Profit MaximizationSingle variable unconstrained optimization

Econ 494

Spring 2013

Hands Ch. 1.1-1.2

Agenda

• Single variable, unconstrained optimization• First and second order conditions

• Application: Profit maximization• Silb., Section 1.5

• Problem set 2 due Mon, Feb 4

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Calculus Review:Single variable optimization

Maximum Minimum

1. Objective Function

2. FONC(First-order necessary conditions)

3. SOSC (Second-order sufficient conditions)

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2( ) 0

d zg x

dx

2

2( ) 0

d zg x

dx

( )xMax z g x

( ) 0dz

g xdx

Application:Firm’s behavior• For most of this course, we will be making assertions about

the behavior of economic agents, and then developing refutable hypotheses that logically follow from these assertions.

• Usually, we will assert that economic agents optimize some objective function, , possibly with some constraints.

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Examples

• Some optimization examples:• Firms maximize profits• Firms maximize total revenue• Firms minimize costs• Individuals maximize utility subject to a budget constraint

• Each of these theories can be tested by developing refutable hypotheses and testing whether results conform to the predictions.

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Also see Silb. Examples 1-5, p. 16-21

Profit Maximization

• Assertion: Firms maximize profits , where equals total revenue minus total costs.

• Some definitions. Let:• = firm’s total output (firm chooses )• = total revenue function• = total cost function

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Generic Profit Maximization Problem

• Step 1: Set up the objective function

• Step 2: Find FONC

• Step 3: Find SOSC

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( ) ( ) ( )yMax y R y C y

( ) ( ) ( ) 0y R y C y

( ) ( ) ( ) 0y R y C y

( )xMax z g x

( ) 0dz

g xdx

2

2( ) 0

d zg x

dx

KEEP THIS SLIDE HANDY !

Profit Maximization

• The preceding problem was in a general form. Most profit max problems we encounter will be variations on this theme.• *** A key to most economic problems will be correctly specifying the

objective function. ***

• What would the objective function look like if:• the firm sells its output in a perfectly competitive market (i.e., price

taker)?• OR: there is only one firm supplying many buyers (monopolist)?• OR: the firm faces some sort of tax per-unit of output?• OR: the firm maximizes total revenue?

• Each of these theories about the firm will result in a different objective function, and possibly, a different set of refutable hypotheses.

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Begin with some assertions

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Monopoly Perfect Competition

Firm behavior Maximize profits by combining inputs into outputs

Output market Many buyers, Single seller

Many buyers, many sellers

Input market Many buyers, many sellers

Information Perfect information about prices, technology, costs, etc.

Technology Each firm knows it technological capabilities and is technically efficient.

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Two Alternative Theories:Monopoly vs. perfect competition

Monopolist Perfect competition

Total Rev.

Total Cost

Total taxes paid

Objective Function

Endogenous Vbls (Choice)

Exogenous Vbls. (parameters)

( ; )y pR yp

; , ) ( )yMax y t y yp yp C t

( )C y

t y

, which effects ( )y C y

,p t

; ) ( )( )y

p yMax y t y C y t y

(( )) pR yyy

( )C y

t y

, which effects ( ) and ( )y C y p y

t

Example 1: Perfect Competition

• A firm sells output, , in a perfectly competitive market with exogenous price .

• Firm faces a per-unit tax, .• Cost function , with

• Postulates• Many firms. This is a postulate about how the market operates.• Many consumers. No consumer can exert influence over prices.• Cost function .• Firm maximizes profits.

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Objective function

• Step 1: Set up objective function for a profit-maximizing, perfectly competitive firm facing a per-unit tax.

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Parameters are:p>0; k>0; t>0

Choice variable is:y>0

In general:

; , , ) ( ; ) ( ; )yMax y p k t R y p C y k t y

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In this example:

; , , )yMax y p k t p y k y t y

First-order necessary conditions (FONC)

• Step 2: Find FONC

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2Obj. fctn: Max ; , , ) y

y p k t p y k y t y

In general:

( ; , , ) 0() ;; )(R C y ky p y p MRtk MCt

NOTE: The FONC are the implicit form of the firm’s choice function.

In this exam

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ple:

( ; , , ) 0y p k t tp k y

A note on the FONC

• We do not assert that the “optimum” output for a firm is where • The FONC () is a necessary event, logically deduced from the

assertion of profit max.• What if the FONC are not observed?

• Does this imply disequilibrium or non-optimal behavior?• NO! If the FONC are not observed it constitutes a refutation of the model.• We assert that firms act as if they are obeying the FONC and SOSC, and on

that account we make predictions about their behavior.

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Second-order sufficient conditions (SOSC)

• Step 3: Find SOSC:

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2Obj. fctn: Max ; , , )

FONC: ( ; , , ) 2 0

y

y p k t p y k y t y

y p k t p k y t

In general:

( ) ( ) ( ) 0y R y C y

Remember, the FONC alone do not guarantee maximum profits!

In this example:

( ; ) 2 0 (Recall 0) y k k k

Solve for

• Step 4: Find the explicit choice function • How would you solve for ??

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( ; , , ) *( , , )02

2

y p k t p k y tt p

y p k tk

IMPORTANT!!

The explicit choice function will always be a function of all the parameters in the FONC, but not necessarily all parameters in the original objective function.

2Obj. fctn: Max ; , , )

FONC: ( ; , , ) 2 0

y

y p k t p y k y t y

y p k t p k y t

The FONC implicitly defines Loosely speaking, y*(p,k,t) is not yet

“alone” on the left-hand side (LHS) We could solve for , but have not yet

done so

This explicitly defines

Solve for

• How did you know that you could solve the FONC for ?• When the FONC take a specific form, it’s usually easy to see

that we can solve it.

• But…we will often be dealing with more general forms where the solution may not be so obvious:

• How do you know you can solve this for ?

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( ; , , ) *( , , )02

2

y p k t p k y tt p

y p k tk

Implicit Function Theorem (IFT)

• The FONC are an implicit relationship between the variables and parameters

• If the SOSC are non-zero, by the IFT, we can, in principle, solve the FONC for the explicit choice function.• This is a sufficient condition (not necessary)• For a more thorough discussion, see Silb section 5.3.

• Key: The IFT allows us to assert that a solution to the FONC exists.

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Solve for

• Step 4 revisited: Find choice function

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FONC

In general: ( ; ) ( ; ) 0R y p C y k t

FONC

In this example: 2 0

*( , , )2

p k y t

t py p k t

k

By the IFT, since the SOSC are non-zero, we know we can in principle solve the FONC for the explicit choice function, .

Look at the denominator.Why is the SOSC ≠ 0

important?

A comment on the SOSC

• Note that the SOSC serve some important purposes• Confirm that a solution to the FONC exists• Confirm that the solution is a maximum or a minimum• Later you will see that the sign of the SOSC is useful in determining the

sign of comparative statics

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What have we accomplished?

• We have now solved the FONC for the explicit choice function .

• In many cases, the interesting question isn’t whether or . Instead, we often want to know how the firm’s decisions will be affected by a change in one of the parameters (price, tax, cost).• Model can be useful in predicting changes in choice variables due to a

change in a parameter (refutable hypotheses).• In economics, theories are tested on the basis of these changes.

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Review• Step 1. Set up objective function

• Step 2. Find FONC. Interpret.

• Step 3. Find SOSC. Interpret.

• Step 4. Use IFT to solve FONC for choice function

• Step 5. Comparative statics (next class…)

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; , , ) ( ; ) ( ; )yMax y p k t R y p C y k t y

; , , ) ( ; ) ( ; ) 0y p k t R y p C y k t

; , ) ( ; ) ( ; ) 0y p k R y p C y k

* *( , , )y y p k t

Example 2: Monopolist (from practice quiz)

• A monopolist faces the demand curve .• Inverse demand curve:

• Cost function .• are all strictly positive constants.

• Postulates• Single firm. This is a postulate about how the market operates.• Many consumers. No consumer can exert influence over prices.• Cost function .• Firm maximizes profits.

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Step 1: Set up objective function

( ; , , , ) ( ; , ) ( ; , )yMax y a b c d R y a b C y c d

( ; , , , ) ( ; , ) ( ; , )yMax y a b c d p y a b y C y c d

21( ; , , , ) ( )

y

aMax y a b c d y y c d y

b b

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Step 2: Find FONC

FONC

2( ; , , ) 2 0

ay a b d y d y

y b b

2

2 2

1( ; , , , ) ( )

1( ; , , , ) ( )

y

y

aMax y a b c d y y c d y

b b

aMax y a b c d y y c d y

b b

Note that dropped out.

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Step 3: Find SOSC

0

2

0

2

SOSC

2 1( ; , ) 2 2 0y b d d d

y b b

2 21( ; , , , ) ( )

y

aMax y a b c d y y c d y

b b

FONC

2( ; , , ) 2 0

ay a b d y d y

y b b

y* is a maximum

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Step 4: Solve FONC for

• Note that can also be simplified to , but the way it is expressed above is handy because it retains the SOSC in the denominator.

• Why isn’t a function of as well?

FONC

2( ; , ; ) 2 0

ay a b d y d y

y b b

2

2

SOSC

1( ; , ) 2 0y b d d

y b

/*( , , )

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a by a b d

db

21( ; , , , )

y

aMax y a b d y d y

b bc c

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and are different functions

• is the demand function• It tells you how much consumers would buy at any given price (for a

particular set of parameters ).

• is the firm’s optimal output for any given set of parameters .• It is an equilibrium condition that accounts for both consumer demand

and the firm costs. • It essentially defines the intersection of supply and demand. • When or shifts, this means demand is shifting – which causes the

intersection of supply and demand, , to also change. • Likewise, a change in represents a change in the firm’s costs. • The function is just a mathematical representation of this concept.

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Solving problems with generic notation

• On your own, review handout “Solving problems with generic notation”