1 Optimal Combination of Inputs Now we are ready to answer the question stated earlier, namely, how to determine the optimal combination of inputs As was said this optimal combination depends on the relative prices of inputs and on the degree to which they can be substituted for one another This relationship can be stated as follows: MP L /MP K = P L /P K (or MP L /P L = MP K /P K )

2 Optimal Combination of Inputs: An Example Let’s consider a company that wants to minimize the cost of producing a given output per hour Q Number of workers used per hour is L and number of machines used per hour is K and Q = 10(LK) 0.5. Wage rate is $8 per hour, and the price of a machine is $2 per hour How many workers and machines the company should use, if the company wants to produce 80 units of the output per hour?

3 Optimal Combination of Inputs Example Continues Q = 10(LK) 0.5 Calculate the marginal products: MP L = 0.5(10)K 0.5 L -0.5 = 5(K/L) 0.5 MP K = 0.5(10)L 0.5 K -0.5 = 5(L/K) 0.5 Thus, if MP L /P L = MP K /P K

4 Optimal Combination of Inputs Example Continues Multiplying both sides of the equation by (K/L) 0.5 we get 5K/8L = 5/2, which means that K = 4L. And since Q = 80, 10(LK) 0.5 = 80 10[L(4L) 0.5 = 80  L = 4 and K = 16

5 Isocost Curves Example 3 Assume P L =$100 and P K =$200

6 Isocost Curve and Optimal Combination of L and K Isocost and isoquant curve for inputs L and K 5 10 L K “Q 52” 100L + 200K = 1000

7 Optimal Levels of Inputs The optimality conditions given in the previous slides ensure that a firm will be producing in the least costly way, regardless of the level of output But how much output should the firm be producing? Answer to this depends on the demand for the product (like in the one input case as well)

8 Optimal Levels of Inputs continued The earlier rule, MRP = MLC, can be generalized: a firm in competitive markets should use each input up to the point where P i = MRP i where P i = price of input i MRP i = marginal revenue product of input i

9 So in the two input case, firm’s optimality condition is P X = MRP X and P Y = MRP Y A profit maximizing firm will always try to operate at the point where the extra revenue received from the sale of the last unit of output produced is just equal to the additional cost of producing this output. This is same as MR = MC

10 Units of capital (K) O Units of labor (L) Expansion path TC = £ TC = £ TC = £ The long-run situation: both factors variable Expansion Path: the locus of points which presents the optimal input combinations for different isocost curves

11 Returns to Scale Let us now consider the effect of proportional increase in all inputs on the level of output produced To explain how much the output will increase we will use the concept of returns to scale

12 Returns to Scale continued If all inputs into the production process are doubled, three things can happen: output can more than double increasing returns to scale (IRTS) output can exactly double constant returns to scale (CRTS) output can less than double decreasing returns to scale (DRTS)

13 Returns to Scale An Example: In this production process we are experiencing increasing returns to scale

14 Reasons for Increasing or Decreasing Returns to Scale: Often we can assume that firms experience constant returns to scale: for example doubling the size of a factory along with a doubling of workforce and machinery should lead to a doubling of output why could a greater (or smaller) than proportional increase occur?

15 Reasons for Increasing Returns to Scale: Division of labor (specialization) increased labor productivity Indivisibility of machinery or more sophisticated machinery justified increased productivity Geometrical reasons Decreasing returns to scale can result from certain managerial inefficiencies: problems in communication increased bureaucracy

16 Measurement of Returns to Scale Coefficient of output elasticity E Q = So if, E Q > 1, increasing returns E Q = 1, constant returns E Q < 1, decreasing returns percentage change in Q percentage change in all inputs

17 Measurement of Returns to Scale continued Multiplying the coefficients of the production function: If original production function is Q = f(X,Y) and if the resulting equation after the multiplication of inputs by k is hQ = f(kX, kY) where h presents the magnitude of increase in production

18 Then, if h > k, increasing returns h = k, constant returns h < k, decreasing returns

19 Graphically, the returns to scale concept can be illustrated using the following graphs Q X,Y IRTS Q X,Y CRTS Q X,Y DRTS

20 Constant Returns to Scale Units of capital (K) Units of labor (L) a b c R

21 Increasing Returns to Scale (beyond point b) Units of capital (K) Units of labor (L) a b c R 700

22 Decreasing Returns to Scale (beyond point b) Units of capital (K) Units of labor (L) a b c R