微观经济理论英文版课件：ch04 UTILITY MAXIMIZATION AND CHOICE

1、Chapter 4UTILITY MAXIMIZATIONAND CHOICECopyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.1Complaints about the Economic ApproachNo real individuals make the kinds of “lightning calculations” required for utility maximizationThe utility-maximization model predicts m

2、any aspects of behaviorThus, economists assume that people behave as if they made such calculations2Complaints about the Economic ApproachThe economic model of choice is extremely selfish because no one has solely self-centered goalsNothing in the utility-maximization model prevents individuals from

3、 deriving satisfaction from “doing good”3Optimization PrincipleTo maximize utility, given a fixed amount of income to spend, an individual will buy the goods and services:that exhaust his or her total incomefor which the psychic rate of trade-off between any goods (the MRS) is equal to the rate at w

4、hich goods can be traded for one another in the marketplace4A Numerical IllustrationAssume that the individuals MRS = 1willing to trade one unit of x for one unit of ySuppose the price of x = $2 and the price of y = $1The individual can be made better offtrade 1 unit of x for 2 units of y in the mar

5、ketplace5The Budget ConstraintAssume that an individual has I dollars to allocate between good x and good ypxx + pyy IQuantity of xQuantity of yThe individual can affordto choose only combinationsof x and y in the shadedtriangleIf all income is spenton y, this is the amountof y that can be purchased

6、If all income is spenton x, this is the amountof x that can be purchased6First-Order Conditions for a MaximumWe can add the individuals utility map to show the utility-maximization processQuantity of xQuantity of yU1AThe individual can do better than point Aby reallocating his budgetU3CThe individua

7、l cannot have point Cbecause income is not large enoughU2BPoint B is the point of utilitymaximization7First-Order Conditions for a MaximumUtility is maximized where the indifference curve is tangent to the budget constraintQuantity of xQuantity of yU2B8Second-Order Conditions for a MaximumThe tangen

8、cy rule is only necessary but not sufficient unless we assume that MRS is diminishingif MRS is diminishing, then indifference curves are strictly convexIf MRS is not diminishing, then we must check second-order conditions to ensure that we are at a maximum9Second-Order Conditions for a MaximumThe ta

9、ngency rule is only a necessary conditionwe need MRS to be diminishingQuantity of xQuantity of yU1BU2AThere is a tangency at point A,but the individual can reach a higherlevel of utility at point B10Corner SolutionsIn some situations, individuals preferences may be such that they can maximize utilit

10、y by choosing to consume only one of the goodsQuantity of xQuantity of yAt point A, the indifference curveis not tangent to the budget constraintU2U1U3AUtility is maximized at point A 11The n-Good CaseThe individuals objective is to maximizeutility = U(x1,x2,xn) subject to the budget constraintI = p

11、1x1 + p2x2 + pnxnSet up the Lagrangian:L = U(x1,x2,xn) + (I - p1x1 - p2x2 - pnxn)12The n-Good CaseFirst-order conditions for an interior maximum:L/x1 = U/x1 - p1 = 0L/x2 = U/x2 - p2 = 0L/xn = U/xn - pn = 0L/ = I - p1x1 - p2x2 - - pnxn = 013Implications of First-Order ConditionsFor any two goods,This

12、 implies that at the optimal allocation of income14Interpreting the Lagrangian Multiplier is the marginal utility of an extra dollar of consumption expenditurethe marginal utility of income15Interpreting the Lagrangian MultiplierAt the margin, the price of a good represents the consumers evaluation

13、of the utility of the last unit consumedhow much the consumer is willing to pay for the last unit16Corner SolutionsWhen 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 = 0This means thatany good whose price exceeds

14、 its marginal value to the consumer will not be purchased17Cobb-Douglas Demand FunctionsCobb-Douglas utility function:U(x,y) = xySetting up the Lagrangian:L = xy + (I - pxx - pyy)First-order conditions:L/x = x-1y - px = 0L/y = xy-1 - py = 0L/ = I - pxx - pyy = 018Cobb-Douglas Demand FunctionsFirst-o

15、rder conditions imply:y/x = px/pySince + = 1:pyy = (/)pxx = (1- )/pxxSubstituting into the budget constraint:I = pxx + (1- )/pxx = (1/)pxx19Cobb-Douglas Demand FunctionsSolving for x yieldsSolving for y yieldsThe individual will allocate percent of his income to good x and percent of his income to g

16、ood y20Cobb-Douglas Demand FunctionsThe Cobb-Douglas utility function is limited in its ability to explain actual consumption behaviorthe share of income devoted to particular goods often changes in response to changing economic conditionsA more general functional form might be more useful in explai

17、ning consumption decisions21CES DemandAssume that = 0.5U(x,y) = x0.5 + y0.5Setting up the Lagrangian:L = x0.5 + y0.5 + (I - pxx - pyy)First-order conditions:L/x = 0.5x -0.5 - px = 0L/y = 0.5y -0.5 - py = 0L/ = I - pxx - pyy = 022CES DemandThis means that(y/x)0.5 = px/pySubstituting into the budget c

18、onstraint, we can solve for the demand functions23CES DemandIn these demand functions, the share of income spent on either x or y is not a constantdepends on the ratio of the two pricesThe higher is the relative price of x (or y), the smaller will be the share of income spent on x (or y)24CES Demand

19、If = -1,U(x,y) = -x -1 - y -1First-order conditions imply thaty/x = (px/py)0.5The demand functions are25CES DemandIf = -,U(x,y) = Min(x,4y)The person will choose only combinations for which x = 4yThis means thatI = pxx + pyy = pxx + py(x/4)I = (px + 0.25py)x26CES DemandHence, the demand functions ar

20、e27Indirect Utility FunctionIt is often possible to manipulate first-order conditions to solve for optimal values of x1,x2,xnThese optimal values will depend on the prices of all goods and incomex*n = xn(p1,p2,pn,I)x*1 = x1(p1,p2,pn,I)x*2 = x2(p1,p2,pn,I)28Indirect Utility FunctionWe can use the opt

21、imal values of the xs to find the indirect utility functionmaximum utility = U(x*1,x*2,x*n)Substituting for each x*i, we getmaximum utility = V(p1,p2,pn,I)The optimal level of utility will depend indirectly on prices and incomeif either prices or income were to change, the maximum possible utility w

22、ill change29The Lump Sum PrincipleTaxes on an individuals general purchasing power are superior to taxes on a specific goodan income tax allows the individual to decide freely how to allocate remaining incomea tax on a specific good will reduce an individuals purchasing power and distort his choices

23、30The Lump Sum PrincipleQuantity of xQuantity of yAU1A tax on good x would shift the utility-maximizing choice from point A to point BBU231An income tax that collected the same amount would shift the budget constraint to IIThe Lump Sum PrincipleQuantity of xQuantity of yABU1U2Utility is maximized no

24、w at point C on U3U3C32Indirect Utility and theLump Sum PrincipleIf the utility function is Cobb-Douglas with = = 0.5, we know that So the indirect utility function is33Indirect Utility and theLump Sum PrincipleIf a tax of $1 was imposed on good xthe individual will purchase x*=2indirect utility wil

25、l fall from 2 to 1.41An equal-revenue tax will reduce income to $6indirect utility will fall from 2 to 1.534Indirect Utility and theLump Sum PrincipleIf the utility function is fixed proportions with U = Min(x,4y), we know that So the indirect utility function is35Indirect Utility and theLump Sum Pr

26、incipleIf a tax of $1 was imposed on good xindirect utility will fall from 4 to 8/3An equal-revenue tax will reduce income to $16/3indirect utility will fall from 4 to 8/3Since preferences are rigid, the tax on x does not distort choices36Expenditure MinimizationDual minimization problem for utility

27、 maximizationallocating income in such a way as to achieve a given level of utility with the minimal expenditurethis means that the goal and the constraint have been reversed37Expenditure level E2 provides just enough to reach U1Expenditure MinimizationQuantity of xQuantity of yU1Expenditure level E

28、1 is too small to achieve U1Expenditure level E3 will allow theindividual to reach U1 but is not theminimal expenditure required to do soA Point A is the solution to the dual problem38Expenditure MinimizationThe individuals problem is to choose x1,x2,xn to minimizetotal expenditures = E = p1x1 + p2x

29、2 + pnxn subject to the constraintutility = U1 = U(x1,x2,xn)The optimal amounts of x1,x2,xn will depend on the prices of the goods and the required utility level39Expenditure FunctionThe expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular s

30、et of pricesminimal expenditures = E(p1,p2,pn,U)The expenditure function and the indirect utility function are inversely relatedboth depend on market prices but involve different constraints40Two Expenditure FunctionsThe indirect utility function in the two-good, Cobb-Douglas case isIf we interchang

31、e the role of utility and income (expenditure), we will have the expenditure functionE(px,py,U) = 2px0.5py0.5U41Two Expenditure FunctionsFor the fixed-proportions case, the indirect utility function isIf we again switch the role of utility and expenditures, we will have the expenditure functionE(px,

32、py,U) = (px + 0.25py)U42Properties of Expenditure FunctionsHomogeneitya doubling of all prices will precisely double the value of required expenditureshomogeneous of degree oneNondecreasing in pricesE/pi 0 for every good, iConcave in prices43E(p1,)Since his consumption pattern will likely change, ac

33、tual expenditures will be less than Epseudo such as E(p1,)Epseudo If he continues to buy the same set of goods as p*1 changes, his expenditure function would be EpseudoConcavity of Expenditure Functionp1E(p1,) At p*1, the person spends E(p*1,)E(p*1,)p*144Important Points to Note:To reach a constrained maximum, an individual should:spend all available incomechoose a commodity bundle such that the MRS between any two goods is equal to