Costs

The goal of this chapter is to forge a link between the production process (output) and the cost of production. We know that the amount of output a firm chooses to produce will depend on the

technology it uses, but it also depends on the relative costs of inputs. In fact, there is a direct relationship between the production process and costs.

As an example, farms in North America are very capital intensive, usually with millions of dollars of machinery operated by a few labourers. Farms in Africa are relatively labour intensive, with

many labourers and little capital. Does this difference arise because farmers in Africa are not using the most efficient means of production available to them?

Costs in the Long RunAll inputs are variable

What would the optimal allocation (lowest cost) of inputs be to produce a given level of output?

Isocost Line: The combination of inputs the firm can purchase for a given cost (Sound Familiar?)

Example: C=$200, r=$2, w=$4

Minimizing Cost at a Given Output

This problem is solved in essentially the same way. Cost is minimized at the tangency between the isoquant and some isocost line.

minC = wL + rK s.t. Q = Q̄

Maximizing Output at a Given Cost

Recall our use of isoquants from the Theory of Production. Output is maximized at the tangency between the isocost line and some isoquant.

Solving Mathematically

MRTS= MPK *MPL* = r

w

wMPL* = r

MPK *

pX1MPX1 = pX2

MPX2 = ... = pXNMPXN

Thus when costs are at a minimum, the extra output from the last $ spent on an input must be the same across all inputs.

More generally:

For Two Inputs:

Example: Farming in Africa vs Saskatchewan

Even with a equivalent production functions (same technology) farming in Africa will be relatively labour intensive compared to Saskatchewan

Q = 10L1/2K1/2

Suppose that both farms have the same technology and want to produce 2000 tonnes of grain, but wages are $10 in Africa vs $40 in Sask. Renting capital costs $10.

No Interior SolutionsThe cases of perfect substitutes and perfect complements is similar to consumer choice.

Comparative StaticsHow do input price changes affect cost minimization? Say wage increases.

Optimal Input Choice and Long-Run Costs

Output Expansion Path: The locus of tangencies (minimum-cost input combinations) traced out by an isocost line of given slope as it shifts outward into the isoquant map for a production process

Note that the points S, T, and U identify the output (Q) and total cost (TC) of these optimal input bundles.

S" (Q1,TC1)

T" (Q2,TC2)

U" (Q3,TC3)

The output, total cost combinations from the expansion path (the cost minimization) can then be plotted to give the long run total cost curve. So we move from describing costs as functions of prices (iso-cost lines) to functions of quantity output. Basically, how much would it cost to produce q? Note: The shape of C(q) is determined by returns to scale (ie: how the isoquants are separated)

Cost Functions

Long Run Marginal Cost and Long Run Average Cost

LMC =dC(Q)

dQLAC =

C(Q)Q

Cost Curves for CRS Production

Recall that Constant Returns to Scale means if you increase all of your inputs by a factor of c, then you’re output also increases by a factor of c.

F(cK,cL) = cF(K,L)

Since the slope of LTC is constant, the long run marginal cost (LMC) must be constant. Also, LAC must be constant and the same as LMC.

Cost Curves for DRS

Recall that Decreasing Returns to Scale means if you increase all of your inputs by a factor of c, then you’re output increases by less than a factor of c.

F(cK,cL)< cF(K,L)

LTC curves upwards and grows faster than output, LMC and LAC are both increasing. Note that LMC is always higher than LAC.

Also note that this production process has some limits to how large it can scale its output.

Cost Curves for IRS

Recall that Increasing Returns to Scale means if you increase all of your inputs by a factor of c, then you’re output increases by more than a factor of c.

F(cK,cL) > cF(K,L)

LTC grows slower than output, LMC and LAC are both decreasing. Note that LMC is always lower than LAC.

Also note that this production process has no limits to how large it can scale its output.

The three extreme cases just shown are “pure” cases, recall that a production process may have IRS, DRS or CRS over the range of output.

Costs in the Short Run

Quantity of labour Quantity of output

(person-hr/hr) (bags/hr)

0 0

1 4

2 14

3 27

4 43

5 58

6 72

7 81

8 86

Fixed Cost: Cost that does not vary with level of outputVariable Cost: Cost that does vary with level of output

Total Cost: All costs of production, the sum of FC and VC

FC= rKo

VC= wL

TC= FC+VC= rKo+wL

Let us consider a short run production process with increasing marginal product at low input levels, then decreasing marginal product at high levels.

Ko= 120

Assume that the cost of labour is w=$10 per hour and the cost of capital is the interest rate (or rental value) r=$0.25

Q FC VC TC

0 30 0 30

4 30 10 40

14 30 20 50

27 30 30 60

43 30 40 70

58 30 50 80

72 30 60 90

81 30 70 100

86 30 80 110

Another ExampleThe production function is Q=3KL. The price of capital is $2 per machine-hour. The price of

labour is $24 per person-hour, and capital is fixed at 4 machine-hr/hr.

More Short Run Costs

Average Fixed Costs: Fixed cost divided by output quantity.Average Variable Cost: Variable cost divided by output quantityAverage Total Cost: Total cost divided by output quantityMarginal Cost: Change in Total cost resulting from a unit change in output

AFC= QFC= QrKo

AVC= QVC= QwL

ATC= AFC+AVC= QrKo+wL

MC=DQDTC

=DQDVC

Relationship between MP, AP, MC and AVC (Short-run)

We’ve seen previously that the MP curve cuts AP at its minimum. We’ve also seen that AP cuts AVC at its minimum. There is a direct link between these relationships.

MC =dV C

dQMP =

dQ

dL

MC= MPLw AVC= APL

w

Note: The scale of the Q axis is not constant. L does not bear a constant proportional relationship with Q.

Long Run Costs and Industry Structure

Natural Monopoly: An industry whose market output is produced at the lowest cost when production is concentrated in the hands of a single firm.

These cost curves represent industries that will have a fairly concentrated structure. This industry will be dominated by a few large firms.

These cost curves represent industries that will have a fairly unconcentrated structure. This industry will be dominated by many small firms.

Long Run, Short Run Cost Relationship

Consider a set of Short Run Average Cost curves for a given range of the fixed factor input.The LAC is the “outer envelope” of all of the SAC curve.