Production

1. Production

2. Technology • The physical laws of nature and limits of material availability and human understanding that govern what is possible in converting inputs into output.

3. Inputs, Factors of Production • Land (incl. raw materials) • Labor (including human capital) • Capital (physical capital, like machinery and buildings)

4. Production Function • A firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (K) and labor (L). q = f(K, L) • Producing less than the maximum is always possible and all levels of output below the maximum are feasible and define the “production set.”

5. Production Function q q = f(K, L) K L

6. Production set q = f(K, L) q K All points “under” the production function L

7. Production Function and Isoquants q = f(K, L) q In the long run, all combinations of inputs are possible K Isoquants are horizontal cross sections of the production function projected on the base plane. L

8. Short Run, Long Run • Long Run, quantities of ALL inputs used in production can be varied. • Short Run, the quantity of at least one input used in production is fixed. • ALL production takes place in a short run environment. • You can think of the long run as the ability to move from one short run environment to another. • Actual time it takes to make this move depends on many factors, technical, economic and regulatory.

9. The model • Standard basic model to think of production as a function of K and L. • L variable in the short run while K is fixed.

10. Short run, hold K fixed. q = f(K, L) q In the short run, K is fixed and only L can vary K The cross section of the production function at a fixed K is the short run production function L

11. More, fixed K q = f(K, L) q In the short run, K is fixed and only L can vary K The cross section of the production function at a fixed K is the short run production function L

12. Three levels of K q = f(K=K3, L) q q = f(K=K2, L) In the short run, we assume, the quantity of at least one input used --but not all -- is fixed. q = f(K=K1, L) L

13. L constant q = f(K,L=L3) q L and K are just names for inputs. Either one could be fixed in the short run. Just intuitive that K is fixed and L variable in the SR. q = f(K,L=L2) q = f(K,L=L1) K

14. SR and then LR • First we’ll think about the short run, and then turn to the long run.

15. Marginal Physical Product • Marginal Product is the additional output that can be produced by employing one more unit of that input • holding other inputs constant, so a short run concept

16. Marginal Productivity Assumptions • We assume managers are not going to allow employees in the building if they bring total output down. • However, over the range where profit is maximized, marginal products are positive.

17. Increasing and Diminishing Marginal Product (assumes something is fixed) • Empirically, economists find that most production processes exhibit (as L increases from zero): • Increasing Marginal Returns – each worker added causes output to increase by more than the previous worker (workers are not able to gain from specialization, K is fixed) • And then… • Decreasing Marginal Returns –workers added to production add less to output than the previous worker (workers crowd each other as they try to share a fixed amount of capital)

18. Marginal Productivity Assumptions • Because of IMR and DMR, these are possible: • Whether MP is always diminishing or whether it first increases and then diminishes depends on the context of the economic discussion. • In economics classes, we think of increasing marginal returns and then diminishing marginal returns (need this for a U-shaped MC curve).

19. MP Assumptions • As revenue or profit max means producing where MC is rising (MPL is falling), theoretically, we tend to ignore IMR and assume DMR

20. Malthus and Diminishing Marginal Productivity • He argued that population growth meant declining marginal labor productivity • His mistake was holding all else (except labor, i.e. population) constant. • Ignored technological growth. • Productivity was actually growing exponentially, but at such a slow rate that he did not see it. Per Capita Output Watts’s Steam Engine Economic growth of IR first noticed in the 1830s Essay on the Principle of Population, 1sted (1798) Malthus Dies, 1834 Year 1840 1880 1800

21. Effect of Technology • If we think of higher technology as being like having MORE capital, then you can think of the industrial revolution the result of fLK > 0 and a rapid expansion of K.

22. Average Physical Product • Labor productivity is often measured by average productivity.

23. Specific Function • Suppose the production function for tennis balls can be represented by • To construct MPL and APL, we must assume a value for K • let K = 10 • The production function becomes

24. SR Production Function (K = 10) q L

25. Marginal Product • The marginal product function is • When MPL = 0, total product is maximized at L = 80.

26. SR Production Function (K = 10) q Slope of function is MPL at that level of L L

27. Inflection Point • Output where MPL goes from increasing to decreasing (inflection point)

28. SR Production Function (K = 10) At inflection point, MPL is at its highest q LI L

29. Average Product • To find average productivity, we hold K=10 and solve

30. SR Production Function (K = 10) Slope of ray from origin to curve at any L is = APL Slope of this ray =36,000 So APL =36,000 when L= 60 q L LA

31. MPL and APL • In fact, when L = 60, both APL and MPL are equal to 36,000 • Thus, when APL is at its maximum, APL and MPL are equal • So long as a worker hired has a MPL higher than the overall APL, the APL will continue to rise. • If the MPL = APL, • But if a worker hired has a MPL below the overall APL, the APL will fall.

32. MPL and APL LI LA

33. MPL and APL Where the ray is also tangent, MPL = APL

34. Long Run • All mixes of K and L are possible. • Daily decisions about production always have some fixed inputs, so the long run is a planning time horizon.

35. Isoquant Map • Each isoquant represents a different level of output, q0 = f(K0,L0), q1= f(K1,L1) K q1 = 30 q0 = 20 L

36. Marginal Rate of Technical Substitution (TRS, RTS, MRTS) • The slope of an isoquant shows the rate at which L can be substituted for K, or how much capital must be hired to replace one Laborer. K A KA B KB q0 = 20 L LA LB

37. TRS and Marginal Productivities • Take the total differential of the production function: • Along an isoquant dq = 0, so

38. Alternatively:Implicit Function Rule

39. Diminishing TRS • Again, for demand (this time of inputs) to be well behaved, we need production technology (akin to preferences) to be convex. K Which means, the slope rises, gets closer to zero as L increases. And means the TRS falls as L increases. L

40. Diminishing TRS • To show that isoquants are convex (that dK/dL increases – gets closer to zero) along all isoquants) • That is, either: • The level sets (isoquants) are strictly convex • The production function is strictly quasi-concave

41. Convexity (level curves) • dK/dLincreases along all indifference curves • We can use the explicit equation for an isoquant, K=K(L, q0) and find to demonstrate convexity. • That is, while negative, the slope is getting larger as L increases (closer to zero). • But we cannot always get a well defined equation for an isoquant.

42. Alternatively (level curves) • As above, starting with q0 =f(K,L), • So convexity if

43. Convexity (level curves) • And, that is *Note that fK3 > 0 • What of: • fL > 0, monotonacity • fK> 0, monotonacity • fLL < 0, diminishing marginal returns • fKK < 0, diminishing marginal returns • fLK = ?

44. Strict Quasi-Convexity(production function) • Also, convexity of technology will hold if the production function is strictly quasi-concave • A function is strictly quasi-concave if its bordered Hessian • is negative definite

45. Negative Definite (production function) • So the production function is strictly quasi-concave if • 1. –fLfL < 0 • 2. 2fLfKfLK-fK2fLL -fL2fKK > 0 • Related to the level curve result: • Remembering that a convex level set comes from this • We can see that strict convexity of the level set and strict quasi-concavity of the function are related, and each is sufficient to demonstrate that both are true.

46. TRS and Marginal Productivities • Intuitively, it seems reasonable that fLK should be positive • if workers have more capital, they will be more productive • But some production functions have fKL < 0 over some input ranges • assuming diminishing TRS means that MPL and MPK diminish quickly enough to compensate for any possible negative cross-productivity effects

47. TRS and MPL and MPK • Back to our sample production function: • For this production function

48. IMR and DMR vs. NMR • Pull out a few terms • If K = 10, then MPL = 0 at L=80

49. IMR vs. DMR • Because • fLL> 0 and fKK > 0 if K*L < 400 • fLL< 0 and fKK < 0 if K*L > 400 • If K = 10, then inflection point at L=40

50. Cross Effect • Cross differentiation of either of the marginal productivity functions yields • fLK > 0 if KL < 533 • fLK < 0 if KL > 533 • If K = 10 • fLK> 0 when L < 53.3 • fLK< 0 when L > 53.3