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Price Theory

Lecture 6: Production & Costs

Topics for today’s lecture . . .

1. Production functions

2. Multiple-input production functions

3. Cost minimisation

4. The total cost curve

Production functions

Production as a process of transformation

Factor markets Production technology Product markets

The Firm

Inputs

Labour

Capital

Raw materials

Intermediate goods

Services

Outputs

Goods

Services

Definition: Production technology

The process by which inputs (factors of production), are transformed by a firm into goods

and services.

A firm’s production function describes the maximum quantity of a good or service that can be

produced by a firm, from a given bundle of inputs, given the firm’s production technology.

Production functions

We will typically assume that the inputs into the production process are labour (L) and

capital (K ).

• Labour represents the productive effort of the workers hired by the firm.

• Capital represents the assets used in, but not consumed by, the production process.

We exclude all other inputs from the production function because we only require two inputs

to understand the nature of the trade-offs faced by firms.

The maximum quantity of a good or service that can be produced by a given bundle of inputs

is written Q = f (L,K ).

A production function with one input

Q = 6L2 − L3

0 L

Consider the production function

Q = 6L2 − L3, in which the only input is labour.

• This is a production function in which the firm’s capital stock is fixed.

If Q represents the number of cars the

firm can service per day then:

• L = 2 workers can service Q = 16 cars.

• L = 3 workers can service Q = 27 cars.

Technical efficiency

00 Technically

efficient

Technically

inefficient

Q = 6L2 − L3

0 L

The production function illustrates the

maximum quantity of output that can be

produced.

If the firm is producing at a point on its

production function we say that it is

technically efficient.

If the firm is producing inside its

production function we say that it is

technically inefficient.

Exercise: Inputs & production

0 L

Using the production function illustrated

in the figure, answer the following

questions:

1. Suppose that this firm hires 5 workers.

If the firm services 25 cars per day, is

it technically efficient? Briefly explain.

2. Would this firm be willing to hire 5

workers? Briefly explain.

Exercise solutions

0 L

1. Using 5 workers to service 25 cars per

day is technically efficient.

The point lies on the production

function, and therefore Q = 25 is the

maximum possible output when L = 5.

2. No. The same level of production can

be achieved with fewer workers (and

therefore lower cost).

Definition: Marginal product

The rate at which total output changes as the usage of an input rises, holding constant the

usage of all other inputs.

The marginal product of an input is the partial derivative (slope) of the production function,

with respect to the input in question.

The marginal product of labour

2 3

Q = 6L2 − L3

MPL = 12L− 3L2

MPL

0 L

For a single input production function, the

marginal product of labour is written,

MPL = dQ

dL .

Note: You will always be provided with

marginal products in this course.

• When 2 workers are employed, the marginal product (slope) is 12.

• When 3 workers are employed, the marginal product (slope) is 9.

Increasing marginal returns

MPL

0 L

Increasing

This example has a number of features

common to many production functions:

• If no labour is employed (L = 0), nothing can be produced (Q = 0).

• Initially, as additional workers are employed, output rises at an

increasing rate.

Increasing marginal returns occur at low

levels of labour usage, as more workers

allows greater specialisation.

Definition: Law of diminishing marginal returns

The principle that as the usage of one input increases, past some point the marginal product

of that input will decrease.

Diminishing marginal returns

Diminishing

total returns

MPL

0 L

Increasing Diminishing

Diminishing marginal returns set in once

the gains to specialisation have been

exhausted.

• As workers are added to a fixed capital stock, they begin to get in one

and other’s way, creating congestion.

• With many workers, coordinating their activities also becomes more difficult.

In extreme cases, the production function

may reach a region where there are

diminishing total returns to labour.

Quiz 1

Gregory owns and runs a cafe. When Gregory employs a third staff member he notices that

the cafe is able to serve 100 additional customers each day. When he employs a fourth staff

member the cafe is able to serve a further 80 additional customers each day. From the

information provided, we can conclude that Gregory’s cafe is experiencing,

(a) increasing marginal returns to labour, and increasing total returns.

(b) increasing marginal returns to labour, and decreasing total returns.

(d) decreasing marginal returns to labour, and decreasing total returns.

Multiple-input production functions

A production function with labour & capital

3 L

The single-input production function is

drawn for a given quantity of capital.

Varying the firm’s capital stock also

affects the quantity of output that can be

produced.

The marginal product of labour is the

slope (partial derivative) of the production

function in the direction of an increase in

labour,

MPL = ∂Q

∂L .

Marginal product of capital

3 L

The marginal product of capital is the

slope (partial derivative) of the production

function in the direction of an increase in

capital,

MPK = ∂Q

∂K .

Past some point, capital experiences

diminishing marginal returns.

In extreme cases the marginal product of

capital may become negative.

Discussion: The marginal product of capital

Consider a firm that services cars. How might we explain the following phenomena? (In each

of the following, the number of mechanics employed by the firm is held fixed.)

• Increasing marginal product of capital, when a firm’s capital stock is relatively low.

• Diminishing marginal returns to capital.

• Diminishing total returns to capital, when a firm’s capital stock is relatively high.

Definition: Isoquant

A curve that shows all combinations of labour and capital that can produce a given level of

output.

Using isoquants we can illustrate the shape of the production function in the input-space.

Isoquants

Q = 5

Q = 16

Q = 27

Isoquants are the contour lines of the

production function; the set of all points

at the same height.

Higher isoquants correspond to larger

quantities of output.

Isoquants illustrate the tradeoff between

labour and capital in the production

function.

Isoquants in labour-capital space

Q = 27

Q = 16

Q = 5 K

0 L

Isoquants illustrate production possibilities

on the space of inputs.

Notice that these isoquants are backward

bending.

This occurs if there is a point beyond

which there are diminishing total returns

to labour.

The economic region of production

Q = 27

Q = 16

Q = 5 K

0 L

economic

region

uneconomic

region

A profit maximising firm never wants to

waste inputs.

This firm will not employ 5 workers if it

can produce the same output with 3.

The economic region is the region in

which isoquants are downward sloping.

A firm will not produce in the uneconomic

region, as the same output can be

produced with fewer inputs in the

economic region.

Definition: Marginal rate of technical substitution

The rate at which a firm can substitute labour for capital, while maintaining a constant level

of output.

The marginal rate of technical substitution of labour for capital (written MRTSL,K ) is the

negative of the slope of an isoquant.

Marginal rate of technical substitution

Q = 100

0 L

The marginal rate of technical

substitution can be written as the ratio of

marginal products,

MRTSL,K = MPL MPK

An isoquant displays a diminishing

marginal rate of technical substitution if it

becomes flatter as we move down along

the curve.

Diminishing MRTSL,K implies that

substituting labour for capital becomes

progressively less effective.

Quiz 2

Winston’s Widgets has the production function Q = 200LK . The associated marginal

products are MPL = 200K and MPK = 200L. For this production function the marginal

returns to labour are,

(a) diminishing.

(b) constant.

(d) not possible to assess with the information provided.

Quiz 3

Winston’s Widgets has the production function Q = 200LK . The associated marginal

products are MPL = 200K and MPK = 200L. For this production function, the marginal rate

of technical substitution is,

(a) diminishing.

(b) constant.

(d) not possible to assess with the information provided.

Cost minimisation

A firm’s cost minimisation problem

Winston’s Widgets has the production function Q = 200LK . The associated marginal

products are MPL = 200K and MPK = 200L.

Suppose that the wage (the price of labour) is w = $180 per day, and that the rental price of

capital is r = $240 per day.

What is the lowest cost at which Winston’s Widgets can produce Q = 2400 widgets per day?

The cost of a bundle of inputs

1 12

Q = 2400

0 L

The bundle A, with LA = 8 workers and

KA = 1.5 units of capital, lies on the

required isoquant as,

200LAKA = 200× 8× 1.5 = 2400.

The total cost of bundle A is,

TCA = wLA + rKA

= 180× 8 + 240× 1.5 = $1800.

Does A minimise the firm’s costs?

Definition: Isocost line

The set of all bundles of inputs with the same total cost.

The slope of an isocost line is the negative of the ratio of input prices (−w/r ).

The isocost line through bundle A

1 12

Q = 2400

TCA

7 12

0 L

The bundles of inputs with the same total

cost as A are described by the equation,

wL + rK = TCA,

or 180L + 240K = 1800.

The L-intercept of the isocost line is,

TCA w

= 1800

180 = 10.

The K -intercept of the isocost line is,

TCA r

= 1800

240 = 7.5.

Comparing costs

1 12

Q = 2400

TCA

7 12

0 L

Bundles lying above the isocost line (such

as B) are more expensive than A.

Bundles lying below the isocost line (such

as C ) are less expensive than A.

A bundle of inputs does not minimise a

firm’s cost if it lies on an isocost line that

crosses the isoquant.

The cost-minimising bundle of inputs

Q = 2400

TCA

0 L

The cost-minimising bundle lies on the

isocost line that touches the isoquant at a

single point.

Diminishing marginal rate of technical

substitution implies that the isocost line

passing through the cost minimising

bundle A is tangent to the isoquant,

MRTSL,K = w

r .

At most one bundle of inputs can satisfy

these conditions.

Two conditions for (interior) cost minimisation

• The cost minimising bundle of inputs lies on the isoquant corresponding to the required quantity of output. Therefore, the production function must be satisfied for the required

value of Q.

• At the cost minimising bundle of inputs the slope of the isoquant is equal to the slope of the isocost line, therefore,

MPL MPx

= w

r .

Note: These conditions are predicated on a diminishing marginal rate of technical

substitution.

Exercise: Finding the optimal bundle of inputs

Winston’s Widgets has the production function Q = 200LK . The associated marginal

products are MPL = 200K and MPK = 200L. The price of labour is w = $180 per day, and

the rental price of capital is r = $240 per day.

1. Derive the equation for the Q = 2400 isoquant.

2. Find the firm’s marginal rate of technical substitution, and use it to construct the

tangency condition.

3. Solve the two equations simultaneously to find the optimal bundle of inputs.

4. What is the total cost of the optimal bundle?

Exercise solutions

1. The equation of the isoquant is found by substituting for Q in the production function,

200LK = 2400.

2. Winston’s Widgets marginal rate of technical substitution is,

MRTSL,K = MPL MPK

= 200K

200L =

L .

The tangency condition equates the marginal rate of technical substitution with the ratio

of input prices, MPL MPK

= w

r ⇒

L =

180

240 .

Exercise solutions

3. Rearranging the tangency condition to solve for K gives us,

L =

180

240 ⇒ K =

4 .

Substituting for K into the equation of the isoquant,

200L

( 3L

) = 2400 ⇒ 150L2 = 2400 ⇒ L2 = 16 ⇒ L = 4.

Substituting for L in the tangency condition,

K = 3× 4

4 = 3.

4. The total cost of bundle A is TCA = wL + rK = 180× 4 + 240× 3 = $1440.

Marginal product per dollar

An alternative way of stating the tangency condition is,

MPL w

= MPK

r .

In words, at an interior optimum the marginal product per dollar must be equal for all inputs.

• This condition must hold for every pair of inputs that a firm utilises, regardless of the number of inputs to the firm’s production process.

Intuitively, if the marginal product per dollar is higher for labour than for capital, the firm can

reduce its costs by reducing usage of capital, and increasing usage of labour.

A change in the price of labour

Q = 2400

TCA

TCB

0 L

How does an increase in the wage, from

w1 = $180 to w2 = $320 per day, alter

the cost-minimising bundle of inputs?

After the increase in the wage, isocost

lines are steeper,

slope = − w2 r

= − 320

240 = −

3 .

The firm substitutes capital for labour.

The total cost increases to,

TCB = 320× 3 + 240× 4 = $1920.

Abundant opportunity for substitution

TCA

TCB

0 L

Along the isoquants of this production

function, labour and capital are relatively

interchangeable.

If the MRTSL,K changes gradually, the

production function offers abundant input

substitution opportunities.

The firm responds to an increase in the

wage by significantly increasing the

automation of the production process.

Minimal opportunity for substitution

TCA

TCB

0 L

A B

Along the isoquants of this production

function, there is little potential to

substitute labour for capital.

When the MRTSL,K changes rapidly, the

production function offers limited input

substitution opportunities.

The firm is unable to significantly

reorganise production in response to an

increase in the wage.

Fixed proportions production functions

00 fixed

proportions

TCA

TCB

0 L

The production function Q = min{L,K} requires inputs to be used in fixed

proportions (the inputs are perfect

complements).

The economic region of production

corresponds to the corners of the

isoquants.

The fixed proportions production function

offers no opportunities to substitute

labour for capital.

The total cost curve

Expanding production

4 12

Q = 2400

Q = 5400

TCA

TCB

0 L

In order for a firm to expand production,

it must select a bundle of inputs on a

higher isoquant.

The new cost-minimising bundle B has a

higher total cost than A.

Given that the input prices have not

changed, the isocost line TCB must have

the same slope as TCA.

Definition: Total cost curve

A curve that shows how total cost varies with output, holding input prices fixed and choosing

all inputs to minimise cost.

The total cost curve indicates the lowest cost at which each quantity can be produced.

Plotting the total cost curve

QA QB

TCA

TCB K

A B

TCA

TCB

0 Q

The quantity of output produced by

bundle A, can be plotted against the total

cost of the bundle.

Increasing output requires moving to a

higher isocost line.

It follows that the total cost curve must

be upward sloping.

Winston’s Widgets input demands

The first step in constructing a total cost function for a firm is deriving the input demands.

To find Winston’s Widgets input demands we solve the isoquant equation and tangency

conditions simultaneously for unknown quantity Q, and input prices w and r ,

200LK = Q and K

L =

r .

Rearranging the tangency condition to solve for K gives us K = wL/r .

Substituting for K into the isoquant equation allows us to derive the labour demand function,

200L

( wL

) = Q ⇒ 200

r L2 = Q ⇒ L2 =

200w ⇒ L =

√ Qr

200w .

Winston’s Widgets total cost function

Substituting for L in the tangency condition allows us to derive the capital demand function,

K = w

√ Qr

200w ⇒ K =

√ Qrw2

200wr2 ⇒ K =

√ Qw

200r .

The total cost of producing the quantity Q is therefore,

TC (Q) = wL + rK = w

√ Qr

200w + r

√ Qw

200r =

√ Qwr

200 +

√ Qwr

200 =

√ Qwr

50 .

Notice that for given input prices, the total cost function is a function of the quantity

produced only. Optimal use of inputs is implicit in all cost functions.

Questions?

Key concepts from today’s lecture

You can use these concepts (as search terms) to conduct further research into the topics

covered in today’s lecture:

• Inputs (factors of production)

• Production technology

• Production function

• Technical efficiency

• Marginal product

• Isoquant

• Economic region of production

• Marginal rate of technical substitution

• Cost minimisation

• Isocost line

• Total cost curve

• Total cost function