Comparative staticsSingle variable unconstrained optimization

Multiple parameters

Econ 494

Spring 2013

Agenda

• Quick review of profit max.• Comparative statics

• One variable, one parameter• One variable, multiple parameters

• Problem set 3 due Mon, Feb 11

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3

Aside: Identity vs. equality

Identity(always holds for all

values of x)

Equality(may hold for some

values of x)

Function

1st derivative

2nd derivative

2 2( 1) 2 1x x x

2( 1) 2 2x x

2 2

2

1o

3 0

2

2

rx x

x x

32

2 3 0

x

x

Nonsen

2 0

se !

Operations, such as differentiation, when applied to identities result in identities, but when applied to equations produce nonsensical results in general.

Also see Chiang p. 6-7

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Identities & equalities:Some other examples

Identities Equalities

𝑥+𝑥≡2𝑥

𝑥+1≡2 𝑥+2

2

𝑥+𝑥=2

𝑥+1=3 𝑥

5

Identities, equalities and the FONC

• The FONC, when set equal to zero, are an equality.• Recall the profit-maximizing perfectly competitive firm

• For the FONC, , there is only one value of such that the FONC equal zero: . • For any other value of , the FONC are not zero.• Thus, the FONC are an equality.

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

6

Evaluate FONC at

• FONC: • Now consider what happens when we substitute

back into the FONC:

• Let’s do some algebra to show that this always equals zero, for any given set of parameters …

7

Evaluate FONC at

• Begin with:

• Cancel out :

• Simplify:

• Rearrange terms:

• Simplify:

• Since this always equals zero, for any given set of parameters we have an identity:

8

More on FONC evaluated at

• Let’s look at the FONC for a perfectly competitive firm:

• Substitute back into the FONC:

Explicit ChoicFONC (implicit f e Fctn

*( , , )2

ctn)

( ; , , ) 2 0

t p

y py p k t k tp k y tk

*( , , )

*(; , , 2

22

, ) 0,

0

y p k tyy p ky p k t p k tt

t pp k t

k

Note the identity

Note the identity

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More on FONC evaluated at

• Now let’s look at the FONC for a more general case:

• Substitute back into the FONC:

Explicit Choice FctnFONC (implicit fctn)

; , , ( ; ) ( ; ) *( , ,0 ) y p k t R y p C y y p kt tk

*( , , )

General case:

; , , ( ; ) ( ;*( , , ) *( , ) ) 0,y p k ty

y p ky p k t t y pR p C k tk t

Note the identity

10

More on FONC evaluated at

• Pick one of the three parameters, say .

• Look at the identity, and note that shows up in 2 colors – blue and red.• Blue: this is where shows up explicitly in the objective function and in the FONC

(before we solve for )

• Red: this is where shows up in the identity after we solve for and then substitute

back into FONC

Specific case (perfect competition):

2 02

tkp t

k

p

No longer a function of

General case:

( *( , , ); ) ( *( , , ); ) 0p pR y k pt C y k t k t

Quick 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

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; ) ( ) ( )yMax y t R y C y t y

; ) ( ) ( ) 0y t R y C y t

) ( ) ( ) 0y R y C y

Set up a generic profit max problem. Make no assertions about the market or the cost function. The firm faces a per-unit tax, , on output.

What are theendogenous variables? exogenous parameters?

Also see BBT section 6.6.1

Keep this slide handy

* *( )y y t

Where are we?

• We have used the IFT to solve the FONC for the explicit choice function .

• We often want to know how the firm’s decisions will be affected by a change in one of the parameters (in this case, ).• 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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Comparative statics

• Definition: Mathematical technique by which an economic model is investigated to determine if refutable hypotheses are forthcoming. (see Silb., p. 15)

• We usually want to know if we can say anything about the sign of the comparative static:

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(Choice function) (Endog. vbl.) *( ) or , for example:

(Parameter) (Exog. vbl.)

d d dy t

d d dt

*( )Is 0? 0?dy t

dt

Step 5. Comparative statics(one variable, one parameter)

• It is intuitive to turn to the FONC and SOSC to derive comparative static results.

• Step 5a. Substitute the explicit choice function back into the FONC to get an identity:

• Note…this is not the FONC (an equality). • Rather, it is the FONC evaluated at the optimal solution (an identity).

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*( ): What is the sign of Ou g l ?r oa

dy t

dt

*( )

*( ); ( ) *( )( ) 0y ty

y t y ty t R C t

: ; ) ( ) ( ) 0FONC t R C ty y y

Step 5. Comparative statics

• Step 5b. Differentiate w.r.t. (wrt = “with respect to”) the parameter of interest, .• First, find all instances of in the identity above.• Then, start differentiating…• Recall that:

• If , then

• So, we can differentiate the terms one at a time, and then add together.

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*( ): What is the sign of Ou g l ?r oa

dy t

dtStep 5a. Identity ( *( )) ( *( )) 0R y t C y t t

Step 5b. Differentiate w.r.t.

• Begin with • Apply the chain rule. • Here’s one way to think about this…• affects in two ways:

1. affects

2. affects

• By the chain rule, we multiply them:

• Repeat with …

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*( )( *( ))

dy tR y t

dt

( *( )) ( *( )) 0 R y C t tyt

Step 5b. Differentiate w.r.t.

• Repeat with • Apply the chain rule. • Again, affects in two ways:

1. affects

2. affects

• By the chain rule, we multiply them:

• Repeat with …

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*( )( *( ))

dy tC y t

dt

( *( )) ( *( )) 0 R y C t tyt

Step 5b. Differentiate w.r.t.

• Repeat with And again with .• Apply the power rule: If , then

• In this case, we have , so

• Therefore

• Combine terms to get:

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( *( )) ( *( )) 0 R y C t tyt

1dt

dt

*( ) *( )( *( )) ( *( )) 1 0

dy t dy tR y t C y t

dt dt

• Because this is an identity, we must also differentiate the right-hand side (RHS).

• The derivative of any constant is zero.

Comparative statics

• Step 5c. Use algebra to solve 5b:

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: What is the sign ofOur g l)

o a ?*(dy t

dtStep 5a. Identity ( *( )) ( *( )) 0R y t C y t t

Step 5b. Differentiate ( *( )) ( *( )*

)(

1( )

0) *dy t dy

R y tt

dC y t

t dt

*( ) 1

( *( )) ( *( ))

dy t

dt R y t C y t

*( )( *( )) ( *( )) 1 0

dy tR y t C y t

dt

*( )( *( )) ( *( )) 1

dy tR y t C y t

dt

Step 5d. What is the sign?

Hint: Look at denominator.

Step 6. Interpret

• Recall the SOSC:

• Use SOSC to sign the comparative static:

• Interpret result:• A profit-maximizing firm’s output will decrease (increase) as the tax rate

increases (decreases).

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( *(

*( ) 1

)) ( *( )) R y t C

d

y

y

t t

t

d

( ) ( )) 0 R y C yy

0 by SOSC

*( ) 1

( *( )) ( *(0

))

dy t

dt R y t C y t

What have we accomplished?

• The unobservable postulate of profit maximization has led to the refutable proposition that output will decline as the tax rate increases.

• We made no assumptions about market structure, revenue functions or cost functions.

• We now have a prediction about changes in the choice variable, , in response to a change in the tax rate, .

• Important thing is not the result, per se, rather the methodological framework used to reach the conclusion.

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Review

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Step 1. ) ( ) ( )yMax y R y C y t y

Step 2. ) ( ) ( ) 0y R y C y t

Step 3. ) ( ) ( ) 0y R y C y

Step 5a. *( )) ( *( )) ( *( )) 0y t R y t C y t t

Set up a generic profit max problem. Make no assertions about the market or the cost function. Firm faces a per-unit tax (t) on output.

*( ) *( )Step 5b. ( *( )) ( *( )) 1 0

dy t dy tR y t C y t

dt dt

*( ) 1Step 5c. 0

( *( )) ( *( ))

dy t

dt R y t C y t

See Handout #4 for example with ad valorem tax.

Step 4. Use IFT to find explicit choice function *( )y t

Keep this slide handy

More than one parameter

• The previous example had only a single choice variable () and a single parameter ().

• The next example will also have one choice variable, but there will be more than one parameter.• For comparative statics, we will use partial derivatives.• Otherwise, approach is the same.

• For review of partial derivatives, see SH 11.1-11.2, Hoy 11.1

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Example

• Consider a perfectly competitive firm whose cost function is , with , and who has a total revenue function .

• Assume that this firm pays an advertising fee, , per-unit of output.

• Assume firm faces fixed costs, .

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What are the choice variables? Parameters?

1. Set up optimization problem

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Cost: Total Revenue:

Advertising per unit: Fixed costs:

In general:

; , , , ) ( ; ) ( ; )yMax y p k a F R y p C y k a y F

2

In this specific example:

; , , , )yMax y p k a F p y k y a y F

2. Find FONC

• Interpret FONC:• The advertising fees are really just another cost of doing business.

From the FONC, we get the usual result that a profit maximizing firm will choose the level output such that , where includes advertising fees.

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21. Obj. fctn.: ; , , , )

; , , , ) ( ; ) ( ; )

y

y

Max y p k a F p y k y a y F

Max y p k a F R y p C y k a y F

2. FONC ; , , ) 2 0

; , , ) ( ; ) ( ; ) 0

y p k a p k y a

y p k a R y p C y k a

What happened to ?

2 ( ) ( )p k y a MR y MC y

3. Find SOSC

• FONC alone do not guarantee a maximum.

• The SOSC tell us 2 things. What are they?1. As long as the SOSC hold, we know profits are maximized.

2. We also know that we can use the IFT to solve the FONC for the explicit choice function.

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3. SOSC ; ) 2 0

; , ) ( ; ) ( ; ) 0

y k k

y p k R y p C y k

FONC ; , , ) 2 0

; , , ) ( ; ) ( ; ) 0

y p k a p k y a

y p k a R y p C y k a

4. Find explicit choice function (specific case)

• What will be a function of?• is not a function of because is not in the FONC.𝐹 𝐹

• In this specific case, we have the functional forms for marginal

revenue, and marginal cost, , so we can solve the FONC

explicitly for

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21. Obj. fctn.: ; , , , )yMax y p k a F p y k y a y F

2. FONC ; , , ) 2 0y p k a p k y a

3. SOSC ; ) 2 0y k k

4. Find explicit choice function (general case)

• How do we know exists?• What will be a function of?

• is not a function of because is not in the FONC.𝐹 𝐹• By the IFT, since the SOSC are non-zero, we know that a

solution to the FONC exists and will be a function of all the parameters in the FONC.• Therefore, we can solve the FONC implicitly for

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1. Obj. fctn.: ; , , , ) ( ; ) ( ; )yMax y p k a F R y p C y k a y F

2. FONC ; , , ) ( ; ) ( ; ) 0y p k a R y p C y k a

3. SOSC ; , ) ( ; ) ( ; ) 0y p k R y p C y k

5. Comparative statics

• How will the firm change its output in response to a decrease in costs (decrease in )?

• Why might costs decrease?• Change in technology• Change in input prices

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5. Comparative statics for a change in .

• Step 5a: Substitute explicit choice function back intoFONC to get identity:

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*( , , )

*( , , ) *( , ,

2 0

( ; ))) ( ; 0

y p a k

y p a

p k a

R p C k ak y p a k

2. FONC ; , , ) 2 0

; , , ) ( ; ) ( ; ) 0

yy p k a p k a

y p k a R p C ky y a

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Step 5b: Differentiate wrt (specific case)

• First, find all instances of in the identity.

• Differentiate both sides using product rule:

• Note that the derivative of any constant, i.e., is zero.

4. Identity 2 *( , , ) 0k kp y p a a

*( , , )2 2 *( , , ) 0

y p a kk y p a k

k

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Step 5c: Solve for

• Use algebra to solve:

*( , , )2 2 *( , , ) 0

y p a

kk

kk y p a

*( , , )2 2 *( , , )

y p a kk y p a k

k

*( , , ) 2 *( , , )

2

y p a k y p a k

k k

) 2 0y k What sign?? • Recall that the SOSC for a

max are negative• Compare with denominator.

¿0

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Step 5b: Differentiate wrt (general case)

• First, find all instances of in the identity.

• Start with and apply chain rule

1. affects

2. affects

• By the chain rule, we multiply them:

• Repeat with …

4. Identity ( *( , , ); ) ( *( , , ); ) 0R k ky p a p C y p a ak

Note partial derivative because is a function of

both and .

Because is a function of only one variable, .

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Step 5b: Differentiate wrt

• Repeat with: • This differs from in that appears in two places

• where denotes the derivative of wrt .

4. Identity ( *( , , ); ) ( *( , , ); ) 0R y p a k p C y p a k k a

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Step 5b: Differentiate identity wrt

• Note that the derivative of both and wrt is zero.

• Combine all terms to get:

• Finally, solve for …

*( , ) *( , , )( *( , , ); ) ( *( , , ); ) ( *( , , ); ) 0yk

y a k y p a kR y p a k p C y p a k k C y p a k k

k k

4. Identity ( *( , , ); ) ( *( , , 0); )R y p a k p C y p a k ak

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Step 5c. Solve 5b (general case)

( *( , , ); ) ( *( , , ); ) ( *( , , ); )*( , , ) *( , , )

0ykR y p a k p C y p ay p a k y p a k

k kk k C y p a k k

Use a little algebra…

( *( , , ); ) ( *( , , ); ) ( *( , , ); ) 0*( , , )

yk

y p aR y p a k p C y p a k k C y

kk k

kp a

( *( , , ); ) ( *( , , ); ) ( *( , ,*( , , )

); )ykR y p a k p C y p a k k C yy

p a kp a

kk

k

( *( , , ); )*( , , )

( *( , , ); ) ( *( , , ); )ykC y p a k ky p a k

k R y p a k p C y p a k k

What sign?? • Recall that the SOSC for a max

are negative• Compare with denominator.

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

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What sign for (general case)

• By the SOSC, we know that the denominator is negative.• What about the numerator?

• In the specific example, we knew that .• However, in the more general case, we do not know what looks like.• To determine the sign, we will need to make an assertion about .• Note that is the marginal cost function.• tells us how marginal cost changes as changes• It makes economic sense to assert that marginal costs are increasing

in , hence

0

0 by SOSC

( *( , , ); )*( , , )0

( *( , , ); ) ( *( , , ); )ykC y p a k ky p a k

k R y p a k p C y p a k k

6. Interpret result

• Ceteris paribus (all else equal), a decrease in the firm’s costs will result in an increase in the quantity of output the firm produces.

• Note: This assumes that only decreases, and advertising, , is unchanged (ceteris paribus).

• We can not draw any conclusions about what would happen if both and were to change at the same time.

• Also, we now have a testable hypothesis that logically follows from the assertion of profit max.

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*( , , )0

y p a k

k

5. More comparative statics change in .

• How will the firm change its output in response to an increase in advertising fees (increase in )?

• Step 5a: We already substituted the explicit choice function back into FONC to get identity:

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*( , , )

*( , , ) *( , ,

2 0

( ; ))) ( ; 0

y p a k

y p a

p k a

R p C k ak y p a k

2. FONC ; , , ) 2 0y p a k p k y a

Typo corrected

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Step 5b: Differentiate identity wrt

• First find all the instances of in the identity, then differentiate.

• Specific:

• General:

2 *( , , ) 0

( *( , , ); ) ( *( , , ); ) 0

a

a

p k y p k

R ay p k p C y p k ak

a

*( , , ) *( , , )( *( , , ); ) ( *( , , ); ) 1 0

y p a k y p a kR y p a k p C y p a k k

a a

*( , , )2 1 0

y p a kk

a

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Step 5c. Solve for *( , , )

2 1 0y p a k

ka

Specific:

*( , , ) 1

02

y p a k

a k

General:*( , , ) *( , , )

( *( , , ); ) ( *( , , ); ) 1 0y p a k y p a k

R y p a k p C y p a k ka a

*( , , ) 1

0( *( , , )) ( *( , , ))

y p a k

a R y p a k C y p a k

Again, the SOSC < 0 is the denominator, so we can sign the comparative static.

Interpret: For the firm, advertising fees are another cost of producing output. As these fees increase, the firm will reduce output.

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Notice a pattern???

• The parameter appears in two colors, red and blue.• Blue: This is where appears explicitly in the FONC

(and therefore the identity as well).• Red: This shows only in the identity, not the FONC,

as a result of substituting back into the FONC.

• When we differentiate the identity wrt in blue, we get –1.• When we differentiate the identity wrt in red, we get:

• Now compare this to the comparative static result…

FONC: ( ; ) ( ; ) 0

Identity: ( *( , , ); ) ( *( , , ); ) 0

R y p C y k

R y p k p C y p k ka

a

aa

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*( , , ) 1

0( *( , , ); ) ( *( , , ); )

y p a k

a R y p a k p C y p a k k

Notice a pattern???

( *( , , ); ) ( *(

*(

, , ); )

,0

1, )

R y p a k p C y

y

p a k k

p a k

a

The numerator is from the explicit appearance in the FONC.

The denominator is the SOSC.

Go back and look at the other comparative static result, . You will see that the same pattern holds.

Here is the comparative static result:

On your own…Try finding the comparative static result for a change in price, .

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Calculus review: Optimization with 2 variables

Maximum Minimum

FONC(both must hold)

SOSC(all 3 must hold)

( , )z g x y

( , ) 0; ( , ) 0x yg x y g x y

2 0xx yy xyg g g

0; 0xx yyg g 0; 0xx yyg g

These are the optimality conditions when there are only 2 variables. Later, you will see a more general version of this using matrices.

Calculus review: Young’s theorem

• The order of differentiation does not matter• Result extends to any 2nd partial derivative of a function of

many variables• This symmetry result will come in handy…

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2 2

For any function ( , ) with continuous 2nd derivatives,

( , ) ( , )or ( , ) ( , )xy yx

g x y

g x y g x yg x y g x y

x y y x