Production and Cost functionsProduction and Cost functions

1From: Heady 1957. An Econometric Investigation of the Technology of Agricultural Production Functions Econometrica, Vol. 25, No. 2: 249-268

OutlineOutline

• Production functionProduction function

• The Firm’s problem

• Parallel with ConsumerParallel with Consumer

• From Production function to costfunction to cost functions

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Production function‐factsProduction function facts• Its easy, its engineering data.

Y d fi t h t h h th– You go and figure out what happens when you vary the quantities of inputs.  

– You may get nothing • because some ingredient is necessary (strict complement)

– You may get faster output, or slower…– Tabulate all that data and you can get a set of points thatTabulate all that data and you can get a set of points that represent different ways of producing the same amount.

– Run plant 24 hours a day (three shifts) or Increase number f bl liof assembly lines

– Use your workers for their strength, give them more machines.  (unload ships with cranes or with people)

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Production and Cost functionsProduction and Cost functions

4From: Heady 1957. An Econometric Investigation of the Technology of Agricultural Production Functions Econometrica, Vol. 25, No. 2: 249-268

From facts to TheoryFrom facts to Theory

• When you measure them, you tend to findWhen you measure them, you tend to find• Isoquants are weakly convex 

– marginal diminishing returns in increasing onemarginal diminishing returns in increasing one input.

• They can have decreasing, constant, or y g, ,increasing returns.– Involves issues of indivisibility (example plant)

• Returns increase as you come close to capacity• Then fall because some input is not adjusted.

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Cobb‐Douglas IsoquantsCobb Douglas Isoquants

1

0.8

0.6

0.4

0

0.2

0 0.2 0.4 0.6 0.8 1

6

Substitutes ComplementsSubstitutes Complements

MachinesMachines

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2

LKf

02

LKf LK

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WorkersWorkers

Firms vs consumerFirms vs consumer

• Consumer has • Firm hasConsumer has – Utility function

– Cost of goods

Firm has– Production function

– Costs of inputs

– Faces budget constraint

• Consumer problem 1– Faces production constraint.

• Firm problem 1– Max U sbj to PX≤Y

• Consumer problem 2– Max F(X) sbj to PX≤Y

• Firm problem 2– Min PX sbj to U(X)≥U – Min PX sbj to F(X)≥Q

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If you can solve 1 you can solve 2 if you can solve the consumer’s problem you can solve the firm’s.

More symmetryMore symmetry

• Consumer • FirmConsumer Firm

Ratio of marginal utilities equal to the price ratio

Ratio of marginal product

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g pequal to the price ratio

Short run vs long runShort run vs long run• Marginal product is non‐negative 

f 2 f

• More is better but there are marginal diminishing returns

0kf

.0)( 2

2

kf

• Some inputs (labor, raw materials) more readily changed than others (plant and equipment)…so there is a long run and a short runthere is a long run and a short run.

• Short run, take fixed assets and technology as given– choose the right mix of inputs to run that firm

• Long run chose your plant size, technology.S th i f i f t i fi– Suppose the price of gas increases for a taxi firm

• What should it do?10

Short Run Profit  Maximization(take p as given)(take p as given)

.),( wLrKLKpF

• FOC.*),(0 wLKLFp

L

• Wage (price of input) is equal to the value of its marginal product

*).,()()(

02

2

2

2LKFp

• SOC)()( 22 LL

• Guaranteed if there are marginal decreasing returns

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Graphical DepictionGraphical Depiction

Slope zero at maximum

Slope negative toSlope positive to left of maximum

Slope negative to right of maximum

LL*

L

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Short‐run Effect of a Wage IncreaseShort run Effect of a Wage Increase

.0*),(

wLK

LFp

L

LLI can differentiate the FOC with respect to w (the wage)

,1)(*))(*,()(

0 2

2

wLwLKLFp)(

.01

)(*2

F

wL

))(*,()( 2

2

wLK

L

Fp

P>0, F’’<0

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Aside: Revealed PreferenceAside: Revealed Preference

• Revealed preference is a powerful techniqueRevealed preference is a powerful technique to prove comparative statics

• Works without assumptions about continuity• Works without assumptions about continuity or differentiability

S 1 2 l l• Suppose w1 < w2 are two wage levels

• The entrepreneur chooses L1 when the wage is w1 and L2 when the wage is w2

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Revealed Preference ProofRevealed Preference ProofPrefer L1 to L2 when wage = w1

)()( LwrKLKpfLwrKLKpf

Prefer L2 to L1 when wage = w2

212111 ),(),( LwrKLKpfLwrKLKpf

Sum these two

.),(),( 121222 LwrKLKpfLwrKLKpf

Sum these two 222111 ),(),( LwrKLKpfLwrKLKpf

)()( LKLKfLKLKf 212121 ),(),( LwrKLKpfLwrKLKpf

LwLwLwLw 15

21122211 LwLwLwLw

Revealed Preference, Cont’dRevealed Preference, Cont d

21122211 LwLwLwLw • But remember w1 < w2

21122211

0))(( LL• So the only way for above to be true is

.0))(( 1221 LLww

•• Revealed preference shows that profit 

.)(0 2112 LLLL

maximization implies L falls as w rises.

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Cost MinimizationCost Minimization

• Profit maximization requires minimizing costProfit maximization requires minimizing cost

• Cost minimization for fixed output

c(y) = Min wL + rK

subject tosubject to yLKf ),(

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Cost Minimization, ContinuedCost Minimization, Continued

• Profit maximization:Profit maximization:

• max py – (wL + rK) s.t.

i hi i i l i i i i

yLKf ),(• For given y, this is equivalent to minimizing cost.

• Cost minimization equation:

f

rw

dLdK

fL

f

yLKf

)( rdLK yLKf ),(

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Cost Min DiagramCost Min Diagram

K

Isocost

IsoquantIsoquant f(K,L)=y

L19

Short run vs Long runShort run vs Long run

• Short run fix K1 (Plant size)Short run fix K1 (Plant size)

• Long run chose K1Sh i bj ( ) Q• Short run Min wL+rK2 sbj F(K2,L)>Q

• G(L, K2, λ)= wL+rK2 + λ (Q‐F(K2,L))

FOC

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• Ratio of the first two gives

• Ratio of MP equals price ratio, or

• Plug back into last FOC

A bit of a mess but in fact its linear in Q if α+β=1; its

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Q β ;also decreasing in  K1

Short run14

Γ=K1=1

10

12

10 K1

4

6

8

output

7

8

9

4

8

11

0

2

4

1 2 4

5

66

BC=5

BC=6

BC=7

11 2 3 4 5 6 7 8 9 10 11 12K2

2

3

4BC=8

BC=8.5

BC=9

0

1

0 5 10 15

L

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On the left Cost Min; on the right Max Q

9

10

K1

10

K1

1

7

8

9

7

8

9 2

3

4

4

5

6 6

BC=5

BC=6

BC=74

5

6

6

8

2

3

BC=8

BC=8.5

BC=9

1

2

3

4

0

1

0 5 10 15

L 0

1

‐4 1 6 11 16

L

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Long runh d l l• First notice that L and K2 are optimal solutions given 

any K1 …so we could just search out the optimal K1given L and Kgiven L and K2

What a mess! Notice that the solution (K1*) is going to be

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( 1 ) g gincreasing in Q and γ and declining in r

When α+β=0.5 

• As the target quantity Q, increases you invest more in plant, same if crowding is a big deal (γ) or if capital is cheap.

• Note this does not depend on α+β=0.5ote t s does ot depe d o α β 0 5

25

30

35

25

30

K1=2

K1=3

K1 4

15

20

Total Cost

K1=4

K1=6

K1=8

K1=10

K1=15

5

10

K1 15

K1=20

(LR‐Cost curve

K1Long run cost function is the min of all the possible 

0

0 20 40 60 80 100 120 140 160 180 200

Quantity

cost functions.

Note Marginal cost is constant for each level of K1

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From production function to costsFrom production function to costs

• Rather than look at a production function one pcan summarize the firm’s decision into a simple cost function.

• Note that implies that we are tracing out the• Note: that implies that we are tracing out the optimal input mix given prices, and technology.

• In our last example cost were increasing in aIn our last example cost were increasing in a linear way for each level of K1. So each K1 corresponds to a different short run cost functionfunction.

• Key: Cost functions assume that input prices are stable.

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ReminderReminder

• For every production function and input price set, o e e y p oduct o u ct o a d put p ce set,you can find the vector X ={x1,..,xi,..,xn} that minimizes the cost of producing a given level of 

h f d h foutput Q. The cost of producing Q is there fore PX.  Now one can do this for every Q over the relevant rangerelevant range.

• The function that maps Q into cost exists if the production function is convex. C(Q)production function is convex. C(Q)

• Marginal cost is simply the derivative of the cost function with respect to quantity. p q y

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From production to FirmFrom production to Firm

• One possibility is for the Entrepeneur to solveOne possibility is for the Entrepeneur to solve one big problem– Max π=pF(x x x ) –(p x + +p x p x )– Max π=pF(x1,..,xi,..,xn) –(p1x1+…+pixi.. pnxn )

• Another takes two steps(1) Th i d h i i i i– (1) The engineers to do the cost minimization 

• Min p1x1+…+pixi..pnxn sbj to F(x1,..,xi,..,xn)<Q

(2) M k ti fi d th tit t d– (2) Marketing finds the quantity to produce • Max π= pQ‐C(Q)

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Short‐run CostsShort run Costs

• Short‐run total cost• L varies, K does not• Short‐run marginal costg

– Derivative of cost with respect to output• Short‐run average cost

– average over output– infinite at zero, due to fixed costsh bl• Short‐run average variable cost– average over output, omits fixed costs

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Comparative StaticsComparative Statics

• What happens to L as K rises?at appe s to as ses?

))(*,(2

KLKLK

F

.))(*,(

))(,()(*

2

2KLKF

LKKL

• Remember the lower part of the ratio has to be

))(,()( 2L

Remember the lower part of the ratio has to be negative (condition for a max)

• Thus, L rises if L and K are complements, and fallsThus, L rises if L and K are complements, and falls if substitutes

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Conclusion and Wrap upConclusion and Wrap up

• 1) remember the symmetry of our two key1) remember the symmetry of our two key decision problems (consumer and producer)

• 2) short run problem is exactly symmetric• 2) short run problem is exactly symmetric

• 3) long run problem involves choosing scale

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