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econ11_09_lecture3

# econ11_09_lecture3 - Utility Maximisation Problem Simon...

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Utility Maximisation Problem Simon Board * This Version: September 20, 2009 First Version: October, 2008. The utility maximisation problem (UMP) considers an agent with income m who wishes to maximise her utility. Among others, we are interested in the following questions: How do we determine an agent’s optimal bundle of goods? How do we derive an agent’s demand curve for a particular good? What is the effect of an increase in income on an agent’s consumption? 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 } . 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). * Department of Economics, UCLA. http://www.econ.ucla.edu/sboard/. Please email suggestions and typos 1

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Eco11, Fall 2009 Simon Board 4. The consumer is endowed with income m . The utility maximisation problem is: max x 1 ,...,x N u ( x 1 , . . . , x N ) subject to N X i =1 p i x i m (1.1) x i 0 for all i The idea is that the agent is trying to spend her income in order to maximise her utility. The solution to this problem is called the Marshallian demand or uncompensated demand. It is denoted by x * i ( p 1 , . . . , p N , m ) The most utility the agent can attain is given by her indirect utility function . It is defined by v ( p 1 , . . . , p N , m ) = max x 1 ,...,x N u ( x 1 , . . . , x N ) subject to N X i =1 p i x i m (1.2) x i 0 for all i Equivalently, the indirect utility function equals the utility the agent gains from her optimal bundle, v ( p 1 , . . . , p N , m ) = u ( x * 1 , . . . , x * N ) . 1.1 Example: One Good To illustrate the problem, suppose N = 1. For example, the agent has income m and is choosing how many cookies to consume. The agent’s utilities are given by table 1. In general, we solve the problem in two steps. First, we determine which bundles of goods are affordable. The collection of these bundles is called the budget set . Second, we find which bundle in the budget set the agent most prefers. That is, which bundle gives the agent most utility. Suppose the price of the good is p 1 = 1 and the agent has income m = 4. Then the agent can afford up to 4 units of x 1 . Given this budget set, the agent’s utility is maximised by choosing x * 1 = 4, yielding utility v = 28. 2
Eco11, Fall 2009 Simon Board Units of x 1 Utility 1 10 2 18 3 24 4 28 5 30 6 29 7 26 8 21 Table 1: Utilities from different bundles. Observe that this agent is satiated at 5 units. Next, suppose the price of the good is p 1 = 1 and the agent has income m = 8. Then the agent can afford up to 8 units of x 1 . Given this budget set, the agent’s utility is maximised by choosing x * 1 = 5, yielding utility v = 30. In this example, the consumer can afford 8 units but chooses to consume 5. If the agent’s preferences are monotone, then she will always spend her entire budget.

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