1. Chapter 6 Production Theory and Estimation

3. Ch. 6: Production Function and Estimation Overview 1. The main objective any business is to maximize profits (Value Maximization). Profits = Revenues - Costs (Value) (P, Ad. Quality) (Efficiency)

4. Guiding Principles of Optimal Resource Use 1.The Single Case : Q=f(L) Decision Rule: MRP i = MFC i => # 4, MRP i => marginal revenue product (MR) input i MFC i => marginal factor cost (MC) input i What if MRP i > MFC i ? Should we use more or less of the resource? Why?

5. Guiding Principles of optimal resource use-cont. 2. The multiple inputs case Q=f(L, K) a. Least cost combination rule=> MP L /w=MP K /r  #10 (This is a necessary condition for optimal input use) What if MP L /w < MP K /r? Use more or less labor? Why? b. Given the production function Q=f(L, K), the cost of hiring workers (w) and the cost of capital (r), and the firm’s cost outlay (C= wL+rK). The objective function of the firm may be to: Maximize Q s.t. C=wL+rK, or Minimize C=wL +rK s.t. the constraint of producing a fixed Q

6. Sufficient Condition for Choosing the Optimal Combination of L and K Optimal input use: MP L /MP K =w/r (This rule a sufficient condition for optimal use). What if MP L /MP K use more what? L K k2k2 L2L2 k1k1 L1L1 Q2Q2 Q1Q1 Q3Q3 k3k3 L3L3 C1C1 C2C2 C3C3

7. 1.Production is the process of transforming inputs (resources) into output of goods and services. Inputs include land, labor including entrepreneurial talent, and capital.

8. 2.a) Fixed inputs are those that can not readily be changed during the period under consideration. Note: In the SR, we have both fixed & variable inputs Variable inputs are those whose use changes as needed (Labor, utilities, etc.) b) Short-run is the time period during which at least one input is fixed. Long-run is the time period during which all inputs are variable.

9. 3.A production Function shows the relationship between inputs and some output. Q books =f(workers,machine,paper,ink, others) Q graduation =f(faculty,facilities,personnel, etc)

10. 4.The short-run production function with one variable input(labor). a) The total product of labor (Q) shows the different levels of output produced by different quantities of labor(capital is fixed).

11. Measures of Productivity The average product of labor(AP L ) measures output per worker AP L = Q/L measures production efficiency The marginal product of labor(MP L ) measures the output produced by the last unit of labor. ;

12. - The output elasticity of labor (E L ) measures the percentage change in output as a result of one percent change in the quantity of labor.

13. b) Short-run production of potatoes on one acre of land using labor as a variable input.

14. L Q(lb) AP L =Q/L MP L = Q/ L E L =% Q/% L 0 0 - - - 1 3 3.0 3 1.00 2 8 4.0 5 1.25 3 12 4.0 4 1.00 4 14 3.5 2 0.57 5 14 2.8 0 0.00 6 12 2.0 -2-1.00 Notice that the MP L is diminishing beyond the 2nd unit of labor. Why?

15. 5.a) Some measures of productivity: -Output per worker (AP L ) =Q/L -Marginal productivity per worker(MP L )=ΔQ/ΔL or dQ/dL

16. b) Factors which influence productivity gains:  Increase in the amount of capital available per worker(K/L)  Improvements in technology(the level of applied scientific knowledge).  Changes in institutional rule(union work rules  Changes in government statutes.

17.  Demographic trends may be the driving force behind technological changes, work- rule modification(fast food tech, flexible work hours). c) Importance of productivity gains: -Increased profits for firms. -Better standards of living for society.

18. 6.The three stages of SR Production See, p. 240 a) Stage 1-The fixed factor of production is underutilized. As we add more of the variable input to the fixed input, productive efficiency increases. AP increasing, MP>AP, E L >1.

19. b) Stage 2-Represents a rational boundary of production. In our example, it occurs between 3 and 5 units of labor. c) Stage 3-Total product is declining. Even if wage per hour is 0, a rational producer will not use more than 5 units of labor in our example (Refer to previous table). Notice that the output elasticity of labor is negative in this stage i.e. E L < 0

20. Stages SR Production Function AP, MP Labor AP L Stage I Stage II Stage III L1L1 L2L2 Stage I:0 to L 1 Stage II: L 1 to L 2 Stage III: above L 2 MP L

21. 7.An optimal labor size will be employed when the marginal revenue product (MRP L ) and the marginal factor cost of labor (MFC L ) are equal =>MRP L = MFC L MRP L = MP L * product price (=>D L ) MFC L = the additional cost of hiring an additional worker (S L )

22. Rule: MRP L =MFC L Labor Units MP L MR=Price MRP L MFC L 2.5 4 $10 $40 $20 3.0 3 10 30 20 3.5 2 10 20 20 4.0 1 10 10 20 4.5 0 10 0 20

23. Optimal Units of Labor MRP MFC Labor MFC L MRP L 3.5 $20

24. 8.A production function with two variable inputs Q = f(L,K). a) Iso-quant curve is a curve which shows the various combinations of labor and capital which yield the same level of output (see p. 243). b) MRTS LK is the rate at which labor is substituted for capital. It represents the slope of the iso-quant curve (p. 276). MRTS LK = - K/ L = MP L /MP K

25. Multiple Inputs Production Function c) Ridge lines are lines which separate the relevant (negatively sloped) from the irrelevant portions of the iso-quants. d) Iso-cost line is a line which shows the various combinations of labor and capital which entail the same total cost Total cost(expenditures) = exp. on labor and exp on capital

26. Guiding Principles of optimal resource use-cont. 2. The multiple inputs case Q=f(L, K) a. Least cost combination rule=> MP L /w=MP K /r  #10 (This is a necessary condition for optimal input use) What if MP L /w < MP K /r? Use more or less labor? Why? b. Given the production function Q=f(L, K), the cost of hiring workers (w) and the cost of capital (r), and the firm’s cost outlay (C= wL+rK). The objective function of the firm may be to: Maximize Q s.t. C=wL+rK, or Minimize C=wL +rK s.t. the constraint of producing a fixed Q

27. Sufficient Condition for Choosing the Optimal Combination of L and K Optimal input use: MP L /MP K =w/r (This rule a sufficient condition for optimal use). What if MP L /MP K use more what? L K k2k2 L2L2 k1k1 L1L1 Q2Q2 Q1Q1 Q3Q3 k3k3 L3L3 C1C1 C2C2 C3C3

28. MRTS LK =ΔK/ΔL=MP L /MP K K L Slope of Q 1 at b => ΔK/ΔL=MP L /MP K =w/γ Q1Q1 b C1C1 L1L1 K1K1 Note: K 1 and L 1 are optimal mix of K and L C2C2 Slope of Isocost equation=> K= C/r – wL/r =>w/r taking the derivative of K w.r.t. L

29. C = wL + rK, where w = wage rate r = price of capital (interest rate) C = total cost outlay From the above, the isocost equation can be written as: K = C/r –(w/r)L

30. Given C=$60; w=10; r=$15 Write the isocost equation in which you express K in terms of L. 60= 10L + 15K K = 60/15 - 10/15L=> K= 4 – 2/3L K= 4 – 2/3L=> Iso-cost equation Slope of Isoquant => dK/dL = -2/3=w/r L K 60/15 = 4 6

31. 9.An optimal input combination is obtained when a mix of resources (inputs) for which the MRTS is equal to the slope of the iso-cost line(the input price ratio). MRTS LK =- K/ L=>MP L /MP K => slope of isoquant dK/dL = w/r =slope of isocost Optimal point=> at MP L /MP K =w/r

32. Optimal Input Mix of K and L E F L K Q1Q1 Notice that only point D satisfies the optimal input combination. Why? P. 250 K L C1C1 C2C2 At point D, 1. MP L /W=MP K /r =>LCC 2. MP L /MP K =w/r => sufficient condition for optimal combination D

33. A firm can minimize the cost of producing a given level of output or maximize the output for given cost outlay when MRTS = W/r. This equilibrium condition yields both production and cost efficiency.

34. 10.An expansion path is a line which traces the least cost combination of inputs (i.e. MP L /w=MP K /r). P. 250 Note this takes place in the long-run when all production inputs can vary.

35. 11.The effect of a decrease in the wage rate, the cost of capital remaining the same is the use of a labor intensive technique of production and vice-versa.

36. 12.Returns to scale refer to what happens to output if all the variable inputs are changed by the same proportion (%) in the LR?

37. There are three possibilities: a) Output can increase by the same percentage as inputs => constant returns to scale. b) Output can increase by a greater percentage than the increase in inputs => increasing returns to scale. c) Output can increase by a smaller percentage than the increase in inputs => decreasing returns to scale.(see p.255)

38. 13.a) The Cobb-Douglas production (Q =AK a L b ) is used to estimate production relationships. b) Useful properties of the Cobb- Douglas production: - MP K and MP L depend on the quantities of capital and labor used, respectively. - The exponents of capital (a) and labor (b) represent the output elasticity of capital(E K ) and labor (E L ), respectively. E K = a; E L = b.

39. - The Cobb-Douglas production function can be estimated by linear regression by transforming it into: lnQ = lnA + a lnK + b lnL - The Cobb-Douglas production function can be easily extended to determine the contribution of several inputs.

40. Returns to Scale from Cobb- Douglas Production Function The sum of the exponents of the C-D represent returns to scale Given Q =AK α L β,  If α + β= 1, the production function exhibits constant returns to scale  If α + β> 1, the production function exhibits increasing returns to scale  If α + β< 1, the production function exhibits decreasing returns to scale