week 6 theory of the firm (jehle and reny,...

Week 6Theory of The Firm

(Jehle and Reny, Ch.3)

Serçin �ahin

Y�ld�z Technical University

30 October 2012

Serçin �ahin (Y�ld�z Technical University)Week 6 Theory of The Firm (Jehle and Reny, Ch.3)30 October 2012 1 / 28

Primitive Notions

A �rm is an entity created by individuals for some purpose.

Pro�t is the di�erence between revenue the �rm earns from selling its

output and expenditure it makes buying its inputs and is income to

owners of the �rm.

Pro�t maximisation is the single most robust and compelling

assumption we can make for predicting �rm behaviour.


Production

Production is the process of transforming inputs into outputs.

Technological feasibility constraint is represented by the production

possibility set, Y ∈ Rm, where each vector y = (y1, ..., ym) ∈ Y is a

production plan whose components indicate the amounts of the

various inputs and outputs.

A common convention is to write elements of y ∈ Y so that yi < 0 if

resource i is used up in the production plan, and yi > 0 if resource i isproduced in the production plan.


Production

When there is only one output produced by the �rm, it is more

convenient to describe the �rm's technology in terms of a production

function.

We shall denote the output by y , and the amount of input i by xi , sothat with n inputs, the entire vector of inputs is denoted by

x = (x1, ..., xn).

When we write y = f (x), we mean that y units of output (and no

more) can be produced using the input vector x.


Production

Assumption 3.1. Properties of the Production FunctionThe production function, f : Rn

+ → R+ is

continuous

strictly increasing

strictly quasiconcave on Rn+,

and f (0) = 0.


Production

When the production function is di�erentiable, its partial derivative

∂f (x)/∂xi is called the marginal product of input i and gives the rate

at which output changes per additional unit of input i employed.

If f is strictly increasing and everywhere continuously di�erentiable,

then ∂f (x)/∂xi > 0. for all input vectors.

For any �xed level of output, y , the set of input vectors producing yunits of output is called the y -level isoquant, and it is denoted by

Q(y).

Q(y) ≡ {x ≥ 0 | f (x) = y}For an input vector x, the isoquant through x is the set of input

vectors producing the same output as x, namely, Q(f (x)).


Production

The marginal rate of technical substitution is de�ned as the ratio of

marginal products, and measures the rate at which one input can be

substituted for another without changing the amount of output

produced.


Production


Production Returns to Scale and Varying Proportions

In the short run, the period of time in which at least one input is �xed,

output can be varied only by changing the amounts of some inputs but

not others.

Returns to variable proportions refer to how output responds in this

situation.

In the long run, the �rm is free to vary all inputs, and classifying

production functions by their returns to scale is one way of describing

how output responds in this situation.

Speci�cally, returns to scale refer to how output responds when all

inputs are varied in the same proportion.



Elementary measures of returns to varying proportions include

the marginal product,

MPi ≡ fi (x)

and the average product,

APi ≡f (x)

xiof each input.

The output elasticity of input i , measuring the percentage response of

output to a 1 per cent change in input i , is given by

µi ≡fi (x)xif (x)

≡ MPi ()x

APi (x)


Cost

The �rm's cost of output is precisely the expenditure it must make to

acquire the inputs used to produce that output.

If the object of the �rm is to maximise pro�ts, it will necessarily

choose the least costly, or cost-minimising production plan for every

level of output.

We will assume throughout that �rms are perfectly competitive on

their input markets and that therefore face �xed input prices.

Let w = (w1, ...,wn) ≥ 0 be a vector of prevailing market prices at

which the �rm can buy inputs x = (x1, ..., xn).


Cost


Cost

The cost minimisation implies that the marginal rate of technical

substitution between any two inputs is equal to the ratio of their prices.

We can write x∗ ≡ x(w, y) to denote the vector of inputs minimising

the cost of producing y units of output at the input prices w.

The solution x(w, y) is referred as the �rm's conditional input demand.


Cost


Duality in Production


week 6 theory of the firm (jehle and reny,...

Documents