Lecture 6

Economics 50: Intermediate Microeconomics

Summer 2010

Stanford University

Michael Bailey

Lecture 6: Cost Minimization and Cost Curves

Overview

• The cost minimization problem (CMP) is for the firm to minimize its cost∑wixi subject to a pro-

duction constraint f(x1, ..., xn) = y

• The CMP is solved by the conditional factor demand functions

• The CMP is mathematically equivalent to the expenditure minimization problem

• The cost function gives the lowest cost attainble given the prices and target output

• The output expansion path is the set of conditional factor demands for all values of output

• Using Shephard’s Lemma, the conditional factor demands can be derived from the cost function, and

sometimes can be used to recover the production function

• A profit maximizing firm is a cost minimizing firm, and the firm’s supply function solves max py−C(y)

• A firm has economies of scale if average costs are decreasing in output, ∂AC(y)∂y < 0. A firm has

diseconomies of scale if average costs are increasing in output, ∂AC(y)∂y > 0

Economies of Scale ⇐⇒ Increasing Returns to Scale

Neither ⇐⇒ Constant Returns to Scale

Diseconomies of Scale ⇐⇒ Decreasing Returns to Scale

• The long-run cost is always less than the short-run cost. The long-run cost function is tangent to the

short-run cost function at the point when the short-run firm is constrained at the optimal long-run

level of inputs

1

Cost Minimization

Suppose that instead of maximizing profits, the firm set a target level of output, y, and produced that level

of output in the cheapest way possible. The objective function of such a firm would be given by:

min∑

wixi s.t. f(x1, ..., xn) = y

and is known as the cost minimization problem (sometimes referred to as the CMP). Notice the similarity

between this problem and the expenditure minimization problem that has min∑pixi s.t. u(x1, ..., xn) = u

as an objective function. In fact, the two problems are identical if we just relabel p = w, u() = f(), and

y = u.

Remark 1 The cost minimization problem and the expenditure minimization problem are identical (just

with different labels). The profit maximization problem is different from the utility maximization problem

because the firm is unconstrained.

Recall that the EMP has an optimality condition pxpy

= MUxMUy

. Hence, the optimality condition for the

cost minimization problem is MPLMPK

= wr , or the ratio of marginal products must equal the ratio of input

prices. We would get the same result by solving the lagrangian for the problem:

L =∑

wixi − λ(f(x1, ..., xn)− y)

which has first order conditions:

∂L∂xi

= wi − λ∂f(x1, ..., xn)

xi= 0

=⇒ λ =wi

∂f(x1,...,xn)xi

=wi

MPxifor all i

λ has an analogous interpretation to that of the EMP; if we relax the constraint by one unit, i.e. increase

target output, y, by one unit, the firm will have to increase its costs by wiMPxi

to achieve this new level of

output. If we set two first order conditions equal to each other, we would get:

λ =wi

MPxi=

wjMPxj

=⇒ wiwj

=MPxiMPxj

Hence to solve the CMP, set wiwj

=MPxiMPxj

, and then substitute into the constraint f(x1, ..., xn) = y.

We could derive the same result graphically. If the isoquant is fixed, the firm would minimize cost by

2

getting to the lowest iso-cost line (the budget line for the firm∑wixi = I is called an iso-cost line and is

the set of inputs that have the same cost) given it must produce on its isoquant. The minimum would be

where the iso-cost is tangent to the isoquant. The slope of the iso-cost is −wiwjand the slope of the isoquant

is −TRSxi,xj = −MPxiMPxj

, hence the tangency condition yields the same optimality condition: wiwj =MPxiMPxj

.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

L

K

Figure 1: The lowest iso-cost curve the firm can attain will be tangent to the given isoquant

Recall that the firm’s first order conditions for profit maximization are pyMPxi = wi, the input price is

equal to the marginal revenue product. Taking the ratio of this condtion for two inputs:

pyMPxipyMPxj

=wiwj

=⇒ MPxiMPxj

=wiwj

which is the same optimality condition for the cost-minimizing firm. So the profit maximizing factor

demands satisfy the optimality condition for cost-minimization. Hence a profit maximizing firm produces

at a level that minimizes cost. Given the profit maximizing firm is maximizing revenue minus cost, it will

produce output in a way such that costs are as low as possible. The cost minimizing firm is not necessarily

a profit maximizer, it depends upon whether the constraint y is chosen in a profit maximizing way. We’ll

discuss this in more detail later.

Conditional Factor Demand and the Cost Function

Just as the EMP requires convexity and monotonicity for the Lagrangian Method to work, so does the CMP.

Before solving the CMP, we must check that the production function is concave and monotone. Because the

EMP and CMP are mathematically identical, you can review the problems we solved for the EMP and the

3

answer to the CMP will be the same.

Example 2 f(L,K) = min(L,K)γ , find the level of inputs that produce y units of output at the lowest cost.

What is the cost of producing y in the cheapest way possible?

Note that the solution is where L = K. If L 6= K, then money is being spent on an input that is not

contributing to creating output.

L = K

=⇒ min(L,K)γ = Lγ = Kγ

Substituting this into the production constraint:

Lγ = Kγ = y

=⇒ LC(w, r, y) = KC(w, r, y) = y1γ

The cost of producing y is:

Cost = C(w, r, y) = wL+ rK

= wy1γ + ry

1γ

= (w + r)y1γ

Definition 3 The cost minimization problem is solved by the conditional factor demands:

xCi (w, y) = arg min∑

wixi s.t. f(x1, ..., xn) = y

The conditional factor demands are the cheapest input bundle that can produce y given the production

function at the nput prices.

Definition 4 The cost function is the minimum cost of producing y given prices:

C(w, y) ≡∑

wixCi (w, y)

Just like in the EMP, the factor demands are homogenous of degree r = 0 in input prices (only depend

upon relative prices, slope of iso-cost line) and the cost function is homogenous of degree r = 1 in input

prices.

Example 5 f(L,K) = 4LK, prices are (w, r), find the factor demand functions and the cost function.

4

f(L,K) is monotone in both inputs and is strictly concave. The optimality condition is:

MPLMPK

=4K

4L=K

L=w

r

=⇒ K =w

rL

=⇒ 4L(wrL)

= y (subbing into production constraint)

=⇒ L2 =y

4

r

w

=⇒ LC(w, r, y) =1

2

√yr

w

=⇒ KC(w, r, y) =w

rL =

1

2

√yw

r

C(w, r, y) = wLC(w, r, y) + rKC(w, r, y)

=w

2

√yr

w+r

2

√yw

r

=1

2

√ywr +

1

2

√ywr

=√ywr

Just like compensated demand, the conditional factor demand must slope downward. If the input price

of an input increases, the firm will substitute away from that input to produce y.

Figure 2: The conditional factor demands are downward sloping

5

Output Expansion Path

Definition 6 The output expansion path (OEC) is the set of conditional input demands for all values of

y ∈ R holding input prices constant.

The OEC is analogous to the ICC in consumer theory. The ICC is derived by increasing income and

plotting the optimal demands. The OEC is derived by expanding output and plotting the conditional input

demands. For the ICC, the budget line is expands outward, whereas for the OEC, the isoquants expand

outward. If the output expansion path slopes downward over some range, then one of the inputs is an inferior

input, meaning the firm would use less of that input to reach a higher level of output.

Figure 3: The output expansion path is the set of conditional factor demands for all levels of output

Example 7 f(L,K) = min(aL, bK)γ

Because the conditional input demands are just the set of kink points, aL = bK, the output expansion

path is just the line from the origin through the kink points, i.e the line K = abL.

Example 8 f(L,K) = (aL+ bK)γ

The firm will use either labor or capital exclusively in production, so the OEC is either the x-axis (if

labor is cheaper) or the y-axis (if capital is cheaper). If they are equally expensive, the OEC is every positive

combination of (L,K).

Shephard’s Lemma

Shephard’s Lemma holds for the CMP as well as the EMP and allows us to derive the conditional input

demands from the cost function.

6

Proposition 9 Shephard’s Lemma:∂C(w, y)

∂wi= xCi (w, y)

Example 10 C(w, r, y) =√ywr; given the cost function, derive the conditional factor demands.

∂C(w, r, y)

∂w=

1

2

√yrw−1/2

=1

2

√yr

w

= LC(w, r, y)

∂C(w, r, y)

∂r=

1

2

√ywr−1/2

=1

2

√yw

r

= KC(w, r, y)

Recovering the Production Function

We can use Shephard’s Lemma to recover the production function from the cost function. Shephard’s Lemma

gives us a system of equations in prices, output, and factor demand. By factoring out input prices, we are

left with a function of just factors and output which can be solved for the production function.

Example 11 C(w, r, y) =√ywr; solve for the production function that gives rise to this cost function.

By Shephard’s Lemma we have LC(w, r, y) = 12

√y rw and KC(w, r, y) = 1

2

√ywr . Solving both of these

functions so that they are a function of input prices:

LC(w, r, y) =1

2

√yr

w

=⇒ 2L√y

=

√r

w

KC(w, r, y) =1

2

√yw

r

=⇒√y

2K=

√r

w

We can combine these into an equation that is only a function of inputs and output and solve for the

production function:

2L√y

=

√y

2K

y = 4LK

7

Relationship Between Profit Maximization and Cost Minimization

A profit maximizing firm is also a cost minimizer. This implies that the factor demand functions and the

conditional factor demand functions coincide at the level of output that maximizes profits, which is given by

the supply function:

x∗i (py, w) ≡ xCi (w, y∗(py, w))

Another implication is that the profit maximization problem can be recast in terms of the cost function.

Maximizing profits is equivalent to maximizing the following objective function:

maxy

pyy − C(w, r, y)

Hence the supply function solves:

y∗(py, w) = arg max pyy − C(w, r, y)

and the profit function can be written as:

Π(py, w) = pyy∗(py, w)− C(w, r, y∗(py, w))

This is equivalent to a 2 stage maximization process: (1) find the cheapest way of producing output, i.e.

find the cost function C(w, r, y) (2) maximize profits = revenue − costs = pyy−C(w, r, y). Intuitively, if the

firm sets the target level of output as the profit maximizing one, and then produces it in the cheapest way

possible, profits will be maximized.

Example 12 f(L,K) = min(L,K)γ ; find the cost function and use it to recover the supply function, the

profit function, and the factor demand for labor.

We found that the cost function is (w+ r)y1γ and the conditional input demands are y

1γ . We can find the

8

profit function and supply function by maximizing the objective function:

maxy

pyy − C(w, r, y)

= pyy − (w + r)y1γ

= y − (w + r)y1γ (normalize py = 1)

=⇒ 1 =1

γ(w + r)y

1−γγ (FOC)

=⇒ y1−γγ =

γ

(w + r)

=⇒ y =

[γ

(w + r)

] γ1−γ

=

[(w + r)

γ

] γγ−1

= y∗(py, w, r)

pyy∗(py, w)− C(w, r, y∗(py, w))

=

[(w + r)

γ

] γγ−1

− (w + r)

[(w + r)

γ

] 1γ−1

=

[(w + r)

γ

] 1γ−1

[(w + r)

γ− (w + r)

]= Π(py, w, r)

LC(w, y∗(py, w, r)) =

([(w + r)

γ

] γγ−1) 1γ

=

[(w + r)

γ

] 1γ−1

= L∗(py, w, r)

Cost Curves

Because a profit-maximizing firm is a cost-minimizing firm, the cost functions of the firm inform us about

optimal firm behavior. By breaking down the firm’s cost function into its various components, we will be

able to draw insight about firm supply and general firm behavior in the short and long run. First, some

definitions:

Definition 13 Econonomic cost includes the opportunity cost of a decision, whereas accounting cost includes

only the numerical cost of the decision.

For example, suppose that the firm purchases a pile of lumbar for $100,000, and now has to decide whether

to use it in production, or resell it. The accounting cost of the lumbar is $100,000, what the firm paid for it.

The economic cost is the opportunity cost, or the next best use of that lumbar, which would be the amount

for which the firm could sell the lumbar. Economic costs are decision specific and thus always take into

account what you are giving up by making a decision. If the firm could resell the lumbar for $80,000, then

9

the economic cost of using the lumbar in production would be $80,000. If the cost of the lumbar went up to

$110,000, then the economic cost of using the lumbar would go up to $110,000, whereas the accounting cost

is always stuck at $100,000.

Notice that the economic cost is the same as the input price, as the input price changes, the opportunity

cost of using that input changes. In this class, all our costs include economic costs. If a firm has a machine,

the rental rate, r, is the value the firm could get by renting the machine to another firm. The wage of the

owner, which is often not included in accounting costs because the owner does not "pay" himself, would be

included in economic costs as the wages the owner would earn working at the next best firm. In a competitive

market, it is not uncommon for the firm to earn 0 economic profit. What accountants would refer to as

positive profit, is often just the unadjusted return to capital and owners for not including in economic costs.

Definition 14 Fixed costs are the costs incurred by the firm in the short run due to its fixed factors of

production. There are no fixed costs in the long run because no factor of production is fixed. For example, if

capital is fixed at K = K, and the rental rate of capital is r, then the fixed costs of the firm in the short run

are rK.

Definition 15 Quasi-fixed costs are those costs that are incurred if the firm produces, but are not dependent

upon the level of output. For example, if the firm has costs for opening the factory that do not depend upon

the output of the factory, those would be quasi-fixed costs.

Definition 16 Variable costs are those costs that vary with the amount of output produced and are equal to

0 when output is 0.

Definition 17 Sunk costs are those costs that are non-recoverable. For example, if the firm buys a machine

for $100,000, and can resell it for $80,000, then $20,000 of the cost of the machine is sunk. Sunk costs do

not alter the decisions of the firm since they are lost no matter what the firm does. Notice that fixed costs

do not necessarily equal to sunk costs, like in the previous example.

Suppose that input prices are not changing, we can treat them as exogenous variables and write the cost

function as C(w, y) = C(y). Notice that Cost is the sum of fixed and variable costs:

C(y) = Fixed Costs + Variable Costs

= FC + V C(y)

10

The average cost is the average per-unit cost of producing output y :

Average Cost ≡ AC(y) =C(y)

y

=FC

y+V C(y)

y

The marginal cost is the cost of producing the marginal unit of output:

Marginal Cost ≡ MC(y) =∂C(y)

∂y

=∂FC

∂y+∂V C(y)

∂y

=∂V C(y)

∂y

Notice that if we integrate marginal cost (take the area under the marginal cost curve), we get variable

cost:

∫MC(y) =

∫∂V C(y)

∂y= V C(y)

=⇒ C(y) = FC +

∫ y

0

MC(y)

Example 18 Suppose we wrote the short-run cost function generically as C(y) = wL(y) + rK. What is the

marginal cost of production?

MC(y) = w∂L(y)

∂y=

w∂y

∂L(y)

=w

MPL

Intuitively, the marginal cost of production is the wage (the cost of increasing the amount of labor) times

by the amount of labor needed to produce one more unit ( 1MP ).

Example 19 C(y) = y3 + 7. What are the fixed costs of production? The variable costs? What are the

marginal and average costs of production?

FC = 7

V C(y) = y3

AC(y) =y3 + 7

y= y2 +

7

y

MC(y) = 3y2

Remark 20 If the marginal cost is above the average cost, the average cost is increasing. If the marginal

11

0 1 2 3 4 50

10

20

30

y

Cost

Figure 4: Graph of FC = 7, V C(y) = y3, and C(y) = y3 + 7. The difference between cost and variable costis the fixed cost

cost is below the average cost, the average cost is decreasing. The marginal cost goes through the minimum

of the average cost.

12

0 1 2 3 4 50

10

20

30

y

Cost

Figure 5: Graph of FCy = 7y , AV C(y) = y2, and AC(y) = y2 + 7

y . Notice that average fixed costs are alwaysfalling, average variable costs are always increasing, and average costs are decreasing and then increasing

0 1 2 3 4 50

2

4

6

8

10

12

14

y

Cost

Figure 6: Graph of AC(y) = y3+7y and MC(y) = 3y2

13

Returns to Scale and Cost Functions

If a firm operates with a constant returns to scale technology, then the lowest cost of producing λ units of

output, is λC(1) where C(1) is the lowest cost of producing 1 unit of output. This is because the firm can

just scale up its cost minimizing inputs that produced one of output in the cheapest way and achieve an

output on the same scale. If a firm has constant returns to scale, we can write its average cost and marginal

cost as:

AC(y) =C(y)

y=C(1)y

y= C(1)

MC(Y ) =∂C(y)

∂y=∂(C(1)y)

∂y= C(1)

If a firm has CRTS, its marginal cost and average cost are equal to each other and are constant.

Example 21 A firm has a production function f(L,K) = (LK)12 , what is it’s average and marginal cost

when all input prices are equal to one?

The cheapest way this firm can produce one of output is by using one unit of labor and unit of capital,

thus C(1) = 1 + 1 = 2. Since the production function has CRTS, we know that AC = MC = 2. If we

had solved for the cost function explicity from the CMP, we would get that C(w, r, y) = 2y√wr = 2y, and

MC = AC = 2.

Economies of Scale

An important question is how do average costs change as output changes, or how do costs change as the firm

"scales up". A firm whose costs per unit are falling as scale is increasing is said to have economies of scale.

Definition 22 A firm has economies of scale if average costs are decreasing in output, ∂AC(y)∂y < 0. A firm

has diseconomies of scale if average costs are increasing in output, ∂AC(y)∂y > 0.

Economies of Scale ⇐⇒ Increasing Returns to Scale

Neither ⇐⇒ Constant Returns to Scale

Diseconomies of Scale ⇐⇒ Decreasing Returns to Scale

To see why this relationship between returns to scale and economies of scale holds, suppose that the firm

doubled its inputs. This change would double its costs (∑

wi2xi = 2∑

wixi). If the firm has CRTS, it

will end up producing exactly twice as much output, so average costs are constant. If it has IRTS, it will

more than double its output, and so average costs per unit must decrease. Similarly, if it has DRTS costs

per unit must increase.

14

An alternative way of determining economies of scale is to look at the elasticity of cost with respect to

output:

εC,y =%∆C(y)

%∆y=∂C(y)

y

y

C(y)= MC(y) · 1

AC(y)

If MC(y) > AC(y), AC(y) must be increasing, and if MC(y) < AC(y), AC(y) must be decreasing.

Therefore:

Economies of Scale ⇐⇒ εC,y < 1

Neither ⇐⇒ εC,y = 1

Diseconomies of Scale ⇐⇒ εC,y > 1

Example 23 C(y) = y3 + 7, AC(y) = y2 + 7y

∂AC(y)

∂y= 2y − 7y−2

∂AC(y)

∂y< 0 =⇒

2y < 7y−2

=⇒ y3 <7

2

The firm has economies of scale when y <(72

) 13 , and diseconomies of scale otherwise.

0 1 2 3 4 50

2

4

6

8

10

12

14

y

Cost

Figure 7: The firm has economies of scale until y =(72

) 13

Economists tend to characterize firms as initially having economies of scale, and then reverting to disec-

onomies of scale. For small amounts of input, the firm can benefit greatly from specialization in production,

15

but eventually those benefits and other constraints means the firm faces decreasing returns to scale.

Short-run versus Long-run Cost

The short-run cost is always larger than the long-run cost, except at the output when the factors that are

fixed in the short run are optimal for the long-run problem. It should be intuitive that the firm can always

achieve a lower cost in the long run because it can freely choose all the inputs of production whereas in the

short run some factors of production are fixed.

Figure 8: The short-run cost conditional factor demands (black) when capital is fixed always lie on a higheriso-cost then the long-run conditional factor demands (red). At the level of output when the fixed level ofinput is optimal in the long-run, the short-run and long-run costs coincide

16

Figure 9: The short-run cost is always higher than the long-run cost, except at the level of output when thefixed factor is the optimal long-run choice

The long-run cost curve will thus lie below every short-run cost curve (where the set of short run cost

curves is the set of curves C(y,K) for all values of the fixed input K) and be equal to each short run cost

curve at the level of output such that the fixed level of output for that short-run curve is the optimal level ot

the input in the long-run problem. Thereofre, the long-run curve is just the lower envelope of the short-run

curves. The short-run average cost and long-run average cost curves thus have the same relationship. as the

short and long-run cost curves.

The long-run marginal cost goes through the minimum of the long-run average cost curve. Each short-run

marginal cost curve intersects the long-run marginal cost at the output when the short-run average cost is

tangent to the long-run average cost.

17

Figure 10: Short-run and long-run average cost

Figure 11: The long-run cost is tangent to every short-run cost curve, and is thus the lower envelope of allthe short-run cost curves

18

Figure 12: Each short-run marginal cost curve goes through the minimum of short-run average cost. Thefirm operates on the marginal cost curve for the short-run average cost that is tangent to the long-run averagecost

Figure 13: The long-run average cost curve goes through the minimum of the long-run average cost, and isequal to the short-run marginal cost where the short-run and long-run average cost curves are tangent

19

20

Example 24 C(y,K) = K + y2

K

To find the long-run cost function, we find the level of the fixed factor K such that firm achieves the

lowest cost:

minK

K +y2

K

=⇒ 1− y2K−2= 0

=⇒ K = y

=⇒ C(y) = y +y2

y= 2y

0 1 2 3 4 50

1

2

3

4

5

y

Cost

Figure 14: Long-run cost and a set of short-run cost functions

AC(y) = 2

AC(y,K) =K

y+

y

K

Example 25 f(L,K) = 4LK; find the short-run and long-run cost function.

21

0 1 2 3 4 50

1

2

3

4

5

y

Cost

Figure 15: Long-run average cost and a set of short-run average cost functions

Notice that the short-run factor demand for L has to satisfy the output constraint:

4LK = y

=⇒ LC(y,K) =y

4K

=⇒ C(y,K) = wL+ rK

= wy

4K+ rK

To find the long-run cost function, we find the value of the fixed factor that minimizes the short-run cost

function:

minK

wy

4K+ rK

=⇒ −wy4K−2

+ r = 0

=⇒ K2

=wy

4r

=⇒ K =1

2

√wy

r

=⇒ C(y) = wy

4 12√

wyr

+ r1

2

√wy

r

=1

2

√wyr +

1

2

√wyr

=√wyr

22

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10

y

Cost

Figure 16: Long-run cost and a set of short-run cost functions when w = r = 1

AC(y) =

√wr

y

AC(y,K) =w

4K+rK

y

0 10 20 30 40 500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

y

Cost

Figure 17: Long-run average cost and a set of short-run average cost functions when w = r = 1

23