Chapter 8Costs Functions

The economic cost of an input is the minimum payment required to keep the input in its present employment. It is the payment the input would receive in its best alternative employment. This cost concept is closely related to the opportunity cost concept (not talking about accounting costs).We usually assume that inputs are hired in perfectly competitive markets. The firm can get all the input it wants without affecting prices. The supply curve for an input is horizontal at the prevailing price.

L

w

Supply of the input = price of the input.Firm’s demand for the input

Total Cost, Revenue, and Profit• Total cost = C = wL + vK (with only 2

inputs, capital and labor)• TR = pq (with only 1 output)• Then, economic profit is:

• Thus, economic profit is simply a function of K and L, given that all prices (p, w, and v) and technology are fixed.

vKwL)L,K(pfvKwLpq

Cost Minimizing Input Choices (for given q)• Assume for now that output has been

determined to be q0 and the firm wishes to minimize its cost. That is, the firm must choose a specific point on the q0 isoquant.

K

L

q0

• Cost will be minimized by choosing the point where RTSLK (-slope of isoquant) equals the ratio of input prices (w/v). This happens when the rate of substitution in production equals the rate of substitution in the market.Min: C = wL + vKst: q0 = f(K, L) or q0 – f(K, L) = 0= wL + vK + (q0 –f(K, L))

0)L,K(fq

0Kfv

K

0Lfw

L

0

is the increase in C when q0 increases by one unit. is marginal cost, MC.

FOC

The SOC require diminishing RTSLK or q=f(K, L) strictly quasi-concave or K = f(L,q0) strictly convex.

ThendLdKRTS

MPMP

KfLf

vw

K,LK

L

So, RTSLK should equal w/v for the minimum cost combination of inputs, or the slope of the isoquant equals the slope of the isocost line

.MCMP

vMP

w

KL

).vw(

the marginal product per dollar spent is equal for all inputs.

Also, = marginal cost is the inverse of the above,

1v

MPw

MP KL

)dLdK(

K

vC 1

K*

0 L*wC 1 L

q0

C1 C2 C3

Minimum cost is C1, which is the minimum cost to achieve q0.The isocost line shows combinations of K and L that can be purchased with fixed total cost.

The solution can be a corner point, but not usually unless the inputs are close substitutes (close to linear isoquants). Assuming perfect substitutes and RTSLK > w/v, which input would not be used? (K!)

C3 > C2 > C1

Graphically

line.isocost Lvw

vCK

Dual: Output maximization subject to a cost constraint:Max: q=f(K, L)s.t.: C1 = wL + vK or C1 – wL – vK = 0= f(K, L) + D(C1 – wL – vK)D represents the marginal product of one additional

dollar of expenditure on inputs. It equals 1/.Solving the FOC yields K* and L* as did the primal.

Output MaximumK

L0

K*

L*

C1

qoq-1

Can we derive a demand curve for L by changing price (w) and looking at

the resulting change in L*?

• The answer is “yes”, but the logic is somewhat different from the consumer’s demand for a good. As w changes and L*

changes, the output level changes, which will change the market for q, which will change p (price of q). We cannot investigate the demand for an input without also considering the interaction of supply and demand for the output. The demand for the input is derived from the output market. Along the demand curve for L, v and p are held constant.

• Therefore, the analogy between consumer and firm optimization is not exact. (Isoquants are not directly interpretable as revenue whereas indifference curves represent utility.)

ΔUpy for consumer

qw for firm p in market for q. Demand for L is a derived demand from the market for q.

• Expansion Path – As the firm expands q, the cost minimization points trace out the expansion path.

L

The Expansion Path shows how optimal input usage changes as output expands with v, w, and technology constant (isocost lines are parallel because v and w are constant).

K

0

q4q3q2q1

C1

C2

C3

C4

K

0 L

If the production function is homothetic, the Expansion Path will be linear. The shape of the isoquants determines the shape of the Expansion Path. The Expansion Path shows points of equal RTSLK on the isoquants because the isocost lines are parallel.

An Expansion Path that slopes toward an axis indicates an inferior input on the other axis; ie., use of the input actually declines as q increases. For example, as you produce larger and larger acreages of vegetables, your use of hand-held implements would decrease; hoes and unskilled labor are inferior inputs.

Cost Functions come directly from the production function and prices. • Total cost: C = C(v, w, q)

Minimum Total Cost is a function of input prices and output quantity. Thus, the C function represents the minimum cost necessary to produce output q with fixed input prices. C represents the minimum isocost line for any level of q. It reflects the cost minimizing combination of inputs (K*, L*) for any given q.

A total cost function is analogous to an expenditure function in consumer theory. Someone define an expenditure function.

• Average Cost

• Marginal CostqCAC

AC is the cost per unit of output; AC = AC(v, w, q).

qCMC

MC is the change in C as output changes. It is the added cost for producing an additional unit of output. MC = MC(v, w, q).

Initially, we will hold v and w constant and look at how cost varies as q changes. This will give us standard two-dimensional graphs.

With a production function that shows constant returns to scale (homogeneous of degree 1, or linear homogeneous), C will be linear with fixed input prices. MC will be constant and equal to AC.

AC = MC

q

$$

q

C

However, typical (in theory) cost curves are sloped as follows:

$

$

Inflection point

C

q

AC = the slope of a cord from the origin to any point on C.

C is cubic 0thenand0C0C

''

'

Based on C, can draw .qCMC

Thus, when C is concave, MC is declining; when C is convex, MC is increasing; and MC is at a minimum at the C inflection point.AC = MC for first unit of q. AC reflects lower MC of first units of q. Once MC crosses AC, AC turns up reflecting higher MC of later units of q. MC crosses AC at minimum AC.q

MCAC

Slope of the cord = slope of C.

C is concave; C convex

Changes in Input Prices• When input prices change relative to each other, the expansion path

changes and the cost curves shift.• But C is homogeneous of degree 1 in input prices, so doubling all input

prices doubles C. This doubling of input prices would not affect q, L*, K*

or the Expansion Path,.• Because C is homogeneous of degree 1, MC and AC will be also.• “Pure inflation” will not affect input combinations or q, but it will affect

C, AC, and MC.

Mathematically and GraphicallyIf C1 = vK1 + wL1 and v and w are multiplied by m, C2 = mvK1 + mwL1 = mC1.

If v, w, and C double, optional point remains at A.

If w and v double, this isocost line changes from C1 to 2C1 = C2, but L*, K* and q* do not change.

LKRTSvw

2v2w

K

L

AC1/w = 2C1/2w =C2/2w is the intercept.

C2/2v=C1/v

Changes in a Single Input Price(relative prices of K and L change).

• The slope of the isocost line will change, resulting in changes in the optimal input combination and changes in the expansion path.Input Substitution (q constant)– Deals with how the optimal combination of K and L (or K/L) changes as w/v changes with q constant. Total Effect and its Direction (q changes)– An increase in v or w will

increase C. AC will also rise. MC will rise if the input is not inferior.From Footnote 9 on page 228 and Footnote 7 on page 226:

.constraintoutput an subject to problemon minimizaticost thefromfunction Lagrangian

optimal theis and k,v MC,λq Theorem Envelope thefrom where

,0v

MC inferior, isk if so ,qk

vMC ;

qk

q)(

qvvqvq)(

vMC 22

v

.

vw%ΔLK%Δ

LKvw

vwLK

s

An alternative elasticity of substitution is

with q, v, and w constant (Partial Elasticity of Substitution).

s is positive in a two-input world, but can be negative if three or more inputs.

This elasticity is similar to the elasticity of substitution () developed earlier from the production function if we remember that at the optimal combination of K and L

In practical application, which would be easier to estimate, or s?

A large s means the optimal input combination changes a lot as the price ratio changes, suggesting close to linear isoquants (close substitutes).

.dLdKRTS

vw

LK

vw

L

K Large s implies a large change in K/L for a change in w/v -close substitutes.

L

K

Small s implies a small change in K/L for a change in w/v – not close substitutes.

vw

q0q0

K/L

RTSRTS

K/L LK

LK

For many inputs

j

i

i

j

i

j

j

i

ij

xxww

wwxx

s

This formula gives a partialapproach for use where many inputs are involved.

Other input usages (other than inputs i and j) are not held constant in sij but they are in ij. sij does not have to be non-negative. Other inputs’ usages may change to give a net negative sign on sij; q isconstant.

Contingent Demand for Inputs and Sheppard’s Lemma

Quantity of output is under the firm’s control and actual input demand changes as output changes. However, cost minimization, subject to an output constraint, creates an implicit demand for inputs with quantity of output held constant. Contingent demand for an input (output-constant input demand) holds output constant similar to compensated demand for a good.

Sheppard’s Lemma is a result of the Envelop Theorem for constrained optimization. Sheppard’s Lemma is that the partial derivative of C with respect to an input price gives the contingent demand function for that input (q constant). The envelop theorem and Sheppard’s Lemma says that when the Lagrangian expression is at its optimum,

q)w,(v,Lw

λ)q,w,(v,w

q)w,C(v,

q)w,(v,Kv

λ)q,w,(v,v

q)w,C(v,

)],L,f(Kq[λwLvK If

c*

c*

******

w./wC/ and v/vC/ ** Use the Lagrangian method to find K* and L*. Or if you are given C, just take the partial derivative of C with respect to w and v to get the contingent demand functions for L and K, respectively.

Size of Shifts in Cost curves – two factors:• Cost Share – The more important the input, the

larger the cost curve response to a change in the input price. If the input makes up a large portion of total cost, an increase in its price will raise total cost substantially.

• Input substitutability – High substitutability with other inputs will reduce the effect on the cost curves of a change in the price of single input, ie., large partial elasticity of substitution (s) will reduce the effect on C of a change in v or w.

Short-Run versus Long-Run Costs• Short run is a time period in which some inputs are

variable, but at least one is fixed. This brings about the “Law of Diminishing Returns” to the variable inputs.

• Long run is a time period in which all inputs are variable. So far, evaluation of Economies of Scale and cost minimization have assumed the long run.

• In short run, we have some fixed cost, while in the long run all costs are variable.

• Not the same time period for all firms and industries. • Not a specific time unit.

• The text considers K as the fixed input in the short run.• SC = vK1 + wL where K is fixed at K1.• Then vK1 is short-run fixed cost, because it will not change

in the short run. wL represents short-run variable cost because L can be changed in the short run to affect q.

• SFC has to be paid regardless of level of q (0 to ). SVC changes as L changes to change q.

• If q = 0, SVC = 0 because L = 0.• If q = 0, SFC = vK1.

wLSVCvKSFC 1 So SC = SFC + SVC

Assuming any level of q requires some L.

SVC

This SC represents minimum short-run cost for each level of output, but not minimum C for each level of output.

SFC same over all q.

SVC = 0 if q = 0.

SC = SFC + SVC.

SFC has no effect on the shape of SC.

$

SFC

q0vK1

vK1

SC

Only one of the output levels in this graph shows the minimum total cost (C not SC) for K1. Because K is fixed at K1 and there is only one level of L that is optimal for K1, there will be only one point on SC where total cost (C) is minimum for K1.

With K fixed at K1, only at output q1 is total cost (C) minimized because only at point A is w/v = RTS. At any other quantity along the line K = K1, SC is minimized but C is not minimized. For C minimization, we would need to use less K and more L for points on K = K1 to the left of L1and more K and less L for points on K = K1 to the right of L1.

K

L

K1 A q3q2q1

q0

SC0 > C0 SC1 = C1

L2

SC2 > C2

Expansion Path

C0

SC0

SC1 = C1

SC2

C2

L0 L1

Per Unit Costs

qSVC

qSCSMC

qSCSAC

The concepts and relationships between SC and SAC and SMC are the same as described earlier for C, AC, and MC.

Only variable costs affect MC.

If SFC = α and SVC = q2, then SC = α + q2, SAC = α/q + q, and

qSVCq 2β

qSCSMC

$

0 qSAFC

SAVCSACSMC

βqqα

qSCSAC

qα

qSFCSAFC

βqq

SVCSAVC

SAC can be separated into fixed and variable parts.

SAVCSAFCSAC

•SAFC is a rectangular hyperbola that is asymptotic to 0. Thus, as q increases, SAVC approaches SAC, ie., SAVC is asymptotic to SAC.

•SMC passes through minimum SAVC and SAC.

•The cost curves (total and variable) reflect the shapes of corresponding physical product curves (production function, etc.).

SAFC (Same vertical distance)

Switching to the Long RunBy changing the fixed factor (K) we can draw a large number of SC curves. SC0 is drawn for K = K0

SC1 is drawn for K = K1SC2 is drawn for K = K2

For a constant returns to scale production function, C would be a ray from the origin.

C$

q

SC0

SC1SC2

Optimal q and C for K = K0

Optimal q and C for K = K1

•At the q that corresponds to optimal L for the fixed level of K, the SC curve will be tangent to C curve. At all other points, the particular SC is above C, reflecting the fact that SC represents costs higher than would be necessary if K were variable.

•The long run C is the “envelope” of the SC curves when K is allowed to vary.

•Every point on C is taken from a SC at its optimal combination of K and L.

q0 q1 q2

Average and marginal curves - For every level of K (fixed) there exists different SAC and SMC.

•If the firm has K0 and is producing q0, it could reduce cost per unit of output by increasing labor to L0

' at minimum SAC0; but q0' could be produced at a lower cost

with K0' reflected by SAC0

'. Further reduce average cost by increasing L to minimum SAC0′, which q could be produced with more K, etc. •Each SAC is tangent to AC at the optimal amount of L for that level of fixed K, but that point is not the minimum point for that SAC except at point A where SAC is tangent to the AC curve at minimum AC and at minimum SAC1.

q

$

SAC0 SMC0

SAC0

q0 = q0(L0,K0)

SAC0'

SMC1AC

MCSMC2

SAC2SAC1

q1 = q1(L1,K1) q2 = q2(L2,K2)q0' = q0

'(L0',K0)

q0' = q0

'(L0',K0

')

Level of K indicates the size of “plant”.K2 > K1 > K0' >K0

A

• All SMC curves cross their respective SAC curves at minimum SAC.

• SMC curves cross MC at the C minimizing level of L for the fixed level of K.

• MC is made up of many optimal points from the SMC curves. At q1, minimum SAC1 equals minimum AC, and SMC1 equals MC.

• The most efficient scale is K1 capital with q1 produced.• The slopes of the SMC curves are greater than the slope of the MC

curve because of diminishing returns with fixed K. The slopes of the SAC curves reflect diminishing returns also.

• Only at q1 will the firm be “satisfied” in the long run because, if the firm is at a point other than the minimum on any SAC curve, it can reduce costs for that level of fixed K by changing L and q to minimum SAC. But for all SAC curves other than SAC1, the q produced at the minimum point on the SAC curve could be produced with another level of K at a lower cost.