Economic Analysis Economic Analysis for Businessfor Business

Session XVII: Production Session XVII: Production Function and Factor MarketsFunction and Factor MarketsInstructorInstructorSandeep BasnyatSandeep Basnyat98418922819841892281Sandeep_basnyat@yahooSandeep_basnyat@yahoo.com.com

Recall: Profit MaximizationRecall: Profit MaximizationAs stated before,Firms will hire extra labour and capital until:

MRPL = w and MRPK = r

PxMPL = w and PxMPK = r

Dividing,

MPL / MPK = w / r

MPL / w = MPK / r ……………..(i)

Equation (i) is known as efficiency condition. Or

Least cost combination input.

Meaning: if a firm is maximizing in the above condition, then it is efficiently operating.

Numerical ProblemsNumerical Problems

1) Consider the Production function: Q = 3LK + 2KPrice per unit of K and L are 76 and 6 respectively. Per unit selling price for output is 2. Find:

a)Amount of factors demandedb)Amount of product producedc)Amount of profit or Loss generated

Solution Numerical Prob. Solution Numerical Prob. (1)(1)Q = 3LK + 2KMPK = 3L+2MPL = 3KProfit Maximizing condition,P x MPL = w and P x MPK = r2(3K) = 6 and 2 (3L+2) = 76K = 1 and L = 12Q = 3(12)(1) + 2(1) = 38Profit = TR –TC = 2(38) –[6(12) + 76(1)] = - 72

(Loss)

Numerical 2Numerical 2Assume the production function:

Q = 100K0.5L0.5

a)If the price of labour and capital are 4 and 2 respectively, find the efficient input combination (least cost 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 2Solution to Numerical 2a) Q = 100K0.5L0.5

MPK = 50(L0.5 / K0.5)MPL = 50(K0.5 / L0.5)Efficiency condition,MPL / MPK = w / r

K = (w/r)L …………….(i)Substituting the value of K in Production function,1000 = 100 ((w/r)L )0.5L0.5 = 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 = 100 L (4/4) 0.5 ; L = 10 and K = 10c) The firm responded to the higher price of capital

by substituting labour for capital

Numerical 3Numerical 3Suppose 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 = 2KLMRTS = MPL / MPK = K / 2LK / 2L = 10 / 20...therefore K=L

To produce 27000 unitsK²L = 27000....therefore K³ = 27000, so K=30, L=30

Numerical 3Numerical 3Suppose 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 iswL + rK = (10)(30) + (20)(30) = $900

Numerical 3Numerical 3Suppose 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 3Numerical 3Suppose 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:wL + rK = 1200We know that L=K, w=10, r=20, so (10)(L) + (20)(L) = 1200, therefore L=40, K=40

Numerical 3Numerical 3Suppose 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 MarketSummary of Factor Market

Factor marketFactor marketDerived 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◦VMPK = P x MPK = MR (or MC) x MPK = r

MPL and VMPL ExampleMPL and VMPL Example

L (number of workers)

The VMPL curve

0

1,000

2,000

3,000

4,000

5,000

$6,000

0 1 2 3 4 5

$2,500

Supply Curve for factor market-eg. Supply Curve for factor market-eg. LabourLabour

W

L

S1

W1

L1

W2

L2

Equilibrium in the Factor MarketEquilibrium in the Factor Market

W

L

D

S

W1

L1

Linkages Among the Factors of Linkages Among the Factors of ProductionProductionIn 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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