Lecture 3: Producer Theory-Perfect Competition

Daniel Zhiyun LI

Durham University Business School (DUBS)

September 2014

Plan of the Lecture

Introduction

The Case of Single Input

The Case of Two Inputs

Producer Surplus

Introduction

A �rm: a machine having the magical ability to convert one type ofgoods (Inputs) into another type of goods (Outputs)

In the simple case of perfect competition, �rms are price takers.In the next lecture, we will study the cases where �rms have controlsof prices to some degree, such as monopoly

The Problem of a Perfectly Competitive Firm

The Problem

to choose: input, (l , k), and output, x , levelsto maximize: �rm�s pro�t, π = px x � wl � rksubject to: technology constraints and the market prices, x = f (l , k)

Mathematically, pro�t maximization

maxfx ;l ,kg pxx � wl � rks.t. x = f (l , k)

The Case of Single Input Labor

The production function f (l), with the common assumptions:

f (l) � 0: output is non-negative∂f (l) /∂l > 0: marginal output is positive∂2f (l) /∂l2 < 0: diminishing marginal return of output

The Problem is then

maxfx ;lg pxx � wl s.t. x = f (l)

The Case of Single Input Labor (2)

Solution 1: A graphical solution

the iso-pro�t line: π = px x � wl ) x = πpx +

wpx l

moving northwesterly, π ")moving as much as possibletangent condition again, where

f 0 (l�) =wpx

The Case of Single Input Labor (3)

Solution 2: a direct mathematical solution

maxfx ;lg pxx � wl s.t. x = f (l)

substituting x ) maxflg px f (l)� wl , and the foc condition is

px f 0 (l)� w = 0) f 0 (l�) =wpx

the second order su¢ cient condition is

px f 00 (l) < 0

which is satis�ed under our assumption.

A graphical illustration of the solution

The Case of Single Input Labor (4)

An intuition:

Revenue= px f (l), and marginal revenue MR (l) = px f 0 (l);Cost= wl , and marginal cost MC (l) = w ;Optimal condition: marginal revenue equals marginal cost

MR (l) = MC (l)

The solution is l� (px ,w), the comparative statics results are

∂l� (px ,w)∂px

> 0∂l� (px ,w)

∂w< 0

(exercise!)

The Supply Function is

x (px ,w) = f (l� (px ,w))

which is upward sloping w.r.t. px . (proof!)

Comparative Statics: A Graphical Illustration

The Case of Single Input Labor (4)

A third solution: substituting l

from x = f (l)) l = f �1 (x)pro�t function is now: π (x) = px x � wf �1 (x)we introduce a cost function, c (w , x) = wf �1 (x), which is the cost ofproducing x at wage w

π (x) = px x � c (w , x)

the �rst order condition

px =∂c (w , x)

∂x

marginal revenue equals marginal cost!

MR (w , x) = MC (w , x)

Average vs. Marginal Cost (1)

Average Cost:

AC (w , x) =c (w , x)x

pro�t is positive i¤ price is greater than average cost.

If MC (w , x)�s are higher than AC (w , x), average costs must berising, and visa versa. (proof!)

Average vs. Marginal Cost (2)

The Case of Two Inputs

Pro�t Maximization

maxfx ,l ,kg pxx � wl � rks.t. x = f (l , k)

Cost Minimization

�rst, to �nd the cheapest way to produce x

minfl ,kg wl + rks.t. x = f (l , k)

) cost function c (x ; r ,w) = wl (x ; r ,w) + rk (x ; r ,w)second, to �nd the optimal x to maximize the pro�t

maxfxg px x � c (x ; r ,w)

Two Assumptions of the Technology

Two Assumptions:

Monotonic: i.e. for l 0 � l and k 0 � k

f�l 0, k 0

�� f (l , k)

Convex: for (l 0, k 0) and (l , k) such that f (l 0, k 0) = f (l , k), then forλ 2 [0, 1]

f�λl 0 + (1� λ) l ,λk 0 + (1� λ) k

�� f (l , k)

Remarks:

the same assumptions as we make for preferences) the iso-quantitycurve for f (l , k) is like the indi¤erence curve for preference;we later will study non-convex production functions.

The Case of Two Inputs: Cost Minimization

c = wl + rk )iso-cost curve l = cw �

rw k

MRTS (Marginal Rate of Technical Transformation)

MRTSl ,k = �dldk=(∂f /∂k)(∂f /∂l)

=MPkMPl

Tangent condition: rw = MRTSl ,k .

The Case of Two Inputs: An Example

Example

f (l , k) = kαl β, α+ β < 1. 1) cost minimization

minfl ,kg wl + rk s.t. x = f (l , k)

the Lagrangian

L (l , k,λ) = wl + rk � λ [x � f (l , k)]

the �rst order conditions are Ll = w + λfl = 0, Lk = r + λfk = 0 andLλ = x � f (l , k) = 0, from which we get

rw=fkfl= MRTSl ,k =

α

β

lk

The Case of Two Inputs: An Example

Example

(continued) substituting it into x = kαl β, we get the factor demandfunctions

l (x , r ,w) =�

βrαw

� αα+β

x1

α+β k (x , r ,w) =�

αwβr

� βα+β

x1

α+β

and the cost function c (x ; r ,w) = wl (x ; r ,w) + rk (x ; r ,w). 2) pro�tmaximization

maxx

pxx � c (x ; r ,w)

the �rst order condition is thus

px =∂c (x ; r ,w)

∂x

marginal revenue equals marginal cost.

Technology: Economy of Scale

De�nitionLet f (l , k) be the production function of a �rm, for t > 1: 1) the �rmexhibits increasing returns to scale if f (tl , tk) > tf (l , k); 2) the �rmexhibits decreasing returns to scale if f (tl , tk) < tf (l , k); 3) the �rmexhibits constant returns to scale if f (tl , tk) = tf (l , k).

Examples

In the case of f (l , k) = kαl β

f (tl , tk) = tα+βf (l , k)

Obviously, α+ β > 1,IRS, α+ β < 1,DRS and α+ β = 1,CRS.

Producer Surplus

In above examples, we already know that the supply curve satis�es

px (x) = MC (x) =∂c (x ; r ,w)

∂x

Thus, integrating along the supply curveZ x �

0px (x) dx =

Z x �

0

∂c (x ; r ,w)∂x

dx = c (x�)

Producer Surplus is thus

p�x x� � c (x�)

which is equal to the pro�t of the �rm.

Producer Surplus

Producer Surplus