Linear expenditure system
From Hobson's Choice
The linear expenditure model is a form of utility function which allows economists to relax the assumption that expenditures on any particular category of good are unaffected by the price. Using a LEM, one can allow for the possibility that that expenditures on good x are decreasing in px or increasing in py (where y is any other good). Usually economists use the Cobb-Douglas function.
Context: the Cobb-Douglas Function
A basic staple of microeconomics is the utility function, which is usually presented to students thus:
Let U represent consumer utility as a function of goods x and y. Maximize U(x,y) subject to I = p1x + p2y. U(x,y) = ln(xy).
|Map of Cobb-Douglas Utility function|
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|Linear Expenditure Utility function|
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U(x,y) = α ln(x) + β ln(y), where α and β are exponents of any value.
The Lagrangian for this equation is
L(λ,x,y) = αln(x) + β ln(y) + λ(I — p1x — p2y)
And the first order conditions are
= α/x — λp1 = 0
= β/y — λp2 = 0
= I — p1x — p2y = 0
These simplify to x* = and y* = , which means that expenditure on either x or y would always be exactly the same given a constant income; the actual amount of the good itself consumed would vary inversely with price.
This is unsatisfying because the result is that there is no feedback on demand from price; demand remains unaffected by the satisfaction that a dollar spent on x yields. In real life, an increase in the price of a thing, such as energy or shelter, will cause one to consume not only less of that thing, but ultimately, seek maximization strategies in which one spends less on that thing. Most of the famous exceptions to this aren't exceptions at all; they involve an increase in income (I).
An additional point to be made is that virtually any statement that can be made about utility functions can be made about production functions. Cobb-Douglas functions are used for estimating production functions as well, with the same pitfalls.
Solution of a Two-Good Linear Expenditure Utility Function
An alternative to the Cobb-Douglas Utility function is the linear expenditure function. This modifies the utility function to
U(x,y)= α ln (x - x0) + β ln (y - y0)
where x0 and y0 are given constants, and α + β = 1.
In this case, the optimal values of x*, y* are
p1x = αI + βp1x0 — αp2y0 p2y = βI — βp1x0 + αp2y0
which can be contrived to alter the shape of indifference curves; say, if we wanted to discuss fuel versus everything else, and then changed the subject to another type of good that is a superior good.
Solving a Three-Good Linear Expenditure Utility Function
Usually, the concept is explained with two goods so it can be illustrated (x and y being goods, and z—the vertical axis—standing for utility). But we can't illustrate this one fully because we are interested in cases where there are actually more than two goods determining utility.
Let U be utility as a function of x, y, and z, where x refers to everything one buys other than software, y is cheap software, and z is costly software (α, β, and γ are arbitrary constants; x0, y0, and z0 are threshold levels of consumption) .
U(x, y, z) = αln(x-x0) + βln(y-y0)+ γln(z-z0)
I = pxx + pyy + pzz .
where I is income and p refers to the price of the respective good.
The Lagrangian will be
L = αln(x-x0) + βln(y-y0)+ γln(z-z0) + λ(I - pxx - pyy - pzz)
and first order conditions will be
First we solve for x, y, and z in terms of the constants (and λ)
and then we solve for the Lagrangian multiplier λ:
And we substitute the values for x, y, and z into the equation for the Lagrangian multiplier.
Now, so far this has just been a generic solution of a symmetric 3-good constrained optimization problem, and it can be made even more general for a very large number of goods:
where gj is any good, cj is the corresponding constant I've been representing with Greek letters, I is income, and i is the counter for summation within the equation (So, for example, pj refers to the price for the good gj whose optimal amount g* you're trying to determine, while pigi refers to the amount expended on any individual good listed in the summation from 1 to n goods).
- ↑ Regarding the utility function: I prefer to use the linear expenditure model instead of the Cobb-Douglas model everyone else uses, because the CD utility function leads to rigid expenditures between x, y, and z. If a researcher wanted to perform regression analysis of "observed preferences" to establish what the coefficients were, the existence of threshold levels of consumption would correspond to y-intercepts for each good.
- Avinash K. Dixit, Optimization in Economic Theory, Oxford University Press (1990)