Daniel l. Rubinfeld

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138 Part 2 Producers, Consumers, and Competitive Markets 4 Individual and Market Demand 11. b. Suppose that she is given a tax rebate of 515000 to ease the effect of the tax~ What would her consumption of food be now c. Is she better or worse off when given a rebate equal to the sales tax payments Discuss. . Suppose that you are the consultant to an agncultural cooperative that is deciding whether members should cut their production of cotton in half next yea.I. Tl:e cooperate wants your advice as to whether this will that cotton (c) and watermelons (lUJ both compete for ag~icultural land in the South, you estimate the demand tor cotton to be where Pc is the price of cotton, V,· the price of watermelon, and i income. Should you support or oppose the plan Is there any additio~1a.l.informati~n that would help you to provide a definitive answer. This appendix presents a mathematical treatment of the basics of demand theory. Our goal is to provide a short overvie'w of the theory of demand for stud~nts who have some familiarity with the use of calculus~ To do this, we will explain and then apply the concept of constrained optimization. Maximization The theory of consumer beha\'ior is based on the assumption that consumers maximize utility subject to the constraint of a limited budget. We saw in Chapter 3 that for each consumer, we can define a lltility jllllctioll that attaches a level of utility to each market basket. We also saw that the IIlmginalutility of a good is defined as the change in utility associated with a one-unit increase in the consumption of the good. Using calculus, as \ve do in this appendix, marginal utility is measured as the utility change that results from a very small increase in consumptiorL Suppose, for example, that Bob's utility function is given by U(X, Y) = log X + log Y, ,,>'here, for the sake of generality, X is now used to represent food and Y represents clothing. In that case, the marginal utility associated with the additional consumption of X is given by the pm·tial derivative oj the utility jUllction witiz respect to good K Here, MU x' representing the marginal utility of good X, is aiven by tJ ~ In §3J, we explain that a utility function is a formula that assigns a level of utility to each market basket In §3~ 2, marginal utility is described as the additional satisfaction obtained by consuming an additional a~ount of a good CiU(X, Y) ax a(log X + log Y) ax 1 X In the following analysis, \ve will assume, as in Chapter 3, that while the level of utility is an increasing function of the quantities of goods consumed, marginal utility decreases with consumption. When there are h'\'o goods, X and Y, the consumer's optimization problem may thus be written as Maximize U(X, Y) (A4.1) subject to the constraint that all income is spent on the hvo goods: (A4.2) Here, U( ) is the utility function, X and Y the quantities of the two goods purchased, P x and Py the prices of the goods, and I income. 1 To determine the individual consumer's demand for the two goods, we choose those values of X and Y that maximize (A4.1) subject to (A4.2)~ When we know simplify the mathematics, we assume that the utility function is continuous (with continuous and that goods are infinitely divisible
Part 2 Producers, Consumers, and Competitive Markets 4 Individual and Market Demand the particular form of the utility function, we can solve to find the consumer's demand for X and Y directly, However, even if we write the utility flmction in its general form U(X,y), the technique of cOllstraincd optillli2ntioll can be used to describe the conditions that must hold if the consumer is maximizing utility rewritten as MUx The three equations in (A4A) can be AP\ method of Lagrange multipliers Technique to maximize or minimize a function subject to one or more constraints. Lagrangian Function to be maximized or minimized, plus a variable (the Lagrange multiplier) multiplied by the constraint. The method of Lagrange multipliers is a technique that can be used to maximize or minimize a function subject to one or more constraints. Because vve will use this technique to analyze production and cost issues later in the book, we will provide a step-by-step application of the method to the problem of finding the consumer's optimization given by equations (Atl) and (A4.2). First, we write the Lagrangian for the problem. The Lagrangian is the function to be maximized or minimized (here, utility is being maximized), plus a variable which we call A times the constraint (here, the consumer's budget constraint). We will interpret the meaning of A in a moment. The Lagrangian is then cD = U(X,Y) - A(PxX + PrY 1) Note that we have written the budget constraint as P.\X + PrY - I = 0 (A4.3) Le., as a sum of terms equal to zero. Vie then insert this sum into the Lagrangian. If we choose values of X and Y that satisfy the budget constraint, then the second term in equation (A4.3) will be zero. Maximizing will therefore be equivalent to maximizing U(X,Y), By differentiating cD with respect to X, Y, and A and then equating the derivatives to zero, we can obtain the necessary conditions for a maximum The resulting equations are MU r = API PxX + PrY = I Now vYe, c~n solve these three equations for the three unknowns. The resultirw '"a lues ot X '1' and Yare .. the . solution to the consumer's optimization pI'oble m, . Tl 1ev ~ are t I 1e uti Ity-maxlmIzmg quantities. " The Marginal The third equation , above is the consumer's budaet 1::> C011stI'al' n t WI 'tl 1 VV l' 1IC 1 we sta.rted, Tl~e .frrst two equ~tions ~e~l us that each good will be consumed up to the pO.ll1t a,t \\ hrch the margmal. util~ty from consumption is a multiple (A) of the ~nce ot the ~ood, To see the ~mphcation of this, we combine the first two conditions to obtam the eqzwillwrgllwi prillciple: A = MUx(X,Y) P x MUy(X,Y) Py (A4.5) In other words, the m,arginal utility of each aood divided by its pl'I'Ce' tI, D b f"', "0 - IS le saIne. o e op muzlr:g, t zc COIlSUlIlcr IIlllst bc gettillg the snlllc lltilit!/ frallz the lnst dollnr spellt btl _ COIlSZII/llll\!: ,,' cztizer X 01 Y If this were not t11e case , co nsummg ~ - . more ot ' one good and less of the other ,,,'ould increase utility. . To ch~rac~erize the individual's optimum iI~ more detail, 'we can rewrite the mformatlOn m (A±.5) to obtain MUx(X,Y) MUy(X,Y) (A4.6) (icD aX - = MUx(X Y) APx = 0 "' aeI) ~ = MUy(X,Y) - APr = 0 dl acD - = p"X + Pi Y - I = 0 aA ., (A4.4) Here as before, MU is short for IIlnrgillnllltility: In other words, MUx(X,Y) == aU(X,y)/ax, the change in utility from a very small increase in the consumption of good X. C These conditions are necessary for an "interior" solution in which the consumer consumes positive amounts of both goods The solution, however, could be a "corner" solution in which all of one good and none of the other is consumed In other words, the ratio of thc IIlnrgillnlutilities is equnl to tlze ratio of the plices. Marginal Rate of Substitution We can use equation (A4.6) to see the link between utility functions and indifference m curves ' that was spelled . out' m eh ap t er J, all '") A n mdl ' 'f ference curve represents . , arket baskets that gIve the consumer the same level of utilitv. If U* is a fixed Utihty y level, the indifference curve that corresponds to that utility _ level is aiven b b U(X,Y) = II' t A~ the market baskets are changed by adding small amounts of X and subractmg small amounts of Y, the total change in utility m.ust equal zero. -iherefOl'e MUx(X,y)dX + MUr(X,Y)dY = dU* = 0 (A4.7) In §3.3, \\'e show that the marginal rate of substihttion is equal to the ratio of the marginal utilities of the two "oods being consumed. '"
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Daniel l. Rubinfeld

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