10.1 The Behavioral Postulates
A fundamental problem in economics is deriving the consumer's demand function from the behavioral postulate of utility maximization. The central theme will be studying the structure of models in order to discover what refutable hypotheses can be derived. This means it is mainly methodological; we wish to find what it is about the postulate of utility maximization subject to constraints that either leads to or fails to generate refutable hypotheses.
The classic problem of the consumer is maximizing happiness with respect to scarcity, that is,
Max \(U(x_1...x_n)\)
Subject to \(\sum p_i x_i = M\)
This hypothesis is referred to as rational behavior, which we extend to all consumers. Although utility was original conceived of as a cardinal quantification (i.e. 7 utils from a new sweater), now we understand that consumers generally rank all commodity bundles, thus weighing opportunity costs in mutually exclusive situations:
- X is preferred to Y.
- Y is preferred to X.
- X and Y are equally preferred.
Only one category will apply at any time, and if it should change, we say that the consumer's preferences have changed. To characterize the real world more closely, we also assume differentiability of utility functions, that consumers would ideally have a varied flow of goods and services.
Now we can assert important features of the utility functions:
- Nonsatiation, or "more is preferred to less": More of any good is always preferred to less of it. Mathematically, this means that the marginal utility of any good is positive: \(U_i = \frac{\partial U}{\partial X} > 0\) .
- Substitution: At any point, the consumer is willing to give up some of one good for an additional increment of another good. Mathematically, this corresponds to an indifference curve with a negative slope which shows the consumer is willing to make tradeoffs.
- Diminishing marginal value: This means that along any indifference surface, the marginal value of any good decreases as more of that good is consumed. Mathematically, this is shown by a positive second derivative which displays the increasing rate of substitution along the amount: \((\frac{\partial ^2 X}{\partial Y^2}) > 0\).
- All can be summed up by this: All consumers possess utility functions that are differentiable everywhere, strictly increasing, and strictly quasi-concave.
10.2 Utility Maximization
Now we will begin with the problem at hand. In going through the steps of utility maximization per the Lagrangian, we arrive at two propositions:
- Proposition 1: The demand curves implied by the utility function and budget constraint are identical to those derived when \(U(X, Y) \) is replaced by \(V(X, Y ) = F(U(X, Y))\) where \(F'(U) >0\).
- Proposition 2: The demand curves \(X = X^M (p_X, p_Y, M)\) are homogenous to the degree 0. That is \(X^M (tp_X, tp_Y, tM) \equiv X^M (p_X, p_Y, M)\).
Interpretation of the Lagrange multiplier
From the first-order relations we know that
\(\lambda ^M = \frac{U_X}{p_X} = \frac{U_Y}{p_Y}\)
We also know from the envelope phenomenon that the rate of change of the objective function with respect to a parameter is the same whether or not the decision variable adjust to that change:
\(\lambda ^M = \frac{U_X X^M + U_Y Y ^M}{M}\)
Hence we can say that lambda is the marginal utility of money income:
\(\lambda^M = \frac{\partial U}{\partial M}\)
Roy's Identity
Using the envelope theorem, we can find an important relation regarding the rate of change of the maximum utility with respect to price:
\(\frac{\partial U}{\partial p_X} = \frac{\partial \mathcal{L}}{\partial p_X} = - \lambda ^M X^M\)
Moreover, solving for X will give us
\(X ^M = - \frac{\frac{\partial U}{\partial p_X}}{\frac{\partial U}{\partial M}}\)
Note that in the case of the demand derived from profit maximization and those derived from constrained cost minimization, the choice functions are the partial derivatives of the indirect objective function with respect to the prices.
We can also employ Young's theorem to analyze the reciprocal relation regarding responses to changes in income:
\(\frac{\partial \lambda ^M}{\partial p_X} = -X ^M \frac{\partial \lambda ^M}{\partial M} - \lambda ^M \frac{\partial X^M}{\partial M}\)
This expression is used in the analysis of consumer surplus.
Finally, we can draw the income consumption path:
Regarding the income elasticities, we know that
\(\epsilon _M = \frac{M}{X^M} \frac{\partial X^M}{\partial M}\)
10.3 The Relationship Between the Utility Maximization Model and Cost Minimization Model
Whereas our initial first-order condition demand curves,
\(X = X^M (p_X, p_Y, M)\)
are called the "money income held constant" demand curves or "income-compensated" demand curves, we will now turn to curves that will hold utility, or "real income," constant. These come from the simultaneous solutions to the first-order relations:
\(X = X^U (p_X, p_Y, U^0)\)
\(Y = Y^U (p_X, p_Y, U^0)\)
\(\lambda = \lambda^U (p_X, p_Y, U^0)\)
These functions are also commonly referred to as Hicksian demands. The partial derivatives with respect to price represent pure substitution effects since utility is held constant and the consumer remains on the same indifference level. The differences are displayed in these graphs:
However, the fundamental contribution to the theory of the consumer is the Slutsky equation, which relates the rates of change of consumption with respect to price changes when money income is held constant to the corresponding change when real income is held constant. That is, a relationship is given between the income and substitution effects.
10.4 The Slutsky Equation
It is apparent from the structure of the utility maximization model that no refutable hypotheses are strictly implied on the basis of the maximization hypothesis alone. All of the parameters appear in the constraint, and furthermore, no testable implications appear in any model for any parameter in the constraint function. Graphically, we will now show the substitution and income effects of a price change:
It relates the finite adjustment of consumption to small changes in the price, though it is not the same as the Slutsky equation which deals with instantaneous rates of change. The general result for the Slutsky equation will be:
\(\frac{\partial X ^M}{\partial p_Y} = \frac{\partial X^U}{\partial p_Y} - Y^M \frac{\partial X^M}{\partial M}\)
This equation shows that the response of a utility-maximizing consumer to a change in price can be conceptually split up into two parts: first, a pure substitution effect which is a response to a price change holding the consumer on the original indifference curve, and second, a pure income effect, wherein income is changed, holding prices constant, to reach a tangency on the new indifference curve.