Chapter 21Profit Maximization and Derived DemandSlide 3Marginal Revenue ProductMarginal ExpenseSlide 6An Alternative DerivationSlide 8Slide 9Price Taking in the Output MarketComparative Statics of Input DemandSingle-Input CaseSlide 13Slide 14Single-Input DemandSlide 16Two-Input CaseSlide 18Substitution EffectSlide 20Output EffectSlide 22Substitution and Output EffectsCross-Price EffectsMathematical DerivationSlide 26Constant Output Demand FunctionsSlide 28Output EffectsSlide 30Slide 31Slide 32Decomposing Input DemandSlide 34Slide 35Slide 36Slide 37Slide 38Slide 39Responsiveness of Input Demand to Changes in Input PricesSlide 41Slide 42Slide 43Elasticity of Demand for InputsSlide 45Slide 46Slide 47Slide 48Slide 49Competitive Determination of Income SharesSlide 51Factor Shares and the Elasticity of SubstitutionMonopsony in the Labor MarketSlide 54Slide 55Slide 56Monopsonistic HiringSlide 58Monopoly in the Supply of InputsSlide 60Slide 61Important Points to Note:Slide 63Slide 64Slide 65Slide 66Chapter 21FIRMS’ DEMANDS FOR INPUTSCopyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONSEIGHTH EDITIONWALTER NICHOLSONProfit Maximization and Derived Demand•A firm’s hiring of inputs is directly related to its desire to maximize profits–any firm’s profits can be expressed as the difference between total revenue and total costs, each of which can be regarded as functions of the inputs used = TR(K,L) - TC(K,L)Profit Maximization and Derived Demand•First-order conditions for a maximum are0KTCKTRK0LTCLTRL–the firm should hire each input up to the point at which the extra revenue yielded from one more unit is equal to the extra costMarginal Revenue Product•The marginal revenue product (MRP) from hiring an extra unit of any input is the extra revenue yielded by selling what that extra input produces MRP = MR  MPMarginal Expense•If the supply curve facing the firm for the inputs it hires are infinitely elastic at prevailing prices, the marginal expense of hiring a worker is simply this market wage•If input supply is not infinitely elastic, a firm’s hiring decision may have an effect on input pricesMarginal Expense•For now, we will assume that the firm is a price taker for the inputs it buysTC/K = vTC/L = w•The first-order conditions for profit-maximization becomeMRPK = vMRPL = wAn Alternative Derivation•The Lagrangian expression associated with a firm’s cost-minimization problem isL = vK + wL + [q0 - f(K,L)]•First-order conditions areL/K = v - (f /K) = 0L/L = w - (f /L) = 0L/ = q0 - f (K,L) = 0An Alternative Derivation•The first two equations can be written asvMPKfKwMPLfLAn Alternative Derivation•Since  can be interpreted as marginal cost in this problem, we haveMC  MPK = vMC  MPL = w•Profit maximization requires that MR = MC so we haveMR  MPK = MRPK = vMR  MPL = MRPL = wPrice Taking in theOutput Market•If a firm exhibits price-taking behavior in its output market, MR = P•This means that at the profit-maximizing levels of each inputP  MPK = vP  MPL = w–sometimes P multiplied by an input’s MP is called the value of marginal productComparative Statics ofInput Demand•We will focus on the comparative statics of the demand for labor–the analysis for capital would be symmetric•For the most part, we will assume price-taking behavior for the firm in its output marketSingle-Input Case•It is likely that L/w < 0–this is based on the presumption that the marginal physical product of labor declines as the quantity of labor employed rises•a fall in w must be met by a fall in MPL for the firm to continue maximizing profits (because P is fixed) –this argument is strictly correct for the case of one inputSingle-Input Case•Taking the total differential ofP  MPL = w yieldsdwwLLMPPdwLwLLMPPL1LMPPwLL/1Single-Input Case•If we assume that MPL /L < 0 (MPL falls as L increases), we haveL/w < 0•A fall in w will cause more labor to be hired–more output will be produced as wellSingle-Input Demand•Suppose that the number of truffles harvested in a particular forest isLQ 100•Assuming that truffles sell for $50 per pound, total revenue for the owner isLQPTR 000,5Single-Input Demand•Marginal revenue product is given by2/1500,2LLTR•If truffle searchers’ wages are $500, the owner will determine the optimal amount of L to hire by2/1500,2500 L25LTwo-Input Case•If w falls, both L and K will change as a new cost-minimizing combination of inputs is chosen•When K changes, the entire MPL function changes–labor has a different amount of capital to work with•However, we still expect that L /w < 0Two-Input Case•When w changes, we can decompose the total effect on the quantity of L hired into two components–substitution effect–output effectSubstitution Effect•If output is held constant and w falls, there will be a tendency to substitute L for K in the production process–cost-minimization requires that RTS = w/v–a fall in w means that RTS must fall as well•because isoquants exhibit a diminishing RTS, the cost-minimizing level of labor hired risesSubstitution EffectLKq0-AThe substitution effect is shown holding output constant at q0K1L1vw1 slope -BAs w falls, the firm will substitute L for K in the production processK2L2vw2 slope Output Effect•A change in w will shift the firm’s expansion path•This means that the firm’s cost curves will also shift–a drop in w will lower MC and lead to a higher level of output•This increase in output will lead to a higher level of L being demandedThe output effect is shown holding relative input prices constantvw2 slope vw2 slope Output EffectLKq0-BK2L2q1CK3L3-Since a drop in w leads to a decline in MC, optimal output will rise and the firm will demand more LSubstitution andOutput EffectsLKq0-AK1L1-BK2L2q1Both the substitution effect and the output effect lead to a rise in the quantity of L demanded when w fallsCK3L3-Cross-Price Effects•No definite statement can be made about how capital will change when w changes•The substitution and output effects move in opposite directions–a fall in w