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Unformatted text preview: Economics 50: Intermediate Microeconomics Summer 2010 Stanford University Michael Bailey Lecture 4: Expenditure Minimization Problem; Substitution and Income E/ects Overview & The Expenditure Minimization Problem (EMP) has the same &rst order condition as the Utility Max- imization 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 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: min p ¡ x such that x 2 X and u ( x ) = u for u 2 R 1 The lagrangian for the problem is: L = p & x ¡ & ( u ( x ) ¡ u ) which has &rst-order conditions: @ L @x i = p i ¡ & @u ( x ) @x i = 0 8 i @ L @& = u ( x ) ¡ u = 0 Solving for the lagrange multiplier, we see that & = p i MU x i : 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 x i ; at a cost of p i per unit. Each x i that is purchased will provide us with MU x i units of additionally utility. Thus we would have to buy 1 MU x i units of x i to get that additional unit of utility and the total cost would be p i MU x i : At the minimum, this condition would need to hold for every good x i , otherwise we could buy more of one good, and less of another, and get the same utility for a cheaper cost. This can be seen by rearranging the &rst order conditions for goods x i and x j : & = p i MU x i = p j MU x j = ) p i p j = MU x i MU x j = MRS x...
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- Spring '11
- Microeconomics, Px, emp