Cost

OverviewIn this section we want to translate the production data into cost data. In other words, we will want to understand how the cost of producing various units of output might change as different amounts of inputs are used.

Fixed/variable inputsInputs can be classified as either fixed or variable.

A variable input is one that can be changed as the level of output is changed .A fixed input is one that can not be changed as the level of output is changed.

We often think of labor as a variable input and capital or land as a fixed input.

Short run/long runThe notion of a fixed or variable input is related to the time frame of production.The short run is that period of time when at least one input is fixed in amount. The long run is that period of time in which all inputs are variable.As an example of this consider fast food in Wayne. About any store in town could remodel and increase floor space in about 3 months. So after 3 months we have the long run, all inputs can vary - even floor space. But less than three months is the short run because there is only so much floor space to use.

short run costsIn the short run we will consider the fixed and variable costs of production and how they change as more of the variable input is used.Definitions:Total cost (TC) = Total variable cost(TVC) + Total fixed cost (TFC).Marginal cost(MC) = (change in TC)/(change in output). where change in output = 1 when possible.Average cost (AC) = TC/Q.Average variable cost(AVC) = TVC/Q.Average fixed cost(AFC) = TFC/Q.Note that in the short run fixed costs must be paid whether output is zero, 100,000, or any other number of units.

exampleLet’s take the production example. Say the fixed costs is $1000 per unit of capital and we have two units and the cost of labor is $400 per unit.The next screen shows the example with an added TP or output amount.

QL Q MPL APL TFC TVC TC

0 0 2000 0 2000

1 76 76 76 2000 400 2400

2 248 172 124 2000 800 2800

3 492 244 164 2000 1200 3200

4 784 292 196 2000 1600 3600

5 1100 316 220 2000 2000 4000

6 1416 316 236 2000 2400 4400

7 1708 292 244 2000 2800 4800

8 1952 244 244 2000 3200 5200

9 2124 172 236 2000 3600 5600

10 2200 76 220 2000 4000 6000

11 2156 -44 196 2000 4400 6400

QL Q MPL APL TFC TVC TC AFC AVC ATC MC

0 0 2000 0 2000

1 76 76 76 2000 400 2400 26.32 5.26 31.58 5.26

2 248 172 124 2000 800 2800 8.06 3.23 11.29 2.33

3 492 244 164 2000 1200 3200 4.07 2.44 6.5 1.64

4 784 292 196 2000 1600 3600 2.55 2.04 4.59 1.37

5 1100 316 220 2000 2000 4000 1.82 1.82 3.64 1.27

6 1416 316 236 2000 2400 4400 1.41 1.69 3.11 1.27

7 1708 292 244 2000 2800 4800 1.17 1.64 2.81 1.37

8 1952 244 244 2000 3200 5200 1.02 1.64 2.66 1.64

9 2124 172 236 2000 3600 5600 0.94 1.69 2.64 2.33

10 2200 76 220 2000 4000 6000 0.91 1.82 2.73 5.26

COST CurvesTotal cost curve

01000200030004000500060007000

0 1000 2000 3000

TP or Q or output

Do

llars

of

Co

st

TFC

TVC

TC

Costs per unit

05

1015

2025

3035

0 500 1000 1500 2000 2500

Quantity of Output

Dolla

rs p

er u

nit

Idealized graph of per unit costs in the short run

$/unit

Q

MC

AC

AVC

Note AVC and AC equal MC when AVC and AC are at their minimum values.

When you look back at the marginal product and average product curves note that the horizontal axis is measuring labor units used and the curves are inverted u-shaped curves.

When you look back at the marginal cost and various average cost curves note that the horizontal axis is measuring output units and the curves are u-shaped curves.

There is a relationship between these two graphs. When you move to the right by adding labor in the one graph you are moving to the right in the other by having output increased.

How much output should the firm make? How much of the variable labor should be hired?

We are not yet ready to answer this question. We are only talking about cost. Will have to talk revenue and profit later before we can answer these questions.

Two production facilities, one type of output.

Say you can make output in 1 of 2 facilities. Let’s also say the MC in each (plant a MC) MCa = 12Qa and (plant b MC) MCb = 4Qb.

For the firm the first unit of output has MC = 12 in plant a and 4 in plant b. So the first unit should come from plant b.

For the firm the second unit could be the first from plant a or the second from plant be. The respective MC’s are 12 and 8. Have it come from plant b.

The third unit will also come from plant b. But the 4th unit for the firm will be the 1st unit from plant a because of comparing MC’s, 12 in a and 16 in b.

Two production facilities, one type of output.

We would keep apply this logic to see which plant the next until of output should come from.

A summary of the firm level MC is found by adding the two MC’s in the following way:

MCa = 12Qa and MCb = 4Qb. Re-arrange these in Q from like the following (just express each with Q term on the left) Qa = (1/12)MCa and Qb = (1/4)MCb.

Q = Qa + Qb = (1/12)MC + (1/4)MC = ((1/12) + (3/12))MC = (1/3)MC. So we have Q = (1/3)MC or MC = 3Q. If you want Q = 32, MC = 96 at the firm level. IN plant a if MC = 96, Qa = (96/12) = 8 and in plant b Qb = (96/4) = 24. Note Qa + Qb = 32.

Isocost linesAn isocost line includes all possible combinations of labor and capital that can be purchased for a given total cost.In equation form the total cost is

TC = wL + rK,where TC = Total cost,

w = the wage rate,L = the amount of labor taken,r = the rental price of capital, andK = the amount of capital taken. This

equation can be re-expressed asK = TC/r - (w/r) L.

exampleAs an example say labor is $6 per unit and capital is $10 per unit. Then if we look at a total cost of $100 we see various combinations of inputs:L = 10 and K = 4 or L = 0 and K = 10 or L = 16.67 and K = 0, amoung others.

On the next screen we can view the isocost line in a graph.

graph of isocost lineK

L

This is the isocost line at $100. If we wanted to see higher costs we would shift the line out in a parallel shift and a lower cost we have a shift in.

cost and outputK

L

On this slide I want to concentrate on one level of output, as summarized by the isoquant. Input combination L1, K1 could be used and have cost summarized by 4th highest isocost shown. L2, K2 would be cheaper, and L*, K*

K1

K2

K*

L1 L2 L*is the lowest cost to produce the given level of

output. Herethe cheapest cost of the output occurs at a tangency point.

cost and outputK

L

On this slide I want to concentrate on one level of cost, as summarized by the isocost line. Input combination L1, K1 could be used and have this cost but more output would be obtained if L*, K* were used.

K1

K*

L1 L*Here, the most output for a given cost occurs at a

tangency point.

cost and outputOn the last two screens we have seen the tangency of an isoquant and isocost line shows either1) the cheapest way to produce a certain level of output, or2) the most output that can be obtained for a given amount of cost. These two things are different sides of the same coin and profit maximizing firms would be expected to reach the tangency positions.The exception to reaching the tangency would be the short run when the amount of some input can not be changed to reach the tangency. In the long run all inputs can be changed in amount and thus the tangency point could be reached.

short runK

L

Here the cheapest way to produce the output level as depicted in the isoquant would be to hire L*, K*. But the firm as committed to having K1 units of capital. Thus the cost of this output is indicated by the fourth highest isocost line.

K1

K2

K*

L1 L2 L*We could follow K1 out and see costs of other levels

of output(by putting in more isoquants).

Tangency = equal slopesOur optimal point for the firm in general was slope of isoquant = slope of isocost (or MRTS = w/r) and this gives

MPl/MPk = w/r. We can rearrange this to get

MPl/w = MPk/r. This last form has an interesting economic interpretation: The extra output from the last dollar spent on labor has to equal the extra output from the last dollar spent on capital. Remember this occurs at the lowest cost level.

Example: say MPl = MPk = 4, and w = 4 and r = 2. Note we do not have MPl/w = MPk/r. Since MPl/w = 4/4 = 1/1 taking a dollar away from labor means we lose roughly 1 unit of output. Also since MPk/r = 4/2 = 2/1 = 1/.5 we only have to spend 50 cents on capital to add back the unit of output we lost from having less labor. When the ratio is not equal costs are too high!

Cost concepts in a graph

Q

unit costs$ MC ATC

AVC

Q1

MC1

AVC1

ATC1

a

b

Interpretation

• I have picked Q1 arbitrarily and have drawn a line from this Q up to the highest cost curve.

• MC1 is the MC of the this unit.

• AVC1 is the AVC of all the units.

• ATC1 is the ATC of all the units.

Interpretation continued

• Since TC = TFC + TVC, ATC = AFC + AVC or AFC = ATC - AVC.

• So in the diagram, AFC1 = ATC1 - AVC1.

• Area a = AVC1 times Q1 = TVC1.

• Area b = (ATC1 - AVC1) times Q1 = TFC1.

Interpretation continued

• Area a + b =TVC1 + TFC1 = TC1.

• The concept of diminishing returns is the primary force driving costs in the short run. The rest of the ideas are definitions. The u - shape of the curves are due to the diminishing returns concept.

Long Run

• The above example assumed we could only have one unit of capital. Now let’s imagine we can have two units of capital.

• We would have a similar table of numbers and graphs as we did when only one unit of capital was available.

Long Run continued

• When we switch from one unit of capital to two units, we have the long run because all inputs are then variable.

• But with the two units we would have short run curves for that level of capital.

• Now we have two sets of cost curves, one for one unit of capital and one for two units of capital.

Long run continued

• Thus the graph of the long run is really just a bunch of curves, one for each plant size.

• I will draw two ATC curves, each with a different amount of capital used.

Long Run GraphsATC

Q

ATC1

ATC2

Q

Interpretation

• If output is going to be less than Q1 in the long run then only one unit of capital would be wanted because those units would be produced cheapest with one unit of capital.

• Greater than Q1 would be produced cheapest with two units of capital.

Interpretation

• The long run curve is parts of the short run curves. For each range of output the long run curve is the segment of the short run curve that is the lowest, representing the cheapest way to produce that range of output in the long run. The final long run curve is smooth. Let’s see.

Smooth long run curveATC

Q

Reason for long run shape

• The long run cost curve is said to be u - shaped, just as in the short run, but for a different reason. In the short run we had diminishing returns. In the long run we have economies of scale.

Reason continued

• The basic idea of economies of scale is that at least for a while when the plant size is increased the average cost curve is pushed down, implying average costs are lowest in a bigger plant. It may be that further increases in plant size push the average cost curve back up. This would technically be called diseconomies of scale.