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We shall assume that all relevant functions are well

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We shall assume that all relevant functions are “well-behaved,” that we are con- cerned with profit-maximization within a single period, that there is only a single out- put ( X , measured in kg) and a single variable input ( L, measured in labour-days). The price of the output ( P X ) is in $/kg and the price of the input ( P L ) is in $/labour-day. The condition for profit-maximization in the output market is that the firm’s Marginal Revenue (the increment to Total Revenue from the last unit produced and sold) is equal to its Marginal Cost (the increment to Total Cost from the last unit produced and sold): MR ∆TR/∆ Q X = ∆TC/∆ Q X MC. (M.4.3) The condition for profit-maximization in the input market is that the firm’s Marginal Revenue Product of Labour (the increment to Total Revenue from sale of the addition- al output resulting from the last labour-day purchased) is equal to its Marginal Factor M4-6 MATH MODULE 4: USING ECONOMIC UNITS
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MATH MODULE 4: USING ECONOMIC UNITS M4-7 Cost of Labour (the increment to Total Cost resulting from the last labour-day pur- chased): MRP L ∆TRP L /∆ L = ∆TFC L /∆ L MFC L . (M.4.4) These two conditions look quite distinct, but we can show that they are in fact equiv- alent: two different ways of expressing the same profit-maximization condition. If we define the Marginal Product of Labour (MP L = ∆ Q X /∆ L ) as the increment to output resulting from the last labour-day purchased, then (as outlined in Chapters 10 and 14) the relations in Equations M.4.5 and M.4.6 hold: MC = MFC L ÷ MP L , or MFC L = MC MP L . (M.4.5) ($/kg) = ($/ L- day) ÷ (kg/ L- day) ($/ L- day) = ($/kg) (kg/ L- day) In the left-hand version of this relation, the labour-days cancel, while in the right-hand version, the kilograms cancel. Notice also that MRP L = MR MP L , or MR = MRP L ÷ MP L . (M.4.6) ($/ L- day) = ($/kg) (kg/ L- day) ($/kg) = ($/ L- day) ÷ (kg/ L- day) Here, in the left-hand version of this relation, the kilograms cancel, while in the right- hand version, the labour-days cancel. Using Equations M.4.3 and M.4.5, we have as our profit-maximizing condition MR = MC = MFC L / MP L . (M.4.7) Rearranging terms in Equation M.4.7 and comparing with Equation M.4.6, we get MR MP L MRP L = MFC L . (M.4.8) Thus, beginning with the profit-maximization condition in the output market, we have generated the profit-maximization condition in the input market: the two conditions are equivalent. In the special case where the firm is a perfect competitor in the output market and faces a horizontal demand curve for its output, MR = P X . Hence in this case, MRP L = MR x MP L = P X MP L . (M.4.9) We give the term P X x MP L the special name, Value of the Marginal Product of Labour (VMP L ). For a firm facing a downward -sloping demand curve for its output, MR < P X . If the firm is a perfect competitor in the input market and faces a horizontal supply curve for its input, then MFC L = P L . Hence MC = MFC L /MP L = P L /MP L , or MC MP L = P L . (M.4.10) If the firm faces an upward-sloping input supply curve, then MFC L > P L .
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