Economic Analysis for Business Session XVII Production Function

Economic Analysis for Business Session XVII: Production Function and Factor Markets Instructor Sandeep Basnyat 9841892281 Sandeep_basnyat@yahoo. co m

Recall: Profit Maximization As stated before, Firms will hire extra labour and capital until: MRPL = w and MRPK = r Px. MPL = w and Px. MPK = r Dividing, MPL / MPK = w / r MPL / w = MPK / r ……………. . (i) Equation (i) is known as efficiency condition. Or Least combination input. Meaning: if a firm is maximizing in the above condition, then it is efficiently operating.

Numerical Problems 1) Consider the Production function: Q = 3 LK + 2 K Price per unit of K and L are 76 and 6 respectively. Per unit selling price for output is 2. Find: a) Amount of factors demanded b) Amount of product produced c) Amount of profit or Loss generated

Solution Numerical Prob. (1) Q = 3 LK + 2 K MPK = 3 L+2 MPL = 3 K Profit Maximizing condition, P x MPL = w and P x MPK = r 2(3 K) = 6 and 2 (3 L+2) = 76 K=1 and L = 12 Q = 3(12)(1) + 2(1) = 38 Profit = TR –TC = 2(38) –[6(12) + 76(1)] = - 72 (Loss)

Numerical 2 Assume the production function: Q = 100 K 0. 5 L 0. 5 a) If the price of labour and capital are 4 and 2 respectively, find the efficient input combination (least combination input) for producing 1000 units of output. b) How would the input mix change if the price of capital increased to r = 4 and the firm still wanted to produce 1000 units of output? c) Interpret the result of (b).

Solution to Numerical 2 a) Q = 100 K 0. 5 L 0. 5 MPK = 50(L 0. 5 / K 0. 5) MPL = 50(K 0. 5 / L 0. 5) Efficiency condition, MPL / MPK = w / r K = (w/r)L ……………. (i) Substituting the value of K in Production function, 1000 = 100 ((w/r)L )0. 5 L 0. 5 = 100 L (4/2) 0. 5 L = 7. 07. Substituting the value of L in eq. (i), K = 14. 14 b) When k = 4, 1000 = 100 L (4/4) 0. 5 ; L = 10 and K = 10 c) The firm responded to the higher price of capital by substituting labour for capital

Numerical 3 Suppose that a firm’s production function is given by Q = K² L. Further suppose that w = $10 and r = $20 a) Suppose the firm wants to produce 27, 000 units of output. What is the most efficient combination of labor and capital? Solution: MPL = K², MPK = 2 KL MRTS = MPL / MPK = K / 2 L = 10 / 20. . . therefore K=L To produce 27000 units K²L = 27000. . therefore K³ = 27000, so K=30, L=30

Numerical 3 Suppose that a firm’s production function is given by Q = K² L. Further suppose that w = $10 and r = $20 b) Suppose that the firm wants to produce 27, 000 units of output in the most efficient way possible. How much does the firm spend? Solution: Budget constraint is w. L + r. K = (10)(30) + (20)(30) = $900

Numerical 3 Suppose that a firm’s production function is given by Q = K² L. Further suppose that w = $10 and r = $20 c) Suppose that the firm wants to produce 27, 000 units of output and has exactly 10 units of capital in hand. In this situation, how many labor has to be employed? Solution: 27000 = 10²L, so L = 270 units of labour

Numerical 3 Suppose that a firm’s production function is given by Q = K² L. Further suppose that w = $10 and r = $20 d)Suppose that the firm wants to spend exactly $1, 200. What is the most efficient combination of labor and capital ? Solution: w. L + r. K = 1200 We know that L=K, w=10, r=20, so (10)(L) + (20)(L) = 1200, therefore L=40, K=40

Numerical 3 Suppose that a firm’s production function is given by Q = K² L. Further suppose that w = $10 and r = $20 e) Suppose that the firm spends exactly $1200 in the most efficient way possible. How much output can the firm produce? Solution: Substitute K=L=40 into production function (40)²(40) = 64000 units

Summary of Factor Market

Factor market Derived demand – derived from a firm’s decision to supply a good in another market. Production function- provides relationship MRPL/MRPK (VMPL/VMPK) curves demand curve for factor market: determine additional labour or capital hired Profit maximizing or efficiency condition ◦ VMPL = P x MPL = MR (or MC) x MPL = W

MPL and VMPL Example The VMPL curve $6, 000 5, 000 4, 000 3, 000 $2, 500 2, 000 1, 000 0 0 1 2 3 4 L (number of workers) 5

Supply Curve for factor market-eg. Labour W S 1 W 2 W 1 L 2 L

Equilibrium in the Factor Market W S W 1 D L 1 L

Linkages Among the Factors of Production In most cases, factors of production are used together in a way that makes each factor’s productivity dependent on the quantities of the other factors. Example: an increase in the quantity of capital ◦ The marginal product and rental price of capital fall. ◦ Having more capital makes workers more productive, MPL and W rise.

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