Chapter 4 UTILITY MAXIMIZATION AND CHOICE MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright 2002 by South Western a division of Thomson Learning All rights reserved Complaints about Economic Approach No real individuals make the kinds of lightning calculations required for utility maximization The utility maximization model predicts many aspects of behavior even though no one carries around a computer with his utility function programmed into it Complaints about Economic Approach The economic model of choice is extremely selfish because no one has solely self centered goals Nothing in the utility maximization model prevents individuals from deriving satisfaction from doing good 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 A Numerical Illustration Assume that the individual s MRS 1 He is 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 The Budget Constraint Assume that an individual has I dollars to allocate between good X and good Y PXX PYY I Quantity of Y I PY If all income is spent on Y this is the amount of Y that can be purchased The individual can afford to choose only combinations of X and Y in the shaded triangle If all income is spent on X this is the amount of X that can be purchased I PX Quantity of X First Order Conditions for a Maximum We can add the individual s utility map to show the utility maximization process The individual can do better than point A by reallocating his budget Quantity of Y A The individual cannot have point C because income is not large enough C B U3 U2 U1 Point B is the point of utility maximization Quantity of X First Order Conditions for a Maximum Utility is maximized where the indifference curve is tangent to the budget constraint slope of budget constraint Quantity of Y PX PY slope of indifferen ce curve dY dX B U2 PX dY PY dX Quantity of X U constant MRS U constant Second Order Conditions for a Maximum The tangency rule is only necessary but not sufficient unless we assume that MRS is diminishing if MRS is diminishing then indifference curves are strictly convex If MRS is not diminishing then we must check second order conditions to ensure that we are at a maximum Second Order Conditions for a Maximum The tangency rule is only necessary but not sufficient unless we assume that MRS is diminishing Quantity of Y There is a tangency at point A but the individual can reach a higher level of utility at point B B A U2 U1 Quantity of X 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 Y At point A the indifference curve is not tangent to the budget constraint U1 U2 U3 Utility is maximized at point A A Quantity of X The n Good Case The individual s objective is to maximize utility U X1 X2 Xn subject to the budget constraint I P1X1 P2X2 PnXn Set up the Lagrangian L U X1 X2 Xn I P1X1 P2X2 PnXn The n Good Case First order conditions for an interior maximum L X1 U X1 P1 0 L X2 U X2 P2 0 L Xn U Xn Pn 0 L I P1X1 P2X2 PnXn 0 Implications of First Order Conditions For any two goods U X i Pi U X j Pj This implies that at the optimal allocation of income Pi MRS X i for X j Pj Interpreting the Lagrangian Multiplier U X1 U X 2 U X n P1 P2 Pn MU X1 P1 MU X 2 P2 MU X n Pn is the marginal utility of an extra dollar of consumption expenditure the marginal utility of income Interpreting the Lagrangian Multiplier For every good that an individual buys the price of that good represents his evaluation of the utility of the last unit consumed how much the consumer is willing to pay for the last unit Pi MU X i Corner Solutions When corner solutions are involved the first order conditions must be modified L Xi U Xi Pi 0 i 1 n If L Xi U Xi Pi 0 then Xi 0 This means that U X i MU X i Pi Any good whose price exceeds its marginal value to the consumer will not be purchased Cobb Douglas Demand Functions Cobb Douglas utility function U X Y X Y Setting up the Lagrangian L X Y I PXX PYY First order conditions L X X 1Y PX 0 L Y X Y 1 PY 0 L I PXX PYY 0 Cobb Douglas Demand Functions First order conditions imply Y X PX PY Since 1 PYY PXX 1 PXX Substituting into the budget constraint I PXX 1 PXX 1 PXX Cobb Douglas Demand Functions Solving for X yields I X PX Solving for Y yields I Y PY The individual will allocate percent of his income to good X and percent of his income to good Y Cobb Douglas Demand Functions The Cobb Douglas utility function is limited in its ability to explain actual consumption behavior the share of income devoted to particular goods often changes in response to changing economic conditions A more general functional form might be more useful in explaining consumption decisions CES Demand Assume that 0 5 U X Y X0 5 Y0 5 Setting up the Lagrangian L X0 5 Y0 5 I PXX PYY First order conditions L X 0 5X 0 5 PX 0 L Y 0 5Y 0 5 PY 0 L I PXX PYY 0 CES Demand This means that Y X 0 5 Px PY Substituting into the budget constraint we can solve for the demand functions X I PX PX 1 PY I Y PY PY 1 PX CES Demand In these demand functions the share of income spent on either X or Y is not a constant depends on the ratio of the two prices The higher is the relative price of X or Y the smaller will be the share of income spent on X or Y CES Demand If 1 U X Y X 1 Y 1 First order conditions imply that Y X PX PY 0 5 The demand functions are X I PY PX 1 PX 0 5 Y I PX PY 1 PY 0 5 CES Demand The elasticity of substitution is equal to 1 1 when 0 5 2 when 1 0 5 Because substitutability has declined these demand functions are less responsive to changes in relative prices The CES allows us to illustrate a wide variety of possible relationships …