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Using Full Duality to Show that Simultaneously Additive Direct and Indirect Utilities Implies Unitary Price Elasticity of Demand

Paul A. Samuelson

Abstract

The "indirect utility function," a concept associated with Hotelling, Roy, Houthakker, and others, gives the maximized value of consumer's ordinal utility in function of the prices and income of his budget constraint: namely, $\phi^{\ast} (p_{1} \I, \ldots, P_{n}\I) = \max \phi (x_{1}, \ldots, x_{n})$ with respect to $x$'s satisfying the budget constraint $\Sigma_{1}^{n} (p_{j} / I) x_{j} = 1$. Writing $p_{j} / I = y_{j}, \Phi (Y) = -\phi^{\ast} (Y)$ it follows that $N (X;Y) = \phi (X) + \Phi (Y) \lesseqqgtr 0$ along $\Sigma^{n}_{1} x_{j} y_{j} = 1$, equalling zero only along the equilibrium demand relations $X = X(Y), Y = Y(X) \equiv X^{-1}(X).$ It is shown that $\phi (X)$ and $\Phi (X)$ are completely dual functions, one possessing all the general properties of the other. Just as $/phi$'s partial derivatives give $\phi_{i} \phi_{j} = y_{i}/y_{j}$, $\varPhi$'s give $\varPhi_{i}/\varPhi_{j} = x_{i}/x_{j}$, etc. Numerous theorems are proved, such as: if either of $\phi$ and $\varPhi$ has homothetic contours, so does the other; if both can be stretched into an additive form, they are both homothetic and belong to the so-called constant-elasticity-of-substitution family of Solow et al., a result already anticipated by Bergson in 1936; if the above can hold with no stretching required, we are in the "pure Bernoulli-Marshall" or Cobb-Douglas case of unitary own-elasticity and other demonstrated equivalent properties. Finally, dual functions of mixed variables, $\phi (y_{1}, \ldots, y_{r}; x_{r+1}, \ldots, x_{n})$ and $\varPhi (x_{1}, \ldots, y_{r+1}, \ldots, y_{n}$, are defined by Legendre transformations and the properties of "demand under rationing" are deduced from them.