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**Unformatted text preview: **Expenditure Minimisation Problem Simon Board This Version: November 3, 2008 First Version: October 27, 2008. The expenditure minimisation problem (EMP) looks at the reverse side of the utility maximisa- tion problem (UMP). The UMP considers an agent who wishes to attain the maximum utility from a limited income. The EMP considers an agent who wishes to find the cheapest way to attain a target utility. This approach complements the UMP and has several rewards: • It enables us to analyse the effect of a price change, holding the utility of the agent constant. • It enables us to decompose the effect of a price change on an agent’s Marshallian demand into a substitution effect and an income effect. This decomposition is called the Slutsky equation. • It enables us to calculate how much we need to compensate a consumer in response to a price change if we wish to keep her utility constant. 1 Model We make several assumptions: 1. There are N goods. For much of the analysis we assume N = 2 but nothing depends on this. 2. The agent takes prices as exogenous. We normally assume prices are linear and denote them by { p 1 ,...,p N } . 1 Eco11, Fall 2008 Simon Board 3. Preferences satisfy completeness, transitivity and continuity. As a result, a utility func- tion exists. We normally assume preferences also satisfy monotonicity (so indifference curves are well behaved) and convexity (so the optima can be characterised by tangency conditions). The expenditure minimisation problem is min x 1 ,...,x N N X i =1 p i x i subject to u ( x 1 ,...,x N ) ≥ u (1.1) x i ≥ 0 for all i The idea is that the agent is trying to find the cheapest way to attain her target utility, u . The solution to this problem is called the Hicksian demand or compensated demand. It is denoted by h i ( p 1 ,...,p N , u ) The money the agent must spend in order to attain her target utility is called her expenditure. The expenditure function is therefore given by e ( p 1 ,...,p N , u ) = min x 1 ,...,x N N X i =1 p i x i subject to u ( x 1 ,...,x N ) ≥ u x i ≥ 0 for all i Equivalently, the expenditure function equals the amount the agent spends on her optimal bundle, e ( p 1 ,...,p N , u ) = N X i =1 p i h i ( p 1 ,...,p N , u ) 1.1 Example Suppose there are two goods, x 1 and x 2 . Table 1 shows how the agent’s utility (the numbers in the boxes) varies with the number of x 1 and x 2 consumed. To keep things simple, suppose the agent faces prices p 1 = 1 and p 2 = 1 and wishes to attain utility u = 12. The agent can attain this utility by consuming ( x 1 ,x 2 ) = (6 , 2), ( x 1 ,x 2 ) = (4 , 3), ( x 1 ,x 2 ) = (3 , 4) or ( x 1 ,x 2 ) = (2 , 6). Of these, the cheapest is either ( x 1 ,x 2 ) = (4 , 3) or ( x 1 ,x 2 ) = (3 , 4). In either case, her expenditure is 4 + 3 = 7....

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