Chapter 2

Costs

Outline. Costs in the short run Costs in the long run

Costs in the short run Example: a laundrette uses capital and labour

w= 10€ capital is fixed in the short run

Table 1

Quantity of labour (person-hour/hr)

Quantity of Output (bags/hr)

FC VC TC

0 0 30 0 30 1 4 30 10 40 2 14 30 20 50 3 27 30 30 60 4 43 30 40 70 5 58 30 50 80 6 72 30 60 90 7 81 30 70 100 8 86 30 80 110

Fixed, variable and total costs Fixed cost: do not vary with the level of

production.

FC = r.K0

Variable costs: costs associated with all variable inputs

VC (Q) = w.L(Q)

Total costs TC (Q) = w.L(Q) + r K0

Average costs

The average fixed cost is the fixed cost divided by the quantity of output.

The average variable cost is the variable cost divided by the quantity of output

The average total cost is the total cost divided by the quantity of output

Q

rK

Q

FC)Q(AFC 0

Q

wL

Q

VC)Q(AVC

Q

rKwL

Q

TC)Q(ATC 0

Marginal cost

The marginal cost is the change in total cost that results from producing one additional unit of output.

For very small changes in output:

Given that the fixed cost does not vary with output:

dQ

dTC)Q(MC

dQ

dVC

Q

VC)Q(MC

Q

TC)Q(MC

Firms’ decisions Optimal allocation of a given amount of

output between 2 production processes so that:

MC(Q1) = MC(Q2)

If MC(Q1) > MC(Q2) , shifting 1 unit of output from 1 to 2 would reduce the cost.

If MC(Q1) < MC(Q2) , shifting 1 unit of output from 2 to 1 would reduce the cost.

In both cases, the initial allocation was not cost minimising.

The relationship between marginal product, average product, marginal cost and average variable cost We can show that MC = w/MP

and Q

VCMC

wLVC

Q

L.w

Q

VCMC

MP

wMC

L

QMP

The relationship between marginal product, average product, marginal cost and average variable cost We can show that AVC = w/AP

Q

wL

Q

VCAVC

AP

wAVC

L

QAP

Outline. Costs in the long run

Cost minimisation

In the long run, all inputs are variable.

If the manager of the firm is free to choose his input combination and if he wants to produce a given level of input at the lowest possible cost, which combination is he going to use?

The total cost of production is given by : r.K + w.L

The problem of the firm:

Min (r.K + w.L)subject to : F(K,L) = Q0

Cost minimisation (ctd 1)Solution :

Forming the Lagrangian expression yields:

]Q)L,K(F[wLrKLa 0

The first order conditions for a minimum are:

0K

Fr

K

La

0L

Fw

L

La

and 0Q)L,K(FLa

0

Dividing the first equation by the second one yields:

w

MP

r

MP

w

r

MP

MP LK

L

K

Cost minimisation (ctd 2)The problem can also be solved graphically:

If the total cost of production is given by:

C = r.K + w.L

we can rewrite this expression in the following way:

r

CL

r

wK

Cost minimisation (ctd 3) The least-cost input bundle corresponds to the point of tangency between the isocost

line and the specified isoquant.

o From Chapter 1, we remember that the slope of the isoquant is equal to –

MRTS so that

MRTSMP

MP

L

KSlope

K

L

o At the point of tangency with the isocost line, we have:

r

wtcosSlopeIso

MP

MPantSlopeIsoqu

K

L

r

MP

w

MP

r

w

MP

MP KL

K

L

Cost minimisation (end)

In case there are more than 2 inputs, the condition for cost minimisation is given by

N

N

3

3

2

2

1

1

X

X

X

X

X

X

X

X

P

MP...

P

MP

P

MP

P

MP

An example

Figure 6

K

L

EUS

LUS

KUS Y = 1 ton

-w/rFR

EFR

LFR

KFR

-w/rUS

Optimal input choice and long-run costs The firm's output expansion path is the set of cost

minimising input bundles when the price of inputs is set at w and r and output increases from Q1 to Q3. :

Long-run marginal and average costs

As in the short-run, the long-run marginal cost is the slope of the long-run total cost

curve

Q

LTC)Q(LMC

The long-run average cost is also very similar to what it is in the short run.

Q

LTC)Q(LAC

Returns to scale Constant returns to scale

Returns to scale (ctd) Decreasing returns to scale

Returns to scale (end) Increasing returns to scale

The structure of the industry Long-run costs are important because they

impact the structure of the industry

When there are decreasing long-run average costs throughout the production process, the tendency will be for a single firm to serve the entire market

When the average cost curve reaches a minimum, the industry will be dominated by a small number of firms

Long-run average costs The long-run average curve associated with a

market served by many firms is likely to take one of the three following forms

Long-run and short-run cost curves Consider a firm with plant size = k*. The short run

cost function for a plant of size k* will be STC(y, k*) and the long run cost function will be LTC(y) = STC (y, k(y)).

Let y* be the level of output for which k* is the optimal plant size: k* = k(y*) We know that for y*, LTC(y*) = STC(y*, k*) because at y* the optimal choice of plant size is k*.

For other levels of output, the short-run cost for k = k* is going to be higher than the long-run cost because the plant size is not optimal.

STC(y, k*) > LTC(y)

Long-run and short-run cost curves (ctd 1) If the short run cost is always larger than

the long run cost and they are equal for one level of output, this means we have the same property for average costs:

LAC(y) SAC(y, k*)

and LAC(y*) = SAC(y*, k*)

Long-run and short-run cost curves(ctd 2)

Long-run and short-run cost curves(ctd3) We can do the same sort of construction

for levels of output other than y*. Suppose we pick output levels y1, y2, …, yn and the corresponding plant sizes k1 = k(y1), k2 = k(y2), …, kn = k(yn).

the LAC curve is the lower envelope of the short run average cost curves.

Long-run and short-run cost curves(ctd4)

Long-run and short-run cost curves(ctd5) Note that for the output level at which LAC

= SAC, the long run marginal cost of producing that level of output is equal to the short-run marginal cost : LMC = SMC.

This is due to the fact that :

LAC = SAC LTC = STC

SMCLMCdQ

dSTC

dQ

dLTC

Long-run and short-run cost curves(end)