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Cost Minimization

Cost Minimization

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Cost Minimization. Accountants vs. Economists. Economists think of costs differently from accountants. Economics Cost = Explicit Costs + Implicit Costs Accounting Cost = Explicit Costs Opportunity Cost is synonymous with Cost Measuring implicit opportunity costs is difficult - PowerPoint PPT Presentation

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Page 1: Cost Minimization

Cost Minimization

Page 2: Cost Minimization

Accountants vs. Economists• Economists think of costs differently from accountants.– Economics Cost = Explicit Costs + Implicit Costs– Accounting Cost = Explicit Costs– Opportunity Cost is synonymous with Cost– Measuring implicit opportunity costs is difficult• generally, these are resources “owned” by the firm• Includes the value of the entrepreneur’s time• Includes accounting profit at next best alternative use of

firm’s resources – or, accounting profit from selling all owned resources and investing the proceeds in the best possible alternative investment.

Page 3: Cost Minimization

Fixed Cost and Sunk Cost• Fixed vs. Sunk cost– “fixed” means its cost does not vary with output– “sunk” means its cost cannot be avoided• The security guard outside the parking lot is a fixed cost,

but if they can be fired at a moment’s notice, the cost is not sunk• A long term lease for renting a building is fixed and sunk

– Varian calls a fixed cost that is not sunk a “quasi-fixed cost”

– Sunk, but not fixed? No, because if it varies, it can be avoided.

Page 4: Cost Minimization

Short Run – Long Run• Key distinction in terms of optimizing

behavior if everything can be controlled vs. not everything can be controlled.

• Short run, the quantities of some inputs used in production are fixed and some are variable.– Short run, firms can shut down (q=0), but cannot

exit the industry• Long run, the quantity of all inputs used are

variable.– Long run, firms can enter or exit an industry.

Page 5: Cost Minimization

To Distinguish Short Run from Long Run

•We need two inputs, one always variable, and one that can only be adjusted periodically, but is then re-fixed.• To accomplish that, we assume two inputs

•Homogeneous labor (L), measured in labor per time•Homogeneous capital (K), measured in machine per

time• Entrepreneurial costs assumed to be zero (or

included in fixed costs)• Inputs are hired in perfectly competitive markets, so

price of L, w, and price of K, v, are not a function of L and K

Page 6: Cost Minimization

To be clear

• L is assumed always variable• K is always fixed when the firm is in

production, but can periodically be adjusted to a new fixed amount (so in the long run it can be varied to a new fixed amount).

• That is, production always takes place in a short run situation.

Page 7: Cost Minimization

Notation• K, capital; v, rental rate of capital• L, Labor; w, wage rate

5340 Nomenclature(Typical Intermediate Micro Abbreviations in Parentheses)

Short Run Long Run

Total Cost SC (TC) C (LRTC)

Average Total Cost SAC (ATC) AC (LRAC)

Variable Cost VC -

Average Variable Cost AVC -

Fixed Cost FC -

Average Fixed Cost AFC -

Marginal Cost SMC (MC) MC (LRMC)

Page 8: Cost Minimization

Cost

• Clearly, C can be stated as:C = vK + wL

• But this is meaningless, we need cost as a function of output, not K and L.

• To get that, back to the production function

Page 9: Cost Minimization

Graphically

K

L

q

q = f(K, L)

q0q1

q2q3

q0

q1

q2

q3

An entire range of input combinations can be used to produce every level of output.

Page 10: Cost Minimization

Short Run, K constant

L

q

q = f(K=K1, L)

q = f(K=K2, L)

q = f(K=K3, L)

Three different slices through the production function at different levels of K.

q1

q2

q3

Possible to produce the same level of output at different combinations of K and L.

Page 11: Cost Minimization

Short Run, K constant

L

qLet’s look at just one slice.

q2

L2

At some fixed level of K, for every q, we know how much L

it will take.

2q K ,L

Page 12: Cost Minimization

Production to CostFlip the axis

L

q L

q

2q K ,L

2L L q,K

Page 13: Cost Minimization

Production to Cost

L = L(q,K)L

q

Page 14: Cost Minimization

Production to Cost

VC = w· L(q)Add in FC,SC=SVC+FC

q

FC

SC = w·L(q)+vKMultiply L by w to get labor cost$/time

Page 15: Cost Minimization

Production to Cost

VC = w· L(q)

And we can figure a few things$/time

q

FC

SC = w· L(q)+rK

dSC dVCSMC

dq dqSC VC

SAC ,AVCq q

SC VC FCSAC AVC AFC

Page 16: Cost Minimization

Production to Cost

AVC$

q

AFC

SACSMC

dSC dVCSMC

dq dqSC VC

SAC ,AVCq q

SC VC FCSAC AVC AFC

Page 17: Cost Minimization

Short Run, K constant

L

q

q = f(K=K1, L)

q = f(K=K2, L)

q = f(K=K3, L)

Inflectionpoint, q3

With more K, you can produce more with a given L and the inflection point moves towards more L and q.

What does that mean for cost curves?

Inflectionpoint, q2

Inflectionpoint, q1

Page 18: Cost Minimization

Production to Cost

$

q

SC = w· L3(q)+rK3

SC = w· L2(q)+rK2

SC = w· L1(q)+rK1

Fixed cost is higher with more K, but the inflection point is further to the right, with a slower build-up of crowding.

FC

Inflection Points

Page 19: Cost Minimization

C

$

q

SC = w· L3(q)+rK3

SC = w· L2(q)+rK2

SC = w· L1(q)+rK1

C is the lowest point on any of the SC curves for any q

C

Page 20: Cost Minimization

SAC

$

q

SAC = C1/q

AC is the lowest point on any of the SR ATC curves for any q

LAC

SAC = C2/qSAC = C3/q

Min value at higher q with more K.

CAC

q

Page 21: Cost Minimization

MC

$

q

MC is the slope of the C curve at any q

C

CMC

q

Page 22: Cost Minimization

Short Run – Long Run

• Firms ALWAYS produce in a short run situation with at least one fixed and one variable input.

• Being in the long run simply means the firm has adjusted to an optimal level of capital to minimize the cost of producing any chosen (profit maximizing) level of output.

Page 23: Cost Minimization

Different Shapes for the long run C Curve

$

q

There are four different potential shapes

C

Page 24: Cost Minimization

C -- CRS

$

q

CRS, as K and L are scaled upwards, C rises at the same rate.That is, MC is constant.

C

Page 25: Cost Minimization

C -- CRS

$

q

ATC = TC1/q

AC=MC

ATC = TC2/qATC = TC3/q

Min value at higher q with more K.

CRS, as K and L are scaled upwards, LAC rises at the same rate.That is, LAC is constant.

Page 26: Cost Minimization

C -- IRS

$

q

IRS, as K and L are scaled upwards, C rises more slowlyThat is, MC is decreasing.

C

Page 27: Cost Minimization

C -- IRS

$

q

AC

CRS, as K and L are scaled upwards, AC rises at a slower rate. AC is decreasing.

MC

Page 28: Cost Minimization

C -- DRS

$

q

DRS, as K and L are scaled upwards, TC rises fasterThat is, MC is increasing.

C

Page 29: Cost Minimization

LTC -- DRS

$

q

AC

CRS, as K and L are scaled upwards, AC rises at a faster rate. AC is increasing.

Note, it is not the low point of each ATC curve, but the simply the lowest point on any ATC.

MC

Page 30: Cost Minimization

C – IRS, CRS, DRS

$

q

IRS, then CRS, then DRSC

Page 31: Cost Minimization

C – IRS, CRS, DRS

$

q

AC

So we have a U-shaped AC curve for a completely different reason than the SAC curve is U-shaped.

MC

Page 32: Cost Minimization

Firm Decisions• So which short run curve should we be on?– That is, how much K (and then L) do we want to hire

(with K fixed in the SR)?• Two choices– Profit Maximization: Firms maximize profit by

choosing q*, K and L to minimize cost all at once.– Cost minimization: Firms choose a q*, then choose K

and L to minimize the cost of producing q*.

Page 33: Cost Minimization

Profit Maximization

• Economic profits () are equal to = total revenue - total cost

• Total costs for the firm are given byC = wL + vK

• Total revenue for the firm is given bytotal revenue = R = p·q = p·f(K,L)

• Economic profits () are equal to = p·f(K,L) - wL - vK

Page 34: Cost Minimization

Profit Maximization• Solving to maximize profit means jointly

choosing q = q* along with K* and L*.– Yields profit maximizing factor (input) demand

functions which provide L* and K* that minimizes the cost of producing q*.

– When r, w, and p change, L*, K*, and q* all change.– L*=L(w, v, p); K*=K(w, v, p); q*=q(w, v, p)– These functions allow for a change in q* (the

isoquant) when prices change.

Page 35: Cost Minimization

Cost Minimization• That firms minimize cost is a weaker hypothesis

of firm behavior than profit maximization.– Yields quantity constant (quantity contingent)

factor (input) demand functions which provide L* and K* that minimize the cost of producing q0.

– When r and w change, L* and K* do change, but q does not, stay on one isoquant.

• Why bother? It is how we get the cost functions and curves (i.e. what we use in principles and intermediate micro)

Page 36: Cost Minimization

New Direction in Graphing

• In all the graphs above, we have illustrated the long run as a series of short run curves and traced out the envelope.

• Good for intuition, but not terribly tied to the math of the optimization

• Let’s switch to the isoquant graph.

Page 37: Cost Minimization

Intuitively

L

K Isoquant, all combinations of factors that yield the same output.Slope is -dK/dL

Isocost (total cost), all combinations of factors that yield the same total cost of production.Slope is -w/v

Page 38: Cost Minimization

Cost-Minimizing Input Choices• When K and L change a small amount

• Along an Isoquant, K Ldq f dK f dL

L

K

MPdKRTS

dL MP

Page 39: Cost Minimization

Cost-Minimizing Input Choices• TRS is the change in K needed to replace one L

while maintaining output.• Minimum cost occurs where the TRS is equal to

w/v– the rate at which K can be traded for L in the

production process = the rate at which they can be traded in the marketplace

L

K

MP wMP v

Page 40: Cost Minimization

Intuitively

• Firing one L has MB of $20 and MC of $2.5• Lowers cost by $17.50

L

K

TRS=.25

w=20, v=10w/v=2

Page 41: Cost Minimization

Intuitively

• Firing one L has MB of $20 and MC of $10• Lowers cost by $10

TRS=1

w=20, v=10w/v=2

L

K

Page 42: Cost Minimization

Intuitively

• Firing one L has MB of $20 and MC of $15• Lowers cost by $5

TRS=1.5

L

Kw=20, v=10

w/v=2

Page 43: Cost Minimization

Intuitively

• Firing one L has MB of $20 and MC of $20• Lowers cost by $0• Further reductions in L require an increase in cost as TRS >w/r.

TRS=2

L

Kw=20, v=10

w/v=2

Page 44: Cost Minimization

Intuitively

• Firing one L has MB of $20 and MC of $25• Raises cost by $5

TRS=2.5

L

Kw=20, v=10

w/v=2

Page 45: Cost Minimization

Plan

1. Figure out the quantity of K and L that minimize total cost, holding q constant.

2. Use the resulting factor demand curves to derive the cost functions.

Page 46: Cost Minimization

Cost-Minimizing Input Choices

• We seek to minimize total costs given q =q0

q = f(K,L) = q0

• Setting up the Lagrangian:

FOCs are

0

L

K

0

wL vK q f K,L

w f 0L

v f 0K

q f K,L 0

L

L

L

L

Page 47: Cost Minimization

Cost-Minimizing Input Choices• Dividing the first two conditions we get

LLK

K

fwTRS (L for K)

v f

• The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices

• But also

• Which tells us that for the last unit of all inputs hired should provide the same bang-for-the-buck.

L Kffw v

Page 48: Cost Minimization

Cost-Minimizing Input Choices• The inverse of this equation is also of interest

L K

w vff

• The value of the Lagrange multiplier is the extra costs that would be incurred by increasing the output constraint slightly by hiring enough L or K to increase output by 1.• That is λ = MC of increasing production by one unit.

Page 49: Cost Minimization

Cost-Minimizing Input Choices• SOC to ensure costs are RISING, along the

isoquant, away from the tangency:• Bordered Hessian

L K L K

3 L LL LK L LL LK

K KL KK K KL KK

2 23 L K LK K L KL L KK K LL

2 23 L K LK L KK K LL

0 0 ffH ff f 0

ff

H ( ff f ) ( ff f ) ( ff ) ( ff ) 0

H 2ff ff ff f 0

L L

L L L

L L L

L 22 L

L LL

0 fH ( ) 0

ff

L

The part in brackets is the same condition required for strict quasi concavity of the production function

Page 50: Cost Minimization

Intuitively

• Cost minimized for q0 when L=L* and K=K * L

K

RTS=2

w=20, r=10w/r = 2

K*

L*

Total CostSOC satisfied as moving along the isoquant means increasing TC

Page 51: Cost Minimization

SOC fail, Intuitively

• Cost maximized for q0 when L=L* and K=K * L

K

RTS=2

w=20, r=10w/r = 2

K*

L*

Total CostSOC not satisfied as moving along the isoquant means decreasing TC

Page 52: Cost Minimization

Conditional Factor Demand (aka Constant Output)

• Solve the FOC to derive– K * =Kq (w, v, q0)

– L * =Lq (w, v, q0)• Earlier, we considered an individual’s expenditure-

minimization problem– to develop the compensated demand for a good

• In the present case, cost minimization leads to a demand for capital and labor to produce a constant quantity of output.

Page 53: Cost Minimization

The Firm’s Expansion Path

L

K

The expansion path is the locus of cost-minimizing tangencies

The curve shows how inputs increase as output increases

K=KE(w, v, L)

L* L* L*

K*K*

K*

Page 54: Cost Minimization

The Firm’s Expansion Path• The cost-minimizing combinations of K and L for

every level of output (according to the quantity constant factor demand functions)

• If input prices remain constant for all amounts of K and L, locus of cost-minimizing choices is the expansion path

• Solve the minimization condition for K to get the expansion path

L

K

E

fwv f

K K (w,v,L)

Page 55: Cost Minimization

The Firm’s Expansion Path• The expansion path does not have to be a straight

line– the use of some inputs may increase faster than

others as output expands• depends on the shape of the isoquants

• The expansion path does not have to be upward sloping– if the use of an input falls as output expands,

that input is an inferior input

Page 56: Cost Minimization

Cost Functions• Long Run Cost

• Short Run Cost– The factor demand curves we have just derived are

not appropriate as they each assume L and K are variable.

q q

q q

q q

C* C(v,w,q) v K (v,w,q) w L (v,w,q)

v K (v,w,q) w L (v,w,q)AC AC(v,w,q)

q

d v K (v,w,q) w L (v,w,q)MC MC(v,w,q)

dq

Page 57: Cost Minimization

Short Run Cost Functions• With only two inputs, once K is fixed, the production

function dictates how much L is needed for each q.

* s

s

s

s

s

q f K,L

then, L L K,q

w L K,qVC w L K,q , AVC

q

w L K,q w KSC w L K,q w K, ATC

qdTC dVC

SMCdq dq

I am just using the “s” superscript here to denote Short Run

Page 58: Cost Minimization

Short Run & Long Run• Pick a level of output, q1 and hold w and v constant

1 1 1 1

s1 1 1

Allow K K(q ) such that K is the cost minimizing level of capital to produce q

SC w L q;K w K or SC SC q;K

If we allow K K* = K(q), then we get the long run C* function

C* SC q,K K(q) and since ever

1

1

1

ything is now a function of q, C*=C(q)

We then know SC q,K C(q), for all q

as SC q,K C* q at any q q

and SC q,K C* q at q q

dSMC dMCAlso, when evaluated at q

dq dq

q

C

q1

1SC= C q,K

C* C q,K(q)

Page 59: Cost Minimization

Short Long Relationship

q

$

ACATC

AVC

SMCMC

Page 60: Cost Minimization

SMC vs. LMC

q

$

1K K

When q rises in SR, cost rises by more than in the LR

When q falls in SR, cost falls by less than in the LR;Means from q1 to q2 the SMC is lower as C starts higher at q1.

q1

q2 q3

Page 61: Cost Minimization

Short Long Relationship(Easier to see if assume CRS)

q

$

ATC (K=K1)

AVC (K=K1)

SMC (K=K1)

MC=AC

q1 q2 q3

SMC actually lower from q1 to q2 in the short run as total cost is higher to start

Page 62: Cost Minimization

Fixed Proportions Factor Demand and Cost Functions

• Suppose we have a fixed proportions technology such that q = f(K,L) = min(aK,bL)

• To minimize cost, production will occur at the vertex of the L-shaped isoquants where q = aK = bL (any extra K or L only drives up cost)

• Expansion path: • Factor Demand Curves Kq=q/a and Lq=q/b• Cost function: C(w,v,q) = vK + wL = v(q/a) + w(q/b)

v w v wC(w,v,q) q , MC AC

a b a b

bK L

a

Page 63: Cost Minimization

Cobb-Douglas Cost Minimization• Suppose that the production function is Cobb-

Douglas: q = K L

• The Lagrangian expression for cost minimization of producing q0 is

ℒ = vK + wL + (q0 - K L )• The FOCs for a minimum are

ℒK = v - K -1 L = 0

ℒL = w - K L -1 = 0

ℒ = q0 - K L = 0

Page 64: Cost Minimization

• Dividing the first equation by the second gives us

1L

1K

MPw K Linput price ratio=

v K L MP

• This production function is homothetic– the TRS depends only on the ratio of the two inputs

w Kinput price ratio= TRS

v L

Cobb-Douglas Cost Minimization

Page 65: Cost Minimization

• Expansion path equation:– the expansion path is a straight line

– The K/L ratio is a function of w and v.

wLK

v

K wL v

Cobb-Douglas Cost Minimization

Page 66: Cost Minimization

Cobb-Douglas Input Demand• Using the remaining FOC (production function), solve for

the input demand equationsK* = Kq (v,w,q)L* = Lq (v,w,q)

1q

1q

vL q

w

wK q

v

Page 67: Cost Minimization

Cobb-Douglas Cost• Now we can derive total costs as

– Where

– MC

– AC

1

C(v,w,q) vK wL q Bv w

( )B

1

q Bv wMC(v,w,q)

1

AC(v,w,q) q Bv w

Page 68: Cost Minimization

CES Cost Minimization• Suppose that the production function is CES:

q = (K +L)/

• The Lagrangian expression for cost minimization of producing q0 is

ℒ = vK + wL + [q0 - (K +L )/]• The FOCs for a minimum are

ℒ L = w - (/)(K +L)(-)/()L-1 = 0

ℒ K = v - (/)(K +L)(-)/()K-1 = 0

ℒ = q0 - (K + L)/ = 0

Page 69: Cost Minimization

CES Cost Minimization

• Dividing the first FOC equation by the second gives us

1 ( 1) 1w L L KTRS

v K K L

• This production function is also homothetic

Page 70: Cost Minimization

CES Expansion Path• Expansion path

• Capital Labor ratio

w 1K L, =

v 1

K w 1, =

L v 1

Page 71: Cost Minimization

CES Input Demand• Using the remaining FOC (production function), solve for

the input demand equationsK* = Kq (v,w,q) L* = Lq (v,w,q)

1

q1

1 1

1

q1

1 1

q 1L , =

1v w w

q 1K , =

1v w v

Page 72: Cost Minimization

CES Cost Functions• To derive the total cost, we would use the inputs

demand functions and get

11

1 1

1 11 1 1

1 11 1 1

1 11 1 1

C(v,w,q) vK wL q v w

C(v,w,q) q v w

1where

1

q v wMC(v,w,q)

AC(v,w,q) q v w

Page 73: Cost Minimization

Law of Demand?

• With Cobb-Douglass, CES etc. we can take the partial derivative w.r.t. price of the inputs and see how quantity demanded responds.

• But if we don’t know the functional form, how do we know how demand for L and K will respond to changes in w and v?

• Comparative Statics

Page 74: Cost Minimization

Comparative Statics• Setting up the Lagrangian:

ℒ = wL + vK + [q0 - f(K,L)]• FOCs are

ℒL = w - ·fL = 0ℒK = v - ·fK = 0ℒλ = q0 - f(K,L) = 0

• Solve for L* = Lq(w,v,q)K* = Kq (w,v,q)λ* = λq (w,v,q)

Page 75: Cost Minimization

Comparative Statics

• Plug solutions into FOCw - λ(w,v,q)·fL(Lq(w,v,q), Kq (w,v,q)) ≡ 0

v - λ(w,v,q)·fK(Lq (w,v,q), Kq (w,v,q)) ≡ 0

q0 - f(Kq (w,v,q), Lq (w,v,q)) ≡ 0

• These are identities because the solutions (FOC) are substituted into the equations from which they were solved.

• Whatever prices and output may be, the firm will instantly adjust K and L to minimize cost of that level of output.

Page 76: Cost Minimization

Comparative Statics• Differentiate these w.r.t. w

To get:

q qL

q qK

q q0

2 * 2 * ** *

2 * 2 * ** *

w (w,v,q)·f (L (w,v,q),K (w,v,q)) 0

v (w,v,q)·f (L (w,v,q),K (w,v,q)) 0

q f(K (w,v,q),L (w,v,q)) 0

q L q K q1 - - 0

L L w L K w L wq L q K q

0 - - 0K L w K K w K w

0

* *q L q K 0

L w K w

Notationally, replace, Lq(w,v,q) with L*, etc.

Page 77: Cost Minimization

Matrix Notation and Cramer’s Rule

*

* * * *LL LK L LL LK L*

* * * *KL KK K KL KK K

*L K L K

*LK L

*KK K

2*K K

Lwff f 1 ff fK

ff f 0 , ff f 0w

ff 0 0 ff 0

w

Cramer's Rule

1 ff0 ff0 f 0 ( 1( f ) )L

0w H

Page 78: Cost Minimization

Matrix Notation and Cramer’s Rule

*

* * * *LL LK L LL LK L*

* * * *KL KK K KL KK K

*L K L K

*LL L

*KL K

*L L K

Lwff f 1 ff fK

ff f 0 , ff f 0w

ff 0 0 ff 0

w

Cramer's Rule

f 1 ff 0 f

f 0 0 1( f )( f )K0

w H

Page 79: Cost Minimization

Matrix Notation and Cramer’s Rule

* *LL LK

* *KL KK

* **L K KL K KK L

**KL K KK L

KL K KK L

Cramer's Rule

ff 1ff 0

ff 0 [( 1)( f )( f )] [( 1)( f )( f )]w H

[(f )(f )] [(f )(f )][(f )(f ) (f )(f )] 0

• With a higher wage, the amount of L used will definitely fall. • It is possible that K becomes more productive with less L being used (if fKL< 0). • If this effect is large enough, the higher productivity of capital can more than

compensate for the higher cost of labor and MC can fall.• Unlikely in the real world.

Page 80: Cost Minimization

Comparative Statics• Differentiate these w.r.t. q

To get:

q qL

q qK

q q0

2 * 2 * ** *

2 * 2 * ** *

w (w,v,q)·f (L (w,v,q),K (w,v,q)) 0

v (w,v,q)·f (L (w,v,q),K (w,v,q)) 0

q f(K (w,v,q),L (w,v,q)) 0

q L q K q0 - - 0

L L q L K q L q

q L q K q0 - - 0

K L q K K q K q

1

* *q L q K

0L q K q

Page 81: Cost Minimization

Matrix Notation and Cramer’s Rule*

* * * *LL LK L LL LK L*

* * * *KL KK K KL KK K

L K L K*

*LK L

*KK K

**K LK

Lq

ff f 0 ff fK

ff f 0 , ff f 0q

ff 0 1 ff 0

q

Cramer's Rule

0 ff0 ff1 f 0 [( 1)( f )L

q H

*

K KK L

*LK K KK L LK LK

( f )] [( 1)( f )( f )]

[ff ff ] 0 if f 0 but could be < 0 if f 0

If labor is an inferior input, we COULD minimize cost with more K and less L. I.e., a “backward bending” expansion path.

Page 82: Cost Minimization

Matrix Notation and Cramer’s Rule*

* * * *LL LK L LL LK L*

* * * *KL KK K KL KK K

L K L K*

* *LL LK

* *KL KK

*L K

Lq

ff f 0 ff fK

ff f 0 , ff f 0q

ff 0 1 ff 0

q

Cramer's Rule

ff 0ff 0

ff 1 [( 1)q H

* * *LL KK KL LK

* 2LL KK KL

( f )( f )] [( 1)( f )( f )]

[(f )(f )] [(f ) ] 0

If there is IRS, although total cost is still rising, MC may well be falling along the expansion path.

Page 83: Cost Minimization

Shephard’s Lemma (Again)• Remember how we used Shephard’s Lemma to

derive the compensated demand curves from the expenditure function?

• We can do it again from the cost function

* * * * *0

* q0

* q0

vK wL q f(K ,L )

(v,w,q) C(v,w,q)L L (v,w,q )

w w(v,w,q) C(v,w,q)

K K (v,w,q )w v

L*

L*

L*

Page 84: Cost Minimization

Marginal Cost Function

*(v,w,q) C(v,w,q)(v,w,q)

q q

L*

That is, λ*= λ(v,w,q) tells us the MC of increasing production along the expansion path

And also:

Page 85: Cost Minimization

Fixed Proportions Shephard’s Lemma Results

• Suppose we have a fixed proportions technology• The cost function is

• For this cost function, output constant demand functions are quite simple:

v wC(w,v,q) q

a b

q

q

C(v,w,q) qK (v,w,q)

v aC(v,w,q) q

L (v,w,q)w b

C(v,w,q) v wMC

q a b

Page 86: Cost Minimization

Cobb-Douglas Shephard’s Lemma Result

• Suppose we have a Cobb-Douglas technology• The cost function is

• where

• then the output constant factor demand functions are:

1

C(v,w,q) vK wL q Bv w

( )B

1q

1

C*K (v,w,q) q Bv w

v

w q B

v

1q

1

C* vL (v,w,q) q B

w w

v q B

w

Page 87: Cost Minimization

CES Shephard’s Lemma Result• Suppose we have a CES technology• The cost function is

• The quantity constant demand functions for capital and labor are

1

1 1 1 1C(v,w,q) q v w , where

1

11 1 1

q

11 1 1

q v w (1 )C 1K (v,w,q)

v 1 v

q v w (1 )

v

1q 1 1 (1 )

11 1 1

C 1 (1 )L (v,w,q) q v w

w 1 w

q v wC

w w

Page 88: Cost Minimization

Properties of Cost Functions• Homogeneity– cost functions are all homogeneous of degree one in

the input prices• a doubling of all input prices will double cost.• As derivatives of HD1 functions are HD0, the

contingent demand functions must be HD0, that is, a doubling of w and v will not affect the cost minimizing input mix.• inflation will shift the cost curves up and will not

change Kc, Lc

Page 89: Cost Minimization

Properties of AC and MC functions

• Both AC and MC are HD1 meaning a doubling of input prices means a doubling of MC and AC– True enough, C=C(v, w, q) is HD1 and

– However, C is HD1 in input prices, and MC is the derivative w.r.t. q. So the derivative of an HD1 function being HD0 does not apply here.

CMC

q

Page 90: Cost Minimization

Properties of Cost Functions• The total cost function is non-decreasing in q, v,

and w.• Fixed factor production has a linear cost function

w.r.t. w and v• To the extent that one factor can be substituted

for another, the function will be concave to input prices.

Page 91: Cost Minimization

C*=C(v,w,q1)

Since the firm’s input mix will likely change, actual costs will be less than C such as C*=C(v,w,q1)

C=wL1+v1K1

If the firm continues to buy the same input mix as w changes, its cost function would be Cpseudo

Concavity of Cost Function

w

C

At w1, the firm’s costs are C(v,w1,q1)

C(v,w1,q1)

w1

Page 92: Cost Minimization

Measuring Input Substitution• A change in the price of an input will cause the firm

to alter its input mix• The change in K/L in response to a change in w/v,

while holding q constant is

KL

wv

Page 93: Cost Minimization

Input Substitution• In the Production lecture:

• But more usefully, because RTS = w/v at cost minimum

L

K

L L

K K

fK Kd dln

fL LKff

d dlnLff

K w Kd dln

L v LsKw w

d dlnLv v

Page 94: Cost Minimization

Input Substitution• This alternative definition of the elasticity of

substitution– in the two-input case, s must be nonnegative– large values of s indicate that firms change their

input mix significantly if input prices change– s = 0 for fixed factor production

Page 95: Cost Minimization

Size of Shifts in Costs Curves• The increase in cost caused by a change in the price of

an input will be largely influenced by• the relative significance of the input in the

production process• the ability of firms to substitute another input for

the one that has risen in price – easy substitution means little change in costs.

Page 96: Cost Minimization

Appendix, Envelope Derivation

• The marginal cost function (MC) is found by computing the change in total costs for a change in output produced

C(v,w,q)MC(v,w,q)

q

Page 97: Cost Minimization

Marginal Cost Function and • Back to the cost function

* q q

* *

* ** *

* **

C w L (w,v,q) v K (w,v,q)

C(v,w,q) L Kw v

q q q

C(v,w,q) q L q K from the FOC

q L q K q

C(v,w,q) q L q Kq L q K q

Page 98: Cost Minimization

Marginal Cost Function* *

*

q q

* *

* *

C(v,w,q) q L q Kq L q K q

Take last FOC (i.e. the production function)q f(L,K)

q f(L (w,v,q),K (w,v,q)) 0differentiate w.r.t. q

q L q K1 0

L q K q

q L q K1

L q K q

So as long as cost are being minimized, this is true

Page 99: Cost Minimization

Marginal Cost Function and • And combine:

*

*

* *

*

*

*

q L q KL q K

C(v,w,q)q

1

C(v,w,

q

q L q KL q K

q)1

q

C

q

M