/math] defined over vectors [math]x[/math] of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector [math]x(p, w)[/math] by solving the utility maximization problem, and second, computing the utility [math]u(x(p, w))[/math] the consumer derives from that bundle. The resulting indirect utility function is

[math]v(p,w)=u(x(p,w)).[/math]

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 [math](p^0, w^0)[/math] and [math]\frac{\partial v(p,w)}{\partial w} \neq 0[/math], then

[math] -\frac{\partial v(p^0,w^0)/(\partial p_i)}{\partial v(p^0,w^0)/\partial w}=x_i (p^0,w^0),\quad i=1, \dots, n. [/math]

Indirect utility and expenditure[edit]

The indirect utility function is the inverse of the expenditure function when the prices are kept constant. I.e, for every price vector [math]p[/math] and utility level [math]u[/math]:[1]:106

[math]v(p, e(p,u)) \equiv u[/math]

See also[edit]

References[edit]

  1. Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0-393-95735-7. 

Further reading[edit]

  • Cornes, Richard (1992). "Individual Consumer Behavior: Direct and Indirect Utility Functions". Duality and Modern Economics. New York: Cambridge University Press. pp. 31–62. ISBN 0-521-33601-5. 
  • Jehle, G. A.; Reny, P. J. (2011). Advanced Microeconomic Theory (Third ed.). Harlow: Prentice Hall. pp. 28–33. ISBN 978-0-273-73191-7. 
  • Luenberger, David G. (1995). Microeconomic Theory. New York: McGraw-Hill. pp. 103–107. ISBN 0-07-049313-8. 
  • Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). Microeconomic Theory. New York: Oxford University Press. pp. 56–57. ISBN 0-19-507340-1. 
  • Nicholson, Walter (1978). Microeconomic Theory: Basic Principles and Extensions (Second ed.). Hinsdale: Dryden Press. pp. 57–59. ISBN 0-03-020831-9. 

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