Static Consumer and Firm Behavior - University of...

Static Consumer and Firm Behavior

Economics 3307 - Intermediate Macroeconomics

Aaron Hedlund

Baylor University

Fall 2013

Econ 3307 (Baylor University) Static Consumer and Firm Behavior Fall 2013 1 / 25

Models and Assumptions

Our approach will be to build and test models of the macroeconomy.

A model is a simplified, mathematical description of reality designedto yield insight and generate testable predictions.

Successful models combine simplicity and explanatory power.

All models rely on assumptions.

An assumption is good when it helps build a model that makesaccurate predictions and accounts for observations well.


Consumer Preferences

Starting point: representative consumer, static environment.

Preferences over consumption C of a basket of goods and leisure lgiven by U(C , l). Refer to (C , l) as a consumption bundle.

Some properties of preferences:1 Monotonicity (increasing utility): If C2 > C1 and l2 > l1, then

U(C2, l2) > U(C1, l1).F “More is better.”

2 Convexity (quasi-concave utility):U(Cθ, lθ) ≥ min{U(C1, l1),U(C2, l2)}, where Cθ = (1− θ)C1 + θC2 andlθ = (1− θ)l1 + θl2.

F “Averages are better than extremes.”

3 Normality : Consumption and leisure are normal goods.


Consumer Preferences

An indifference curve containsbundles that give equal utility.

The marginal rate ofsubstitution of leisure forconsumption, MRSl ,C is

MRSl ,C =∂U(C ,l)∂l

∂U(C ,l)∂C

=Ul(C , l)

UC (C , l)

“How much consumption areyou willing to give up to obtain1 more unit of leisure?”

Convexity ⇒ diminishing MRS .


Consumer Budget Constraint

Assume that consumers exhibitcompetitive behavior, i.e. they areprice-takers.

Cashless economy (barter) assumption:exchange labor for consumption.

Time constraint: l + Ns = h.

Budget constraint: C = wNs + π − T⇒ C = w(h − l) + π − T .


Consumer Optimization

Consumers optimize by maximizing utilitysubject to their budget constraint:

maxC ,l

U(C , l) such that C = w(h− l)+π−T

Interior solution:

Ul(C , l)

UC (C , l)= w and C = w(h − l) + π − T

I MRSl,C = slope of budget constraint

Corner solution:

l = h,C = π − T

Comparative Statics


Consumer Optimization: Detailed Solution

The steps to solving the consumer’s problem are as follows:

1 Write down the consumer problem,

maxC ,l

u(C , l)

subject to

C = w(h − l) + π − T

2 Write the Lagrangian,

L = U(C , l) + γ[w(h − l) + π − T − C ]

3 Set up the solution conditions,

First-Order Conditions:

{∂L∂C = 0∂L∂l = 0

Constraints: w(h − l) + π − T − C = 0



Detailed solution steps, continued:

4 Calculate the solution conditions,

First-Order Conditions:

{∂L∂C = 0 = UC (C , l)− γ∂L∂l = 0 = Ul(C , l)− γw

Constraints: w(h − l) + π − T − C = 0

5 Combine the two first-order conditions to eliminate the multiplier γ,

Solution Conditions

First-Order Condition: w =Ul(C , l)

UC (C , l)

Constraint: C = w(h − l) + π − T




6 If given a specific function U(C , l), isolate either C or l in thefirst-order condition, substitute into the budget constraint, and solve.

Example: U(C , l) =√C +√l ⇒ solution conditions are given by

First-Order Condition: w =

√C√l


From the F.O.C., we isolate C to get C = w2l and substitute into thebudget constraint, giving w2l = w(h − l) + π − T ⇒ l = wh+π−T

w(w+1) .




7 Lastly, substitute the variable you solved for to find the variable youoriginally isolated.

From before, solve for C by plugging the solution for l into C = w2l :

l =wh + π − T

w(w + 1)

C =w(wh + π − T )

w + 1

Remember that the goal is to find the consumer’s choices C and l interms of variables the consumer takes as given, w , π, and T .

If given a generic utility function U(C , l), the two solution conditionsare as far as you can go (step 5). Otherwise, complete steps 6 and 7.


Response to a Change in Taxes or Dividends

How do consumers respond to changesin π or T?

Mathematically, what are ∂C∂π and ∂l

∂π?

Using comparative statics, we get

∂C

∂π= −∂C

∂T=

wUCl − Ull

∆> 0

∂l

∂π= − ∂l

∂T=

UCl − wUCC

∆> 0

where ∆ ≡ −Ull + 2wUCl − w2UCC > 0 because of quasi-concave utility.

Income effect only.


Response to a Change in the Real Wage

How do consumers respond to changesin w?

Mathematically, what are ∂C∂w and ∂l

∂w ?

Using comparative statics, we get

∂C

∂w=

wUC + (h − l)(wUCl − Ull)

∆> 0

∂l

∂w=−UC + (h − l)(UCl − wUCC )

∆

Income and substitution effects: consumption increases, but laborsupply ambiguous because ∂l

∂w |subst = −UC∆ < 0.


Labor Supply

Optimal labor supply is Ns(w , π,T ) = h − l(w , π,T ).

When the substitution effect dominates, we get the familiar upwardsloping labor supply curve. Firms


Comparative Statics: Detailed Solution

Comparative statics tell us how choices adjust to a change in avariable that is given, e.g. how C responds to a change in w .

Mathematically, we are looking for ∂C∂w , ∂l

∂w , ∂C∂π , ∂l

∂π , etc.

If we are able to derive an exact equation for C and l , we directlytake the above derivatives and we are done.

If given a generic U(C , l), then to find C and l , we can only go as faras the solution conditions,

First-Order Condition: w =Ul(C , l)

UC (C , l)




The steps to doing comparative statics for general U(C , l) are as follows:

1 Re-write the solution conditions to get two equations equaling zero,

Ul(C , l)− wUC (C , l) = 0

w(h − l) + π − T − C = 0

These conditions hold for all w , π, and T because C = C (w , π,T )and l = l(w , π,T ) always adjust to changes in w , π, and T to ensurethe solution conditions are still satisfied.

Because the solution conditions equal zero for all w , π, and T , weknow their derivatives with respect to w , π, and T equal zero,

∂ [Ul(C , l)− wUC (C , l)]

∂π= 0,

∂ [Ul(C , l)− wUC (C , l)]

∂w= 0, etc.

∂ [w(h − l) + π − T − C ]

∂π= 0,

∂ [w(h − l) + π − T − C ]

∂w= 0, etc.



Detailed comparative statics steps, continued:2 Calculate the derivative of each solution condition with respect to w ,π, or T (any variable the consumer does not choose) and equate itto zero, where C = C (w , π,T ) and l = l(w , π,T ). For example,

∂ [Ul (C , l)− wUC (C , l)]

∂w= 0 =

=∂Ul (C,l)

∂w(chain rule)︷︸︸︷

UlC∂C

∂w+ Ull

∂l

∂w−

=∂[wUC (C,l)]

∂w(product rule and chain rule)︷︸︸︷[

w

(UCC

∂C

∂w+ UCl

∂l

∂w

)+ UC

]∂ [w(h − l) + π − T − C ]

∂w= 0 = h −

[w∂l

∂w+ l

]︸︷︷︸

=∂[wl ]∂w

(product rule)

−∂C

∂w

3 Isolate either ∂C or ∂l in the second equation, substitute into thefirst equation, and solve for the variable you did not isolate.

4 Substitute back to solve for the variable you isolated.



Doing comparative statics with a fully general U(C , l) can get rathermessy but is not as bad with a slightly less general function.

I Examples: U(C , l) = u(C ) + v(l) or U(C , l) = u(C + v(l)) withu′ > 0, v ′ > 0, u′′ < 0, and v ′′ < 0.

Without a specific U(C , l), the best we can usually do is to sign thederivatives of C and l (i.e. < 0, > 0, or ambiguous).

Example: U(C , l) = u(C ) + v(l) ⇒ solution conditions are given by

v ′(l)− wu′(C ) = 0

w(h − l) + π − T − C = 0



To find ∂C∂π and ∂l

∂π , first differentiate through both conditions by π,

∂ [v ′(l)− wu′(C )]

∂π= 0 = v ′′(l)

∂l

∂π− wu′′(C )

∂C

∂π∂ [w(h − l) + π − T − C ]

∂π= 0 = −w ∂l

∂π+ 1− ∂C

∂π

Isolate ∂C∂π in the second equation, plug into the first equation to find

∂l∂π , and substitute ∂l

∂π back to find ∂C∂π ,

∂C

∂π= 1− w

∂l

∂π︸︷︷︸from second equation

⇒ 0 = v ′′(l)∂l

∂π− wu′′(C )

(1− w

∂l

∂π

)

⇒ ∂l

∂π=

<0︷︸︸︷wu′′(C )

v ′′(l)︸︷︷︸<0

+w2u′′(C )︸︷︷︸<0

> 0 and∂C

∂π=

<0︷︸︸︷v ′′(l)

v ′′(l)︸︷︷︸<0

+w2u′′(C )︸︷︷︸<0

> 0



To find ∂C∂w and ∂l

∂w , first differentiate through both conditions by w ,

∂ [v ′(l)− wu′(C )]

∂w= 0 = v ′′(l)

∂l

∂w−[wu′′(C )

∂C

∂w+ u′(C )

]∂ [w(h − l) + π − T − C ]

∂w= 0 = h −

[w∂l

∂w+ l

]− ∂C

∂w

Isolate ∂C∂w in the second equation, plug into the first equation to find

∂l∂w , and substitute ∂l

∂w back to find ∂C∂w ,

∂C

∂w= h − l − w

∂l

∂w︸︷︷︸from second equation

⇒ 0 = v ′′(l)∂l

∂w−[wu′′(C)

(h − l − w

∂l

∂w

)+ u′(C)

]

⇒∂l

∂w=

>0︷︸︸︷u′(C) +

<0︷︸︸︷wu′′(C)

>0︷︸︸︷(h − l)

v ′′(l)︸︷︷︸<0

+w2u′′(C)︸︷︷︸<0

(ambiguous sign) and∂C

∂w=

<0︷︸︸︷−wu′(C) +

<0︷︸︸︷v ′′(l)

>0︷︸︸︷(h − l)

v ′′(l)︸︷︷︸<0

+w2u′′(C)︸︷︷︸<0

> 0


Firms

Starting point: a representative firm with production technologyY = zF (K ,N).

I Inputs are capital K and labor N with total factor productivity z .

Production satisfies:1 Constant returns to scale:

F (xK , xN) = xF (K ,N) for allx > 0.

2 Monotonicity : FK > 0 andFN > 0.

3 Decreasing marginal product:FKK < 0 and FNN < 0.

4 Complementarity : FKL > 0.


Marginal Product and Technological Improvements


Profit Maximization

In the short run K is fixed, but firms choose N to solve

maxN

zF (K ,N)− wN

The optimality condition is zFN(K ,Nd) = w where Nd is optimallabor.

How does Nd respond to changes in w , z , and K?

∂Nd

∂w=

1

zFNN< 0

∂Nd

∂z=−FNzFNN

> 0

∂Nd

∂K=−FKNFNN

> 0


Cobb-Douglas Production

Consider the Cobb-Douglas production function Y = zKαN1−α.

Short-run profit maximizing labor choice is Nd =(

(1−α)zw

)1/αK .

Total wage bill divided by output is wNd

Y = 1− α⇒ constant laborshare of income.

Labor share of income in U.S. has been approximately constant with1− α = 0.64.


Solow Residuals and Labor Productivity

Can use measures of output Y , capital K , and labor N to extractSolow residuals z = Y

K0.36N0.64 .

Total factor productivity different from average labor productivity,YN = z

(KN

)0.36.


Productivity in the 2008-2009 Recession

Given the severity of the 2008-2009 recession, average laborproductivity declined surprisingly little.

Potential causes:1 Recession originated more in housing and financial services.

2 Long term sectoral shifts in employment.


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