Econ 102 SY 2008 2009

Lecture 7

Profit Maximization and Supply

August 21, 24 and 28, 2008

Econ 102 SY 2008 2009

Key concepts

Production process Production function

Econ 102 SY 2008 2009

Outputs and inputs

Production as the transformation of inputs into outputs

Alternative production processes: One input, one output Several inputs, one output Several inputs, several outputs Several tiers of inputs, one output

Substitution among inputs Transformation of one output to another

Econ 102 SY 2008 2009

L

Q

L1

Q1

Q2

Econ 102 SY 2008 2009

Q for local market

Value Added Aggregate Intermediate Input

K Lx1 x2 xn

C OL

Xport

Econ 102 SY 2008 2009

Specification of technology

Technology – how inputs are transformed into outputs.

Net outputs – yiS

– yiy

Production plan - set of all technologically feasible net outputs, i.e. a negative y means it is a net input, while a positive y means it is a net output (yS; – yy,– xy); (y’S; – yy,– xy) such that y’S< yS

Production possibilities set – set of all production plans

Econ 102 SY 2008 2009

Virgin Coconut Oil

a

b

VCO1

C1

Copra

C2

Econ 102 SY 2008 2009

Short run possibilities set

Eventually technologically feasible set

Immediately technologically feasible or short run production possibilities set - et of outputs, yi in the production possibilities set Y, constrained by some resource z - immediately technologically feasible :

{ } k= in Y : k (y -l,-k)) = kY(

{ } in Y (y -l,-k)k) = ,Y(l

Econ 102 SY 2008 2009

Production function

Suppose the firm has one output and uses inputs x, then the production function is

Efficient production plan: output y is efficient if there is no y’ > y that can be produced with input vector x, and y’ is not y.

Transformation function: set of net output vector y. T(y)=0 if and only if y is efficient.

{ } in Yinputoutput of maxis the y in R : = )x(f x-

Econ 102 SY 2008 2009

Input requirement set

Input requirement set – Set of inputs -x that can produce at least any y in Y.

Isoquant: set of all inputs such that they produce the same level of output.

{ } x) is in Y: (y; in RV (y) = L -x +

{ } : in RQ(y) = L y> yfor ) V(yin not is x and (y) V in isx x 00+

Econ 102 SY 2008 2009

L

K

a

La

Ka

bYa

Yb

c

Econ 102 SY 2008 2009

Examples of Convex Input requirement set

Econ 102 SY 2008 2009

Convex input requirement set

Econ 102 SY 2008 2009

Cobb Douglas Production Technology

{ }{ }{ }{ }

==

= = ) T(y;

≤

=

≤=-

+

-

)-α1(2

α121

)-α1(2

α121

2)-α1(

2α1

321

)-α1(2

α1

2+21

121

221

121

321

xx);xx(fy

0xy-x;xx

=zand xxx: y) in R;-x(y;-xY(z) =

x:y=x) in R;x(xQ(y) =

xxy:Rin);x(x)y(V

xxy:)in R;-x(y;-xY

≤

Econ 102 SY 2008 2009

Leontief Production Technology

{ }{ }{ }{ }

=

y = ) T(y;

≤

); bx(axmin);xx(f

); bx(axmin-;xx

=zand x); bx(axmin: y=) in R;-x(y;-xY (z) =

); bx(axmin:y=) in R;x(xQ(y) =

); bx(axmin:y) in R;x(xV(y) =

)2; bx1(axmin: y ≤ ) in R;-xf(y;-xY =

2121

2121

2213

21

212+21

212+21

321

Econ 102 SY 2008 2009

Properties of Production Set

Inaction. Zero production can be a production plan. 0 Y.

Closedness. Given a sequence of production plans yi, i = 1; 2;… n; all plans are in Y and as yi y, then the limit production plan y is also in Y . Guarantees that points on the boundary of Y are feasible. Closedness of Y implies that the input requirement set V (y) is a closed

set for all y>=0. Free disposal or monotonicity. If y Y, then y’ Y for all y’<y.

Free disposal means commodities can be thrown away. A weaker requirement is that we only assume that the input

requirement is monotonic: If x is in V (y) and x’ > x, then x’ is in V (y).

Econ 102 SY 2008 2009

Properties of Production Set

Irreversibility. All production plans except the zero plan are irreversible.

Convexity. Y is convex if whenever y and y’ are in Y , the weighted average ty + (1 - t) y’ is also in Y for any t with 0 <= t <= 1. With convexity of Y, and if all goods are divisible,

two production plans y and y’ can be scaled downward and combined.

Convexity rules out “start up costs" and other sorts of returns to scale.

Econ 102 SY 2008 2009

Properties of Production Set

Strict Convexity. Y is strictly convex if y Y and y’ Y , then ty +(1-t)y’ intY for all 0 < t < 1, where intY denotes the interior points of Y . Guarantees unique profit maximizing production plans. Implies input requirement set is convex.

Convexity of input requirement set. If x and x’ are in V (y), then tx + (1 - t)x’ is in V (y) for all 0 <= t <= 1. Convexity of V (y) implies that , if x and x’ can each

produce y, so can any weighted average of them.

Econ 102 SY 2008 2009

Propositions

Convexity of production and input requirement sets. If production set Y is convex, then its associated input requirement set, V (y) is also convex.

Quasi-concavity of prod function and convexity of input requirement set. Input requirement set V (y) is a convex set if and only if the production function f (x) is a quasi-concave function.

Econ 102 SY 2008 2009

Definitions: Concave and Quasi-concave functions

Let X be a convex set. A function f : X R is said to be concave on X if for any x, x’ X and any t with 0 <= t <= 1, we have f(tx + (1 - t)x’) >= tf (x) + (1 -t)f(x’)

The function f is said to be strictly concave on X if f(tx + (1 - t)x’) > tf (x) + (1 -t)f(x’) for all x, x’ X an 0 < t < 1.

Let X be a convex set. A function f : X R is said to be quasi-concave on X if the set{x R : f(x) >= c} is convex for all real numbers c, for all real numbers c

It is strictly quasi-concave on X if {x R : f(x) >c} is convex for all real numbers c.

Econ 102 SY 2008 2009

x1

x2

x’

x

f(x)=f(x’)=c

t x + (1-t)x’

f(t x + (1-t)x’ )

Econ 102 SY 2008 2009

Returns to scale

Suppose vector of inputs x which produces output y is proportionately scaled up or down, then there are three possibilities for output y. The technology may exhibit:(1) constant returns to scale;

(2) decreasing returns to scale, and

(3) increasing returns to scale.

Econ 102 SY 2008 2009

Definitions

A production function f(x) is said to exhibit : constant returns to scale if f(tx) = tf (x) for all t

>= 0. decreasing returns to scale if f(tx) < tf (x) for all

t > 1. increasing returns to scale if f(tx) > tf (x) for all

t > 1.

Econ 102 SY 2008 2009

x

costs

Econ 102 SY 2008 2009

Returns to scale and homogeneity of production function

CRTS implies production function is homogeneous of degree 1.

Homogeneous production functions means: f(tx) = tk

f(x). In general, a production technology which is

homogeneous of degree k exhibits any of the returns to scale possibilities as follows: If k=1, then it is CRS. If k<1, then it is DRS. If k>1, then it is IRS.

Econ 102 SY 2008 2009

Global versus Local Returns to Scale

Global if the returns to scale applies to all vectors in the input requirement set.

Local if the type of returns to scale applies to a subset of the vectors in the input requirement set only; another type of RTS may apply to other subsets.

Econ 102 SY 2008 2009

Elasticity of Scale and local returns to scale

It is defined as the percent increase in output due to a one percent increase in all inputs; evaluated at t=1.

LOCAL RETURNS TO SCALE. A production function f(x) is said to exhibit locally increasing, constant, or decreasing returns to scale as e(x) is greater, equal, or less than 1.

tdty(t)

dy(t)

= e(x)

Econ 102 SY 2008 2009

Marginal rate of technical substitution Suppose that

technology is represented by a smooth production function. Suppose the firm is at producing y* = f(x*1 ; x*2). If the firm increases a small amount of input 1 and decreases some amount of input 2 to maintain a constant level of output.

MRTS of the two inputs is defined to be the percent decrease in input 2 due to an increase of input 1.

.MP

MP

d

dMRTS

d),

d),

2

1

),

),

0dy

x,x 21

-=

-=x

x=

xx

xf(x+x

x

xf(x= 0

2

21

1

21

x

xf(xx

xf(x

=1

2

22

211

1

21

∂

∂∂

∂

∂

∂

∂

∂

Econ 102 SY 2008 2009

L

K

Econ 102 SY 2008 2009

Econ 102 SY 2008 2009

MRTS of a Cobb Douglas Production Function

1

2

2

1,xx

221

2

21

1

12

11

1

21

x)1(

αx=

xY)1(

xY

=MRTS

x

Y)1(xx)1(=

∂x

),x∂f(x

x

Yxx=

∂x

),x∂f(x

21

--

--

-=-

=

-

--

Econ 102 SY 2008 2009

Elasticity of substitution

Defined as the percentage change of input ratio divided by the percentage change in MRTS.

Measures the curvature of the isoquant.

MRTSlnd

)xx

ln(d

MRTSMRTS

)xx

(

)xx

(

1

2

1

2

1

2

==

Econ 102 SY 2008 2009

Econ 102 SY 2008 2009

Profit Maximization

Economic profit is defined to be the difference between the revenues the firm receives

The revenues may be written as a function of the level of operations of some n actions, R(a1,…, an), and costs as a function of these same n activity levels, C(a1,…, an)

Actions may be in term of employment level of inputs or output level of production expressed in value terms.

Econ 102 SY 2008 2009

Profit maximization

Max (p;w) = p’y-w’x such that {y,-x) is in Y Constrained or restricted or short run profit

maximization Max (p;z) = p’y-w’x such that {y,-x) is in Y(z)

Suppose we have a production function f(x) where x is a vector of inputs then the profit function is: (p,w) = py – w’x and its restricted or short run profit function (p,w;z) = py – w’x.

Econ 102 SY 2008 2009

Conditions for profit maximum

In vector notation Kuhn Tucker condition (which includes border

solution): Y*

wxD =

=

*)(fp

,...,n2,1 i=∀wp ix*)x(f

i∂∂

,...,n2,1 i=∀,0≥x⊥wp iix*)x(f

i ≤∂

∂

Econ 102 SY 2008 2009

Proposition

Proposition. Suppose Y strictly convex. Then, for each given p, w RL

+ the profit maximizing production {y*,-x*} is unique provide it exists.

Econ 102 SY 2008 2009

Iso-profit lines

y=(/p)+(w/p)x profit

maximum when the slope of the iso-profit line equals the slope of the production function.

Labor

y

Y=f (x)

L*

=py-wL

B

Econ 102 SY 2008 2009

End of Lecture 7

Profit Maximization and Supply