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 designed
to 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 makes
accurate predictions and accounts for observations well.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
2 / 25
Consumer Preferences
Starting point: representative consumer, static environment.
Preferences over consumption C of a basket of goods and leisure l
given 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
2
Convexity (quasi-concave utility):
U(Cθ , lθ ) ≥ min{U(C1 , l1 ), U(C2 , l2 )}, where Cθ = (1 − θ)C1 + θC2 and
lθ = (1 − θ)l1 + θl2 .
F
3
“More is better.”
“Averages are better than extremes.”
Normality : Consumption and leisure are normal goods.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
3 / 25
Consumer Preferences
An indifference curve contains
bundles that give equal utility.
The marginal rate of
substitution of leisure for
consumption, MRSl,C is
MRSl,C =
∂U(C ,l)
∂l
∂U(C ,l)
∂C
=
Ul (C , l)
UC (C , l)
“How much consumption are
you willing to give up to obtain
1 more unit of leisure?”
Convexity ⇒ diminishing MRS.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
4 / 25
Consumer Budget Constraint
Assume that consumers exhibit
competitive behavior, i.e. they are
price-takers.
Cashless economy (barter) assumption:
exchange labor for consumption.
Time constraint: l + N s = h.
Budget constraint: C = wN s + π − T
⇒ C = w (h − l) + π − T .
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
5 / 25
Consumer Optimization
Consumers optimize by maximizing utility
subject to their budget constraint:
max U(C , l) such that C = w (h−l)+π −T
C ,l
Interior solution:
Ul (C , l)
= w and C = w (h − l) + π − T
UC (C , l)
I
MRSl,C = slope of budget constraint
Corner solution:
l = h, C = π − T
Comparative Statics
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
6 / 25
Consumer Optimization: Detailed Solution
The steps to solving the consumer’s problem are as follows:
1 Write down the consumer problem,
max u(C , l)
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
∂L
∂l
=0
=0
Constraints: w (h − l) + π − T − C = 0
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
7 / 25
Consumer Optimization: Detailed Solution
Detailed solution steps, continued:
4
Calculate the solution conditions,
First-Order Conditions:
∂L
∂C
∂L
∂l
= 0 = UC (C , l) − γ
= 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
Ul (C , l)
UC (C , l)
Constraint: C = w (h − l) + π − T
First-Order Condition: w =
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
8 / 25
Consumer Optimization: Detailed Solution
Detailed solution steps, continued:
6
If given a specific function U(C , l), isolate either C or l in the
first-order condition, substitute into the budget constraint, and solve.
√
√
l ⇒ solution conditions are given by
√
C
First-Order Condition: w = √
l
Constraint: C = w (h − l) + π − T
Example: U(C , l) =
C+
From the F.O.C., we isolate C to get C = w 2 l and substitute into the
budget constraint, giving w 2 l = w (h − l) + π − T ⇒ l = wh+π−T
w (w +1) .
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
9 / 25
Consumer Optimization: Detailed Solution
Detailed solution steps, continued:
7
Lastly, substitute the variable you solved for to find the variable you
originally isolated.
From before, solve for C by plugging the solution for l into C = w 2 l:
wh + π − T
w (w + 1)
w (wh + π − T )
C=
w +1
l=
Remember that the goal is to find the consumer’s choices C and l in
terms of variables the consumer takes as given, w , π, and T .
If given a generic utility function U(C , l), the two solution conditions
are as far as you can go (step 5). Otherwise, complete steps 6 and 7.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
10 / 25
Response to a Change in Taxes or Dividends
How do consumers respond to changes
in π or T ?
Mathematically, what are
∂C
∂π
and
∂l
∂π ?
Using comparative statics, we get
∂C
wUCl − Ull
∂C
=−
=
>0
∂π
∂T
∆
∂l
∂l
UCl − wUCC
=−
=
>0
∂π
∂T
∆
where ∆ ≡ −Ull + 2wUCl − w 2 UCC > 0 because of quasi-concave utility.
Income effect only.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
11 / 25
Response to a Change in the Real Wage
How do consumers respond to changes
in w ?
Mathematically, what are
∂C
∂w
and
∂l
∂w ?
Using comparative statics, we get
∂C
wUC + (h − l)(wUCl − Ull )
=
>0
∂w
∆
∂l
−UC + (h − l)(UCl − wUCC )
=
∂w
∆
Income and substitution effects: consumption increases, but labor
∂l
C
supply ambiguous because ∂w
|subst = −U
∆ < 0.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
12 / 25
Labor Supply
Optimal labor supply is N s (w , π, T ) = h − l(w , π, T ).
When the substitution effect dominates, we get the familiar upward
sloping labor supply curve. Firms
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
13 / 25
Comparative Statics: Detailed Solution
Comparative statics tell us how choices adjust to a change in a
variable that is given, e.g. how C responds to a change in w .
Mathematically, we are looking for
∂C
∂l ∂C ∂l
∂w , ∂w , ∂π , ∂π ,
etc.
If we are able to derive an exact equation for C and l, we directly
take 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 far
as the solution conditions,
Ul (C , l)
UC (C , l)
Constraint: C = w (h − l) + π − T
First-Order Condition: w =
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
14 / 25
Comparative Statics: Detailed Solution
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 ensure
the solution conditions are still satisfied.
Because the solution conditions equal zero for all w , π, and T , we
know their derivatives with respect to w , π, and T equal zero,
∂ [Ul (C , l) − wUC (C , l)]
∂ [Ul (C , l) − wUC (C , l)]
= 0,
= 0, etc.
∂π
∂w
∂ [w (h − l) + π − T − C ]
∂ [w (h − l) + π − T − C ]
= 0,
= 0, etc.
∂π
∂w
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
15 / 25
Comparative Statics: Detailed Solution
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 it
to zero, where C = C (w , π, T ) and l = l(w , π, T ). For example,
=
∂Ul (C ,l)
(chain
∂w
rule)
=
∂[wUC (C ,l)]
(product
∂w
rule and chain rule)
z }|
{
∂ [Ul (C , l) − wUC (C , l)]
∂C
∂l
∂l
∂C
= 0 = UlC
+ Ull
− w UCC
+ UCl
+ UC
∂w
∂w
∂w
∂w
∂w
∂ [w (h − l) + π − T − C ]
∂l
∂C
=0=h−
w
+l
−
∂w
∂w
∂w
|
{z
}
z
}|
{
∂[wl]
= ∂w (product rule)
3
Isolate either ∂C or ∂l in the second equation, substitute into the
first equation, and solve for the variable you did not isolate.
4
Substitute back to solve for the variable you isolated.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
16 / 25
Comparative Statics: Detailed Solution
Doing comparative statics with a fully general U(C , l) can get rather
messy 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)) with
u 0 > 0, v 0 > 0, u 00 < 0, and v 00 < 0.
Without a specific U(C , l), the best we can usually do is to sign the
derivatives of C and l (i.e. < 0, > 0, or ambiguous).
Example: U(C , l) = u(C ) + v (l) ⇒ solution conditions are given by
v 0 (l) − wu 0 (C ) = 0
w (h − l) + π − T − C = 0
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
17 / 25
Comparative Statics: Detailed Solution
To find
∂C
∂π
and
∂l
∂π ,
first differentiate through both conditions by π,
∂ [v 0 (l) − wu 0 (C )]
∂l
∂C
= 0 = v 00 (l)
− wu 00 (C )
∂π
∂π
∂π
∂ [w (h − l) + π − T − C ]
∂l
∂C
= 0 = −w
+1−
∂π
∂π
∂π
Isolate ∂C
∂π in the second equation, plug into the first equation to find
∂l
∂l
∂C
∂π , and substitute ∂π back to find ∂π ,
∂C
∂l
∂l
∂l
00
00
⇒ 0 = v (l)
=1−w
− wu (C ) 1 − w
∂π}
∂π
∂π
|∂π
{z
from second equation
<0
<0
z }| {
z }| {
∂C
∂l
wu 00 (C )
v 00 (l)
⇒
= 00
> 0 and
= 00
>0
∂π
∂π
v (l) + w 2 u 00 (C )
v (l) + w 2 u 00 (C )
| {z } | {z }
| {z } | {z }
<0
Econ 3307 (Baylor University)
<0
Static Consumer and Firm Behavior
<0
<0
Fall 2013
18 / 25
Comparative Statics: Detailed Solution
To find
∂C
∂w
and
∂l
∂w ,
first differentiate through both conditions by w ,
∂l
∂C
∂ [v 0 (l) − wu 0 (C )]
00
00
0
= 0 = v (l)
− wu (C )
+ u (C )
∂w
∂w
∂w
∂ [w (h − l) + π − T − C ]
∂l
∂C
=0=h− w
+l −
∂w
∂w
∂w
∂C
Isolate ∂w
in the second equation, plug into the first equation to find
∂l
∂C
∂l
∂w , and substitute ∂w back to find ∂w ,
∂C
∂l
∂l
∂l
=h−l −w
⇒ 0 = v 00 (l)
− wu 00 (C ) h − l − w
+ u 0 (C )
∂w
∂w
∂w
∂w
|
{z
}
from second equation
>0
⇒
∂l
=
∂w
<0
<0
>0
z }| { z }| { z }| {
u 0 (C ) + wu 00 (C ) (h − l)
v 00 (l) + w 2 u 00 (C )
| {z } | {z }
<0
Econ 3307 (Baylor University)
(ambiguous sign) and
∂C
=
∂w
<0
>0
v 00 (l) + w 2 u 00 (C )
| {z } | {z }
<0
Static Consumer and Firm Behavior
<0
z }| { z }| { z }| {
−wu 0 (C ) + v 00 (l) (h − l)
>0
<0
Fall 2013
19 / 25
Firms
Starting point: a representative firm with production technology
Y = zF (K , N).
I
Inputs are capital K and labor N with total factor productivity z.
Production satisfies:
Econ 3307 (Baylor University)
1
Constant returns to scale:
F (xK , xN) = xF (K , N) for all
x > 0.
2
Monotonicity : FK > 0 and
FN > 0.
3
Decreasing marginal product:
FKK < 0 and FNN < 0.
4
Complementarity : FKL > 0.
Static Consumer and Firm Behavior
Fall 2013
20 / 25
Marginal Product and Technological Improvements
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
21 / 25
Profit Maximization
In the short run K is fixed, but firms choose N to solve
max zF (K , N) − wN
N
The optimality condition is zFN (K , N d ) = w where N d is optimal
labor.
How does N d respond to changes in w , z, and K ?
1
∂N d
=
<0
∂w
zFNN
∂N d
−FN
=
>0
∂z
zFNN
∂N d
−FKN
=
>0
∂K
FNN
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
22 / 25
Cobb-Douglas Production
Consider the Cobb-Douglas production function Y = zK α N 1−α .
1/α
Short-run profit maximizing labor choice is N d = (1−α)z
K.
w
Total wage bill divided by output is
share of income.
wN d
Y
= 1 − α ⇒ constant labor
Labor share of income in U.S. has been approximately constant with
1 − α = 0.64.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
23 / 25
Solow Residuals and Labor Productivity
Can use measures of output Y , capital K , and labor N to extract
Solow residuals z = K 0.36YN 0.64 .
Total factor productivity different from average labor productivity,
Y
K 0.36
.
N =z N
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
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Productivity in the 2008-2009 Recession
Given the severity of the 2008-2009 recession, average labor
productivity declined surprisingly little.
Potential causes:
1
Recession originated more in housing and financial services.
2
Long term sectoral shifts in employment.
Econ 3307 (Baylor University)
Static Consumer and Firm Behavior
Fall 2013
25 / 25