Lecture 4

Expenditure Minimization Problem
Suppose that instead of maximizing utility subject to the budget constraint, the consumer set a target utility level u and minimized expenditure on bundle x (min p x) such that u(x) = u: This is called the expenditure minimization problem (EMP) and has a very interesting relationship with the utility maximization problem (UMP). Mathematically, the two problems are duals of each other, the UMP is the primal and the EMP is the dual. Formally, the EMP is written:
Economics 50: Intermediate Microeconomics Summer 2010 Stanford University Michael Bailey Lecture 4: Expenditure Minimization Problem; Substitution and Income E¤ects
H
M RSx;y
a py b px
a
=) y (px ; py ; u) = u =) x (px ; py ; u) = u
H
"
b px a py a py b px
Solving for the lagrange multiplier, we see that
=
pi M Uxi
: This has a very intuitive interpretation,
remember that the lagrange multiplier is the value, in terms of increasing the max or decreasing the min, of relaxing the constraint by one unit. In this case, because we are trying to minimize expenditure, the lagrange multiplier is the additional expenditure we would have to provide for raising our target utility by one unit. To get that additional unit of utility, we would need to buy more of good xi ; at a cost of pi per unit. Each xi that is purchased will provide us with M Uxi units of additionally utility. Thus we would have to buy
pi pj
= M RSxi ;xj ; for
the UMP, plug back into the budget constraint, for the EMP, plug back into the target utility constraint u(x) = u: The solution to the EMP, xH i (px ; py ; u) = arg min p x s.t. u(x; y ) = u; is called the compensated demand function1 for good xi and tells us how much of good xi will be demanded at the minimal expenditure level to achieve utility u given prices. Notice that income does not appear anywhere in this problem, we assume
3
y
10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10
x
Figure 2: The EMP: get to the lowest expenditure (budget line) given the target utility constraint (must stay on IC) Example 1 Cobb-Douglas u(x; y ) = xa y b ; min px x + py y s.t. u(x; y ) = u ay px axa 1 y b = = bxa y b 1 bx py a py =) x = y b px a a py y y b = u (plugging into target utility constraint) =) b px = =) y a+b = u
1 Also known as the "Hicksian" demand function, after Sir John Hicks, a nobel laureate in economics. This is why there is an "H" superscript on the compensated demand function. We often call the solution to the UMP "Marshallian" demand after Alfred Marshall, a famous classical economist.
10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10
y
x
Figure 1: The UMP: get to the highest indi¤erence curve subject to the budget constraint (must stay on budget line)
= =)
But this is the same …rst-order condition as in the UMP. It doesn’ t matter whether we are trying to …nd the highest indi¤erence curve given the budget line (UMP) or the lowest budget line given the indi¤erence curve (EMP), the curves will be tangent. This implies that the ratio between two goods will be equivalent in both problems. To solve either the UMP or the EMP, always begin with setting
Overview
The Expenditure Minimization Problem (EMP) has the same …rst order condition as the Utility Maximization Problem (UMP) Compensated demand functions solve the EMP and indicate the cheapest bundle that yields utility u The EMP and UMP have a very speci…c relationship; The indirect utility function and the expenditure function are inverses of each other The substitution e¤ect is the change in demand given a price change if utility is held constant The income e¤ect is the change in demand after a price change once the substitution e¤ect is accounted for The change in consumer’ s surplus is a very popular measure of the welfare loss due to a price change and is the area between the price and under the demand curve EV and CV are two alternative measures of the welfare loss due to a price change and measure the expenditure needed to return to the original utility level. The CV measures the change in expenditure needed to return to original utility using new prices, whereas CV uses old prices
2
the consumer has in…nite income and is trying to …nd the cheapest way to achieve utility u: The demand is called a "compensated" demand because when prices change, we give income or take away income from the consumer such that they achieve u(x; y ) = u: So the consumer is in e¤ect "compensated" for price increases and is brought back to utilitt x 2 X and u(x) = u for u 2 R