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Definition and Propertiesof the Production Function

Lecture II

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Overview of the ProductionFunction

The production function (and indeed allrepresentations of technology) is a purely

technical relationship that is void of economiccontent. Since economists are usuallyinterested in studying economic phenomena,the technical aspects of production are

interesting to economists only insofar as theyimpinge upon the behavior of economicagents. (Chambers p. 7).

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Because the economist has no inherentinterest in the production function, if it ispossible to portray and to predict economic

behavior accurately without directexamination of the production function, somuch the better. This principle, which setsthe tone for much of the following discussion,

underlies the intense interest that recentdevelopments in duality have aroused.(Chambers p. 7).

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A Brief Brush with Duality

The point of these two statements is thateconomists are not engineers and have no

insights into why technologies take on anyparticular shape.

We are only interested in those propertiesthat make the production function useful in

economic analysis, or those properties thatmake the system solvable.

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One approach would be to estimate aproduction function, say a Cobb-Douglas

production function in two relevant inputs:

1 2 y x x

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Given this production function, we couldderive a cost function by minimizing the

cost of the two inputs subject to somelevel of production:

1 2

1 1 2 2

,

1 2

min

. .

x x

w x w x

s t y x x

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1 1 2 21 2

2 1 1

2

L

x w x wx x

L w x w

x

1*2 1

2 2 2 1 21 20 , ,

w wL

y x x x w w y yw w

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1

* 21 1 2

1

, ,w

x w w y yw

1 1

2 11 2 1 2

1 2

12 1

1 2

, ,w w

C w w y w y w yw w

w wy

w w

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Thus, in the end, we are left with a cost functionthat relates input prices and output levels to thecost of production based on the economic

assumption of optimizing behavior. Following Chambers critique, recent trends in

economics skip the first stage of this analysis byassuming that producers know the general shapeof the production function and select inputs

optimally. Thus, economists only need toestimate the economic behavior in the costfunction.

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Following this approach, economists onlyneed to know things about the production

function that affect the feasibility andnature of this optimizing behavior.

In addition, production economics istypically linked to Sheppards Lemma that

guarantees that we can recover theoptimal input demand curves from thisoptimizing behavior.

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Production Function Defined

Following our previous discussion, wethen define a production function as a

mathematical mapping function:

: n m f R R

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However, we will now write it in implicitfunctional form

This notation is sometimes referred to as anetputnotation where we do notdifferentiate inputs or outputs.

0Y z

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Following the mapping notation, we typicallyexclude the possibility of negative outputs orinputs, but this is simply a convention. Inaddition, we typically exclude inputs that are noteconomically scarce such as sunlight.

Finally, I like to refer to the production function asan envelopeimplying that the production functioncharacterizes the maximum amount of output thatcan be obtained from any combination of inputs.

, 0Y y x

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Properties of the ProductionFunction

Monotonicity and Strict Monotonicity:

If , then (monotonicity) x x f x f x

If then (strict monotonicity) x x f x f x

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Quasi-Concavity and Concavity

: is a convex set (qausi-concave)V y x f x y

0 * 0 *1 1 for any 0 1(concave)

f x x f x f x

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Weakly essential and strictly essentialinputs

0 0, where 0 is the null vector (weakly essential)n nf

1 1 1, ,0, , 0 for all (strict esstential)i i n i f x x x x x

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The set V(y) is closed and nonempty for all y> 0.

f(x) is finite, nonnegative, real valued, andsingle valued for all nonnegative and finite x.

Continuity

f(x) is everywhere continuous; and

f(x) is everywhere twice-continuouslydifferentiable.

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Properties (1a) and (1b) require theproduction function to be non-decreasing ininputs, or that the marginal products be

nonnegative. In essence, these assumptions rule out stage III

of the production process, or imply some kind ofassumption of free-disposal.

One traditional assumption in this regard is thatsince it is irrational to operate in stage III, noproducer will choose to operate there. Thus, if wetake a dual approach (as developed above) stageIII is irrelevant.

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Properties (2a) and (2b) revolve aroundthe notion ofisoquantsor as

redeveloped here input requirementsets. The input requirement setis defined as

that set of inputs required to produce at

least a given level of outputs, V(y). Othernotation used to note the same conceptare the level set.

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Strictly speaking, assumption (2a) impliesthat we observe a diminishing rate oftechnical substitution, or that the isoquantsare negatively sloping and convex withrespect to the origin.

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1x

2x

V y

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Assumption (2b) is both a strongerversion of assumption (2a) and an

extension. For example, if we chooseboth points to be on the same inputrequirement set, then the graphical

depiction is simply

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1x

2x

V y

0 0 0 01 1 f x x f x f x

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If we assume that the inputs are on two differentinput requirement sets, then

Clearly, letting approach zero yields f(x)

approaches f(x*

), however, because of theinequality, the left-hand side is less than the righthand side. Therefore, the marginal productivityisnon-increasing and, given a strict inequality, isdecreasing.

0 * 0 * *

*

0 * 0 * *

1

1

f x x f x f x f x

f x f x x x x f x

x

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As noted by Chambers, this is an exampleof the law of diminishing marginalproductivitythat is actually assumed.

Chambers offers a similar proof on page12, learn it.

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The notion of weakly and strictly essentialinputs is apparent. The assumption of weakly essential inputs says

that you cannot produce something out ofnothing. Maybe a better way to put this is that ifyou can produce something without using anyscarce resources, there is not an economicproblem.

The assumption of strictly essential inputs is thatin order to produce a positive quantity of outputs,you must use a positive quantity of all resources.

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Different production functions havedifferent assumptions on essential inputs.It is clear that the Cobb-Douglas form is anexample of strictly essential resources.

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The remaining assumptions are fairlytechnical assumptions for analysis.

First, we assume that the inputrequirement setis closed and bounded.This implies that functional values forthe input requirement setexist for all

output levels (this is similar to thelexicographic preference structure fromdemand theory).

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Also, it is important that the productionfunction be finite (bounded) and real-

valued (no imaginary solutions). Thenotion that the production function is asingle valued map simply implies that

any combination of inputs implies oneand only one level of output.

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Law of Variable Proportions

The assumption of continuous functionlevels, and first and second derivatives

allows for a statement of the law ofvariable proportions.

The law of variable proportionsis

essentially restatement of the law ofdiminishing marginal returns.

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The law of variable proportionsstates thatif one input is successively increase at aconstant rate with all other inputs heldconstant, the resulting additional productwill first increase and then decrease.

This discussion actually follows our

discussion of the factor elasticity from lastlecture

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%

%

dy

y dy x MPPyE

dx x dx y APP

x

d TPP d x APP d APP MPP APP x

dx dx d x

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Working the last expression backward, wederive

1d APP MPP APPdx x

1ii i i i

AP f y

x x x x

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Elasticity of Scale

The law of variable proportionswasrelated to how output changed as you

increased one input. Next, we want toconsider how output changes as youincrease all inputs.

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In economic jargon, this is referred toas the elasticity of scaleand is defined

as

1

ln

ln

f x

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1x

2x

1x

2x

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The elasticity of scale takes on threeimportant values:

If the elasticity of scale is equal to 1, then theproduction surface can be characterized byconstant returns to scale. Doubling allinputsdoubles the output.

If the elasticity of scale is greater than 1, then

the production surface can be characterized byincreasing returns to scale. Doubling allinputsmore than doubles the output.

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Finally, if the elasticity of scale is less than 1,then the production surface can becharacterized by decreasing returns to scale.

Doubling allinputs does not double the output. Note the equivalence of this concept to the

definition of homogeneity of degree k:

k f x f x

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For computational purposes

1 11

ln

ln

n n

i i

i ii

f x f x

x y