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2.The Theory of the Firm

What to produce?

How to produce?

How much to produce?

To whom to produce?

Varian, chap 18

Frank, chap 9

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2.1.Technology Sets

Q = f(x) is the

production

function.

L LInput Level

Output Level

Q

Q

Q = f(L) is the maximal outputlevel obtainable from x inputunits.

One input, one output

Q = f(L) is an output level that is

feasible from x input units.

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2.1. Production Function

Production Function: maximum quantity of output that

can be produced for given quantities of the inputs

Example: Q = f(K,L) = 2KL

5

1 2 3 4 5

1 2 4 6 8 10

2 4 8 12 16 20

3 6 12 18 24 30

4 8 16 24 32 40

5 10 20 30 40 50

K

L

4

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2.1.The Long-Run and the Short-

Runs Fixed and Variable inputs

A short-run is a circumstance in which a firm is restricted

in some way in its choice ofat least one input level. There are many possible short-runs

The long-run is the circumstance in which a firm is

unrestricted in its choice ofall input levels.

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2.1. The Long-Run and the Short-

Runs Examples of restrictions that place a firm into a short-

run:

temporarily being unable to install, or remove,

machinery

being required by law to meet affirmative action

quotas

having to meet domestic content regulations.

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2.1. Production in the Short-Run

Example : Q = f(K,L) = 2KL

where K = K0 = 2 K = K0 =1

Q=2KL=4L Q=2KL=2LQ

Q

LL1 2

4

8

1 2

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Total Product Curve:

Q = f(L)

Average Product

Curve:

APL = Q/L

4 6 8

Q

L

6 L

Q/L

APL

Q(4)

Q(4)/4

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Marginal Product Curve:

MPL = (Q/(L

L(person-hr/week)

L(person-hr/week)

Q

dQ/dL 8

8

4

4

Increasing

marginal

returns

Decreasing

marginal

returns

6

MPL

For L=6, MPL=APL

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2.1. Average and Marginal

Product CurvesIf (Q/(L>Q/L, Q/L is increasingIf (Q/(L

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Effect of technological progress

L

Q

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2.1.Long-run: Technologies with

Multiple Variable Inputs What does a technology look like when there is more

than one input?

The two input case: Input levels are x1 and x2. Output

level is y.

Suppose the production function is

.),( 3/13/1 LKLKfQ !!

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2.1. Long-run: Technologies with

Multiple Inputs

Output, Q

L

K(8,1)

(8,8)

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2.1. Long-run: Technologies with

Multiple Inputs

Output, Q

L

K

Q |

Q |

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2.1. Long-run: Isoquants with Two

Variable Inputs

Output, Q

L

K

Q |

Q |

Q |

Q |

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2.1.Technologies with Multiple

Inputs: isoquant map

x1y

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2.1.Technologies with Multiple

Inputs

L1

Q

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Q=f(K,L) Isoquant : (Q=0

Ex: Q=2K0,7L0,3 Q=5 = > K=2,51/0,7.L0,3/0,7

K

L

(K

(L

Marginal Rate of Technical Substitution, MRTS =-(K/(L

2.1. Production in the Long-Run:

example

Q=5

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2.1. Marginal Product and MRTS

Q=F(K,L)

when L changes, by how much should K vary so that Qdoes not change, ie, dQ=0?

dQ = 0 = (HF/HK).dK + (HF/HL).dL

-dK/dL = MRTS = MPL/MPK

Law of diminishing returns: HMPK/HK

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Monotonicity: More ofany input generates

more output.

Q

L

Q

L

monotonicnot

monotonic

2.1. Production Function: well-

behaved technologies

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K

L

'')1(','')1(' KttKLttL

Q|



L L

K

K

Convexity

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K

L

Convexity implies that the MRTS

decreases as L increases.



K

K

L L

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K

L

Q|Q|

Q|

higher output



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K

L

min{L,2K} = 14

4 8 14

24

7

min{L,2K} = 8min{L,2K} = 4

L = 2K

}2,min{ KLQ !

2.1. Production Function: ParticularCases

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9

3

18

6

24

8

L

K L+ 3K= 18

L+ 3K = 36L + 3K = 48

All are linear and parallel

y x x!

1 2

3


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Cobb-Douglas Production Function


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The Long-Run and the Short-Runs

x2

x1

y

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x1

y

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x1

y

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x1

y

Four short-run production functions.

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2.1. Returns to Scale

Returns to scale (or (dis)economies of scale) : are only

defined in the long-run, when the quantity ofany input

can vary.

This is in contrast to the case ofmarginal returns

(increasing or decreasing) which measure the change

in output when one input varies and the other (s) is

(are) kept constant.

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2.1. Returns to Scale

F(tK,tL) = t F(K,L) Constant returns to scale

t>1 (CRS)

F(tK,tL) = ta F(K,L) Increasing returns to scale

t >1, a>1 or economies of scale

Ex: F(K,L)= AKcLd

F(tK,tL)=A (tcKc)(tdLd)=tc+d(AKcLd)

=> if c+d>( increasing (decreasing) returns

F(tK,tL) = ta F(K,L) Decreasing returns to scale

t>1, a

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Can a technology exhibit increasing returns-to-scale

even if all of its marginal products are diminishing?

A: Yes.

E.g. QK2/3L2/3

Q(tK,tL)=t2/3L2/3t2/3K2/3 =t4/3Q(K,L): increasing returns to

scale

Yet, MPL=2/3K2/3

.L-1/3

HMPL/HL=-1/3.2/3.K2/3.L-4/3

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20101021_1102_1991287664245826