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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
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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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