#### Transcript MICROECONOMIC THEORY

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) = XY • Setting up the Lagrangian: L = XY + (I - PXX - PYY) • First-order conditions: L/X = X-1Y - PX = 0 L/Y = XY-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 Indirect Utility Function • It is often possible to manipulate firstorder conditions to solve for optimal values of X1,X2,…,Xn • These optimal values will depend on the prices of all goods and income X*1 = X1(P1,P2,…,Pn,I) X*2 = X2(P1,P2,…,Pn,I) • • • X*n = Xn(P1,P2,…,Pn, I) Indirect Utility Function • We can use the optimal values of the Xs to find the indirect utility function maximum utility = U(X*1,X*2,…,X*n) • Substituting for each X*i we get maximum utility = V(P1,P2,…,Pn,I) • The optimal level of utility will depend indirectly on prices and income – If either prices or income were to change, the maximum possible utility will change Indirect Utility in the CobbDouglas • If U = X0.5Y0.5, we know that I X* 2Px I Y* 2PY • Substituting into the utility function, we get I maximum utility 2PX 0.5 I 2PY 0.5 I 2PX0.5PY0.5 Expenditure Minimization • Dual minimization problem for utility maximization – allocating income in such a way as to achieve a given level of utility with the minimal expenditure – this means that the goal and the constraint have been reversed Expenditure Minimization • Point A is the solution to the dual problem Expenditure level E2 provides just enough to reach U1 Quantity of Y Expenditure level E3 will allow the individual to reach U1 but is not the minimal expenditure required to do so A Expenditure level E1 is too small to achieve U1 U1 Quantity of X Expenditure Minimization • The individual’s problem is to choose X1,X2,…,Xn to minimize E = P1X1 + P2X2 +…+PnXn subject to the constraint U1 = U(X1,X2,…,Un) • The optimal amounts of X1,X2,…,Xn will depend on the prices of the goods and the required utility level Expenditure Function • The expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular set of prices minimal expenditures = E(P1,P2,…,Pn,U) • The expenditure function and the indirect utility function are inversely related – both depend on market prices but involve different constraints Expenditure Function from the Cobb-Douglas • Minimize E = PXX + PYY subject to U’=X0.5Y0.5 where U’ is the utility target • The Lagrangian expression is L = PXX + PYY + (U’ - X0.5Y0.5) • First-order conditions are L/X = PX - 0.5X-0.5Y0.5 = 0 L/Y = PY - 0.5X0.5Y-0.5 = 0 L/ = U’ - X0.5Y0.5 = 0 Expenditure Function from the Cobb-Douglas • These first-order conditions imply that PXX = PYY • Substituting into the expenditure function: E = PXX* + PYY* = 2PXX* Solving for optimal values of X* and Y*: E X* 2PX E Y* 2PY Expenditure Function from the Cobb-Douglas • Substituting into the utility function, we can get the indirect utility function E U ' 2PX 0.5 E 2PY 0.5 E 0.5 0.5 2PX PY • So the expenditure function becomes E = 2U’PX0.5PY0.5 Important Points to Note: • To reach a constrained maximum, an individual should: – spend all available income – choose a commodity bundle such that the MRS between any two goods is equal to the ratio of the goods’ prices • the individual will equate the ratios of marginal utility to price for every good that is actually consumed Important Points to Note: • Tangency conditions are only first-order conditions – the individual’s indifference map must exhibit diminishing MRS – the utility function must be strictly quasiconcave • Tangency conditions must also be modified to allow for corner solutions – ratio of marginal utility to price will be lower for goods that are not purchased Important Points to Note: • The individual’s optimal choices implicitly depend on the parameters of his budget constraint – choices observed will be implicit functions of prices and income – utility will also be an indirect function of prices and income Important Points to Note: • The dual problem to the constrained utilitymaximization problem is to minimize the expenditure required to reach a given utility target – yields the same optimal solution as the primary problem – leads to expenditure functions in which spending is a function of the utility target and prices