1 Chapter 4 UTILITY MAXIMIZATION AND CHOICE Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.

2 Optimization Principle To maximize utility, given a fixed amount of income to spend, an individual will buy the goods and services: –that exhaust his or her total income –for which the psychic rate of trade-off between any goods (the MRS) is equal to the rate at which goods can be traded for one another in the marketplace

3 A Numerical Illustration Assume that the individual’s MRS = 1 –willing to trade one unit of x for one unit of y Suppose the price of x = $2 and the price of y = $1 The individual can be made better off –trade 1 unit of x for 2 units of y in the marketplace

4 The Budget Constraint Assume that an individual has I dollars to allocate between good x and good y p x x + p y y  I Quantity of x Quantity of y The individual can afford to choose only combinations of x and y in the shaded triangle If all income is spent on y, this is the amount of y that can be purchased If all income is spent on x, this is the amount of x that can be purchased

5 First-Order Conditions for a Maximum We can add the individual’s utility map to show the utility-maximization process Quantity of x Quantity of y U1U1 A The individual can do better than point A by reallocating his budget U3U3 C The individual cannot have point C because income is not large enough U2U2 B Point B is the point of utility maximization

6 First-Order Conditions for a Maximum Utility is maximized where the indifference curve is tangent to the budget constraint Quantity of x Quantity of y U2U2 B

7 Corner Solutions In some situations, individuals’ preferences may be such that they can maximize utility by choosing to consume only one of the goods Quantity of x Quantity of y At point A, the indifference curve is not tangent to the budget constraint U2U2 U1U1 U3U3 A Utility is maximized at point A

8 The n-Good Case The individual’s objective is to maximize utility = U(x 1,x 2,…,x n ) subject to the budget constraint I = p 1 x 1 + p 2 x 2 +…+ p n x n Set up the Lagrangian: L = U(x 1,x 2,…,x n ) + ( I - p 1 x 1 - p 2 x 2 -…- p n x n )

9 The n-Good Case First-order conditions for an interior maximum:  L/  x 1 =  U/  x 1 - p 1 = 0  L/  x 2 =  U/  x 2 - p 2 = 0  L/  x n =  U/  x n - p n = 0  L/  = I - p 1 x 1 - p 2 x 2 - … - p n x n = 0

10 Implications of First-Order Conditions For any two goods, This implies that at the optimal allocation of income

11 Interpreting the Lagrangian Multiplier is the marginal utility of an extra dollar of consumption expenditure –the marginal utility of income

12 Interpreting the Lagrangian Multiplier At the margin, the price of a good represents the consumer’s evaluation of the utility of the last unit consumed –how much the consumer is willing to pay for the last unit

13 Cobb-Douglas Demand Functions Cobb-Douglas utility function: U(x,y) = x  y  Setting up the Lagrangian: L = x  y  + ( I - p x x - p y y) First-order conditions:  L/  x =  x  -1 y  - p x = 0  L/  y =  x  y  -1 - p y = 0  L/  = I - p x x - p y y = 0