Economists consider both explicit costs and implicit costs.

Explicit costs are a firm’s direct, out-of-pocket payments for inputs to its production process during a given time period such as a year.

These costs include production worker’s wages, manager’s salaries, and payments for materials.

However, firms use inputs that may not have an explicit price.

These implicit costs include the value of the working time of the firm’s owner and the value of other resources used but not purchased in a given period.

The economic costs or opportunity cost is the value of the best alternative use of a resource.

The economic or opportunity cost includes both explicit and implicit costs.

If a firm purchases and uses an input immediately, that input’s opportunity cost is the amount the firm pays for it.

If the firm uses an input from its inventory, the firm’s opportunity cost is not necessarily the price it paid for the input years ago.

Rather, the opportunity cost is what the firm could buy or sell that input for today.

The classic example of an implicit opportunity cost is captured in the phrase “There’s no such thing as a free lunch.”

Suppose that your friend offer to take you to lunch tomorrow.

You know that they’ll pay for the meal, but you also know that this lunch will not really be free for you.

Your opportunity cost for the lunch is the best alternative use of your time.

This might be studying, working at a job or watching TV.

Determining the cost of capital, such as land or equipment, requires special considerations.

Capital is a durable good: a good that is usable for years.

Two problems arise in measuring the cost of capital.1. Allocating Capital Costs over Time2. Actual and Historical Costs

An economists amortizes the cost of the truck on the basis of its opportunity cost at each moment of time, which is the amount that the firm could charge others to rent the truck.

Regardless of whether the firm buys or rents the truck, an economist views the opportunity cost of this capital good as a rent per time period: the amount the firm will receive if it rents its truck to others at the going rental rate.

The firm’s current opportunity cost of capital may be less than what it paid if the firm cannot resell the capital.

A firm that bought a specialized piece of equipment that has no alternative use cannot resell the equipment.

Because the equipment has no alternative use, the historical cost of buying that capital is a sunk cost: an expenditure that cannot be recovered.

Because this equipment has no alternative use, the current or opportunity cost of the capital is zero.

In short, when determining the rental value of capital, economists use the opportunity value and ignore the historical price.

A firm’s cost rises as the firm increases its output.

A firm cannot vary some of its input, such as capital, in the short run.

As a result, it is usually more costly for a firm to increase output in the short run than in the long run when all inputs can be varied.

To produce a given level of output in the short run, a firm incurs costs for both its fixed and variable inputs.

A firm’s fixed cost (F) is its production expense that does not vary with output.

The fixed cost includes the cost of inputs that the firm cannot practically adjust in the short run, such as land, a plant, large machines, and other capital good.

A firm’s variable cost (VC) is the production expense that changes with the quantity of output produced.

The variable cost is the cost of the variable inputs – the inputs the firm can adjust to alter its output level, such as labor and materials.

A firm’s cost (or total cost, C) is the sum of a firm’s variable cost and fixed cost:

C VC F Because variable cost changes with the

level of output, total cost also varies with the level of output.

A firm’s marginal cost (MC) is the amount by which a firm’s cost changes if the firm produces one more unit of output. The marginal cost is

( )dC qMC

dq Because only the variable cost changes with

output, we can also define marginal cost as the change in variable cost from a small increase in output.

( )dVC qMC

dq

The average fixed cost (AFC) is the fixed cost divided by the units of output produced

FAFCq

The average fixed cost falls as output rises because the fixed cost is spread over more units:

2 0dAFC Fdq q

It approaches zero as the output level grows very large.

The average variable cost (AVC) is the variable cost divided by the units of output produced

VCAVCq

Because the variable cost increases with output, the average variable cost may either increase or decrease as output rises.

The average cost (AC) – or average total cost – is the total cost divided by the units of output produced:

CACq

Because total cost equals variable cost plus fixed cost, C = VC + F, when we divide both sides of the equation by q, we learn that

C VC FAC AVC AFCq q q

Illustration:A manufacturing plant has a short-run cost

function of 2 3100 4 0.2 450C q q q q

What is the firm’s short-run fixed cost and variable cost function?Derive the formulas for its marginal cost, average fixed cost, average variable cost, and average cost.

The fixed cost is F = 450, the only part that does not vary with q.The variable cost function

2 3100 4 0.2VC q q q qis the part of the cost function that varies with q.Differentiating the short-run cost function or variable cost function, we find that

2 3100 4 0.2 450d q q qdC qMC

dq dq

2100 8 0.6MC q q

450FAFC

q q

2 3100 4 0.2V q q q qAVCq q

2100 4 0.2AVC q q

2 3100 4 0.2 450C q q q qACq q

2 450100 4 0.2AC q qq

AC AVC AFC

q

cost

0 10 15cost per unit

q10 150

80

450

800

1,725

45

115125

80

115

1

1

C

VC

F

MC

ACAVC

AFC

b

a

A

B

The production function determines the shape of a firm’s cost curves.

The production function shows the amount of inputs needed to produce a given level of output.

The firm calculates its cost by multiplying the quantity of each input by its price and summing.

,q f L K g L

If a firm produces output using capital and labor and its capital is fixed in the short run, the firm’s variable cost is its cost of labor.

Its labor cost is the wage per hour, w, times the number of hours of labor, L, employed by the firm: VC = wL

If input prices are constant, the production function determines the shape of the variable cost curve.

Because capital does not vary, we can write the production function as

1C q VC q F wg q F

By inverting, we know that the amount of labor we need to produce any given amount of output is L = g-1(q).

If the wage of labor is w, the variable cost function is VC(q) = wL = wg-1(q)

Similarly, the cost function is

In the short run, when the firm’s capital is fixed, the only way the firm can increase its output is to use more labor.

If the firm increases its labor enough, it reaches the point of diminishing marginal returns to labor, where each extra worker increases output by a smaller amount.

If the production function exhibits diminishing marginal returns, then the variable cost rises more than in proportion as output increases.

The marginal cost is the change in variable cost as output increases by one unit:

dVC qMC

dq In the short run, capital is fixed, so the only

way a firm can produce more output is to use extra labor.

The extra labor required to produce one more unit of output is

1

L

dLdq MP

The extra labor costs the firm w per unit, so the firm’s cost rises by

dLwdq

As a result, the firm’s marginal cost is

dVC q dLMC wdq dq

The marginal cost equals the wage times the extra labor necessary to produce one more unit of output.

Since the marginal product of labor, the amount of extra output produced by another unit of labor, holding other input fixed is

Thus, the extra labor needed to produce one more unit of output is

L

wMCMP

LdqMPdL

1

L

dLdq MP

So the firm’s marginal cost is

For the firm whose only variable input is labor, variable cost is wL, so average variable cost is

VC wLAVCq q

Because the average product of labor isL

qAPL

Average variable cost is the wage divided by the average product of labor

L

wAVCAP

In the long run, a firm adjusts all its inputs so that its cost of production is as low as possible.

The firm can change its plant size, design and build new machines, and otherwise adjusts inputs that were fixed in the short run.

The rent of F per month that a restaurant pays is a fixed cost because it does not vary with the number of meals served.

In the short run, this fixed cost is sunk. The firm must pay F even if the restaurant

does not operate.

In the long run, this fixed cost is avoidable. The firm does not have to pay this rent if it

shuts down. The long run is determined by the length of

the rental contract, during which time the firm is obligated to pay rent.

All inputs can be varied in the long run, so there are no long-run fixed costs (F = 0).

As a result, the long-run total cost equals the long-run variable cost: C = VC.

In the long run, the firm chooses how much labor and capital to use, whereas in the short run, when capital is fixed, it chooses only how much labor to use.

As a consequence, the firm’s long-run costs is lower than its short-run cost of production if it has to use the ‘wrong’ level of capital in the short run.

A firm can produce a given level of output using many different technologically efficient combinations of inputs, as summarized by an isoquant.

From among the technologically efficient combinations of inputs, a firm wants to choose the particular bundle with the lowest cost of production, which is the economically efficient combination of inputs.

To do so, the firm combines the information about technology from the isoquant with information about the cost of labor and capital

The cost of producing a given level of output depends on the price of labor and capital.

The firm hires L hours of labor services at a wage of w per hour, so its labor cost is wL.

The firm rents K hours of machine services at a rental rate of r per hour, so its capital cost is rK.

The firm’s total cost is the sum of its labor and capital costs:

C wL rK The firm can hire as much labor and capital

as it wants at these constant input prices.

The firm can use many combinations of labor and capital that cost the same amount.

These combinations of labor and capital are plotted on an isocost line, which indicates all the combinations of inputs that require the same (iso) total expenditure (cost).

Along an isocost line, cost is fixed at a particular level.

We can write the equation for isocost line with cost fixed at

C wL rKC

Which we can rewrite as

C wK Lr r

By differentiating with respect to L, we find that the slope of any isocost line is

dK wdL r

Thus, the slope of the isocost line depends on the relative prices of the inputs.

41.670 L

K

C = 1,000

C = 2,000

C = 3,000w = 24, r = 8

125

250

375

83.33 125

By combining the information about costs that is contained in the isocost lines with information about efficient production that is summarized by an isoquant, a firm chooses the lowest-cost way to produce a given level of output.

28

41.670 L

K

C = 1,000

C = 2,000

C = 3,000w = 24, r = 8

83.33 125

125

250

375

x100

50

z

11624

303y

q = 100

The firm is minimizing cost subject to the information in the production function contained in the isoquant expression:

,q f L K The corresponding Lagrangian problem is

, , min ,L K

wL rK q f L KL

The first-order conditions are

0fwL LL

0fr

K KL

, 0q f L KL

By rearranging terms, we obtain

L

K

fMPw L

fr MPK

We find that cost is minimized where the factor price ratio equals the ratio of the marginal products.

We could equivalently examine the dual problem of maximizing output for a given level of cost,

, ,

max ,L K

f L K C wL rKL The first-order conditions are

0f w

L LL

0f r

K KL

0C wL rKL

By rearranging terms, we obtain

L

K

fMPw L

fr MPK

The same condition as when we minimized cost by holding output constant.

0 L

K

C = 2,000

w = 24, r = 8

83.33

250

x100

50

q = 100q = 75q = 175

Suppose that the wage falls but that the rental rate of capital stays constant, the firm now minimizes its new cost by substituting away from the now relatively more expensive input, capital, toward the now relatively less expensive input, labor.

Because of the wage decrease, the new isocost lines have a flatter slope.

0 L

K

C = 2,000, w = 24, r = 8

77

52

x100

50

q = 100

v

C = 1,032, w = 8, r = 8

200

0 L

K

C = 2,000

C = 3,000

C = 4,000

w = 24, r = 8

x100

75

z

10050

150y

q = 100q = 150

q = 200

Expansion path

Illustration:What is the expansion path for a constant-

returns-to-scale Cobb-Douglas production function

1a aq AL KShow the special case for A = 1.52 and a = 0.6 given that w = 24 and r = 8.

Use the tangency condition between the isocost and isoquant that determines the factor ratio when the firm is minimizing cost to derive the expansion path.

and 1L Kq qMP a MP aL KSince

The tangency condition is

1L

K

qaMPw Lqr MP aK

Rearranging, we obtain the expansion path function

For the case of a = 0.6, since w = 24 and r = 8, we have

1

w a Kr a L

1 a wK La r

0.4 240.6 8K L

2K L

As the expansion path plot shows, to produce q units of output takes K = q units of capital and L = q/2 units of labor.

Thus the long-run cost of producing q units of output is

2qC q wL rK w rq

24 82 2wC q r q q

20C q q

4,000

0 q

C

X2,000

150

Z

200100

3,000 Y

Long-run cost curve 20C q q

Illustration:Derive the long-run cost as a function of only output and factor prices for a Cobb-Douglas production function

1a aq AL KShow the special case for A = 1.52 and a = 0.6 given that w = 24 and r = 8.

From the expansion path, we know that

1 a

rK wLa

Substituting for rK in the cost identity gives

1 a

C wL wLa

Simplifying shows that

CL aw

Repeating this process to solve for K, we find that

1 CK ar

Substituting L and K into the production function, we have;

1

1a aC Cq A a a

w r

We can rewrite asC q

where

Thus,

1

11a a

aa

w rAa a

For the case of A = 1.52 and a = 0.6 given that w = 24 and r = 8, we have

0.6 0.4

0.6 0.424 8

1.52 0.6 0.4 20

20C q

In the long run, returns to scale play a major role in determining the shape of the average cost curve and the other cost curves.

If a production function exhibits increasing returns to scale at low levels of output, constant returns to scale at intermediate levels of output, and decreasing returns to scale at high levels of output, then the long-run average cost curve must be U-shaped.

A cost function is said to exhibit economies of scale if the average cost of production falls as output expands, as we would expect in the range where the production function had increasing returns to scale.

In the range where the production function has constant returns to scale, the average cost remains constant, so the cost function has no economies of scale.

Finally, in the range where the production function has decreasing returns to scale, average cost increases.

A firm suffers from diseconomies of scale if average cost rises when output increases.