Understanding Production Functions: Short vs. Long Run Dynamics

1. Chapter 1Production

2. Outline. • The input-output relationship: the production function • Production in the short term • Production in the long term

3. Production • Production = any activity that creates present or future utility • Production transforms inputs into outputs • Inputs = capital, labour + … • Ignore intermediate goods K, L → (Intermediate goods), K, L → Q proxied by K, L → Q • Firms face technological constraints: feasible production corresponds to the production set

4. The Production set

5. The production function • The production function is the relationship describing the maximum amount of output that can be produced with given quantities of inputs. • It is the boundary of the production set. • The production function can be represented as: Q = f(K,L)

6. Example: the production of meals. • The production process: Q = 2KL

7. The long vs. short run • We will distinguish between production in the long run and in the short run • The long run: the shortest period of time necessary to alter the amounts of all inputs. Note that • Variable input = the quantity of which can be altered • Fixed input = the quantity of which cannot be altered during the period • The short run: period during which at least one of the inputs is fixed.

8. Outline. • Production in the short term • Production in the long term

9. The production function in the short run • An Example • The production of meals: Q = f(K,L) = 2KL • Assume capital is fixed in the short run at K = K0 = 1. • Then the production function becomes f(K0,L) = 2L

11. A more standard production function

12. The law of diminishing returns • For L > 4, additional units of labour increase output by a decreasing amount: there are diminishing returns to the variable factor. • Very common property in the short run: see Malthus (1798) • Prediction: At some point, agriculture workers won't produce enough to feed the whole population • Has not happened: why? Growth in the agricultural technology.

13. Growth in agricultural technology

14. Marginal product • Definition: It is the change in the total product that occurs in response to a unit change in the variable input (all other inputs being held fixed) • It is also called marginal productivity. For small changes in the amount of labour: • Slope of the production function.

16. Average product • Definition: it is the average amount of output produced by each unit of variable input. This is also called average productivity. • Slope of the line joining the origin to the corresponding point on the production function

18. Example: fishing at both ends of a lake • You are the owner of 4 fishing boats. • 2 of them are fishing at the East end of a very large lake and 2 of them at the West end. • Each boat fishing at the East: 100 pounds of fish per day • Each boat fishing at the West: 120 pounds of fish per day • There is no exhaustion of the fish so that these yields can be maintained forever. Moreover, the fish does not move across the lake. • The question is: should you alter your current allocation of boats? • The intuitive answer is YES because boats at the west end bring more fish than boats at the east end. • However, if the structure of AP and MP are as displayed in Table 2, this answer is wrong.

19. Table 2 • The current allocation is optimal for the owner of the fish company.

20. Maximising production • In order to maximise production: • If resources are not perfectly divisible: allocate the next unit of input where its marginal productivity is highest • If resources are perfectly divisible: allocate the resource so that its marginal product is the same in every activity MP(Input)Activity 1 = MP(Input)Activity 2 • Allocating the boats on the basis of their AP is wrong because the MP of a third boat sent to the west is lower than the previous AP. So, it reduces the AP. • Due to diminishing returns to boats given that fish is a fixed amount • If the MP of boats were constant: all boats would be allocated to the West: corner solution ≠ interior solution

21. Outline. • Production in the long term

22. Isoquants • In the long run, all inputs can be varied. Q = 2KL • Isoquants • Definition: all combinations of variable inputs that yield a given level of output • They are analogous to the indifference curves for the consumer that would display all combinations of consumption yielding a given level of utility. • We get an isoquant map by moving the isoquants to the northeast as the level of production increases.

24. The marginal rate of technical substitution • It is the rate at which an input can be exchanged for the other input without altering the production level. • At any given point, it is the absolute value of the slope of the isoquant • It is decreasing along a given isoquant

25. Graphical definition of the marginal rate of technical substitution

26. Marginal rate of technical substitution and marginal productivity • There is an important relation between the MRTS and the MP • A variation in output can be decomposed as follows • Along a given isoquant:

27. Isoquants when inputs are perfect substitutes/complements

28. Returns to scale • Returns to scale tell us what happens to output when all inputs are increased exactly by the same proportions. • For any a > 1 • Increasing returns to scale: f(aK,aL) > a f(K,L) • Constant returns to scale: f(aK,aL) = a f(K,L) • Decreasing returns to scale: f(aK,aL) < a f(K,L) • Note that in principle, decreasing returnsto scale have nothing to do with the law of diminishing returns.

29. Some examples of production functions • The Cobb-Douglas production function • where a and b are between 0 and 1 and m > 0 • In this case, returns to scale are given by • if a + b > 1 returns to scale are increasing • if a + b = 1 returns to scale are constant • if a + b < 1 returns to scale are decreasing

30. Some examples of production functions (ctd) • Isoquants are described by the following expression: • The Leontieff production function Q = min (aK,bL) • Inputs are perfect complements • Isoquants will lie on a locus the equation of which is K = b/a.L • When inputs are perfect substitutes, the production function has the following form Q = aK + aL