Chapter 1 Production. Outline. The input-output relationship: the production function Production in the short term Production in the long term.

Chapter 1

Production

Outline. The input-output relationship: the

production function Production in the short term Production in the long term

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

The Production set

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)

Example: the production of meals.

The production process:Q = 2KL

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.

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

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

A more standard production function

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.

Growth in agricultural technology

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.

L

QMPL

dL

dQMPL

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

L

QAPL

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.

Table 2

The current allocation is optimal for the owner of the fish company.

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

Outline. Production in the long term

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.

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

L

KMRTS

Graphical definition of the marginal rate of technical substitution

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:

K.MPL.MPQ KL

K.MPL.MP0 KL

K

L

MP

MP

L

K

K

L

MP

MP

L

KMRTS

Isoquants when inputs are perfect substitutes/complements

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 returns to scale have nothing to do with the law of diminishing returns.

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

LmKQ

QaLmKa)aL()aK(m)aL,aK(f

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

/1

mL

QK