FIRMS IN THE
TWO-PERIOD FRAMEWORK
CHAPTER 6
2BASICS
Introduction
 Embed firms in two-period (multi-period) economy
 In each period t, representative firm produces according to a
production technology Atf(kt, nt)
 nt: labor used for production
 kt: capital (“machines and equipment”) used for production
 At: total factor productivity
 A catch-all measure for level of sophistication of technology
 Real Business Cycle (RBC) view: the driving force behind the periodic ups
and downs of macroeconomic activity (Chapter 14)
 For now, suppose At = 1 always (i.e., in both period 1 and 2)
 Broad macro view of the factors of production
 Labor – all types
 Capital
 Machines and equipment
 Trucks
 Factories
 A stock (not a flow…) variable
 Takes time to build capital (simple starting assumption: takes one period)
The function f(k, n) describes
how capital and labor
combine with each other to
yield output (goods)
Can also
think of
education
and other
intangibles
(i.e.,
experience,
brand name)
as “capital”
3PRODUCTION FUNCTION
Model Structure
 Production function f(kt, nt) with all the “usual properties” of
production functions
 Strictly increasing in kt and nt
 Diminishing marginal product in kt and nt
 When allow time-varying At, changes in A cause shifts in
production function
 Source of business cycle fluctuations in RBC theory
kt nt
Atf(kt,nt)Atf(kt,nt)
for any t
Recall from basic micro
rise in A
fall in A
rise in A
fall in Atotal output
(i.e., GDP)
total output
(i.e., GDP)
The extra
output that
results from
using one
additional unit
of input (for A = 1) (for A = 1)
4PRODUCTION FUNCTION
Model Structure
 Production function f(kt, nt) with all the “usual properties” of
production functions
 Strictly increasing in kt and nt
 Diminishing marginal product in kt and nt
 When allow time-varying At, changes in A cause shifts in
production function
 Source of business cycle fluctuations in RBC theory
 For now suppose At = 1 in each period
kt nt
for any t
Recall from basic micro
The extra
output that
results from
using one
additional unit
of input
total output
(i.e., GDP)
Atf(kt,nt)Atf(kt,nt)
(for A = 1) (for A = 1)
5CAPITAL AND INVESTMENT
Macro Fundamentals
 Capital takes time to build
 Firms must decide in period t how much capital they want to use in
the production process in t+1
 Investment
 The change in a firm’s capital stock between two consecutive periods
 Investment: a flow variable
 Analogous to consumers’ savings
 Capital: a stock variable
 Analogous to consumers’ wealth/asset position
 Except k cannot be negative (negative machines?...)
 One of the components of GDP ( = C + I + G + NX )
 Investment comprises ≈ 15% of GDP in U.S.
 Investment comprises ≈ 40% of GDP in China (high I drives rapid growth)
6BASICS
Model Structure
 Timeline of events
 Notation
 k1: capital used for production in period 1 (decided upon in “period 0”)
 n1: labor used for production in period 1
 w1: real wage rate for labor in period 1 (w1 = W1/P1)
 i: nominal interest rate
 P1: nominal price of output produced and sold by firm in period 1
AND nominal price of one unit of capital bought by the firm in period 1 for
use in period 2 (recall time to build…)
Underlying assumption/view of world: capital goods are not necessarily
“distinct” from consumption goods (i.e., computers purchased by both
firms and individual consumers)
Period 1 Period 2
k1 k3
Events during period 1: firm uses existing
capital and hires labor to produce output,
and chooses capital for next period
k2
Start of the
world
End of the
world
Events during period 2: firm uses existing
capital and hires labor to produce output,
and chooses capital for next period
Start of economic
planning horizon
End of economic
planning horizon
7BASICS
Model Structure
 Timeline of events
 Notation
 k2: capital used for production in period 2 (decided upon in period 1)
 n2: labor used for production in period 2
 w2: real wage rate for labor in period 2 (w2 = W2/P2)
 i: nominal interest rate
 P2: nominal price of output produced and sold by firm in period 2
AND nominal price of one unit of capital bought by the firm in period 2 for
use in period 3 (recall time to build…)
Underlying assumption/view of world: capital goods are not necessarily
“distinct” from consumption goods (i.e., computers purchased by both
firms and individual consumers)
Period 1 Period 2
k1 k3
Events during period 1: firm uses existing
capital and hires labor to produce output,
and chooses capital for next period
k2
Start of the
world
End of the
world
Events during period 2: firm uses existing
capital and hires labor to produce output,
and chooses capital for next period
Start of economic
planning horizon
End of economic
planning horizon
8FIRM PROFIT MAXIMIZATION
Model Structure
 A dynamic profit maximization problem
 Because firm exists for both periods
 All analysis conducted from the perspective of the very beginning of
period 1
  Must consider present-discounted-value (PDV) of lifetime (i.e., two-
period) profits
9FIRM PROFIT MAXIMIZATION
Model Structure
 A dynamic profit maximization problem
 Because firm exists for both periods
 All analysis conducted from the perspective of the very beginning of
period 1
  Must consider present-discounted-value (PDV) of lifetime (i.e., two-
period) profits
 Dynamic profit function
 (specified in nominal terms – could specify in real terms…)
1 1 1 1 1 1 1 1 1 2( , )P f k n Pk Pwn Pk  
Period-1 profits
Total revenue in
period 1 (price x
output)
Value of pre-
existing
capital (an
asset for
firms)
Total labor
cost in
period 1
Total cost of
buying capital
for period 2
(time to build 
must purchase
period-2 capital
in period 1)
10
FIRM PROFIT MAXIMIZATION
Model Structure
 A dynamic profit maximization problem
 Because firm exists for both periods
 All analysis conducted from the perspective of the very beginning of
period 1
  Must consider present-discounted-value (PDV) of lifetime (i.e., two-
period) profits
 Dynamic profit function
 (specified in nominal terms – could specify in real terms…)
2 32 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2
( , )
( , )
1 1 1 1
P kP f k n P k Pw n
P f k n Pk Pwn Pk
i i i i
      
   
Period-1 profits
Total revenue in
period 1 (price x
output)
Value of pre-
existing
capital (an
asset for
firms)
Total labor
cost in
period 1
Total cost of
buying capital
for period 2
(time to build 
must purchase
period-2 capital
in period 1)
(PDV of) period-2 profits
Total revenue in
period 2 (price x
output)
Value of pre-
existing
capital (an
asset for
firms)
Total labor
cost in
period 2
Total cost of
buying capital
for period 3
(time to build 
must purchase
period-3 capital
in period 2)
11
FIRM PROFIT MAXIMIZATION
Model Structure
 A dynamic profit maximization problem
 Because firm exists for both periods
 All analysis conducted from the perspective of the very beginning of
period 1
  Must consider present-discounted-value (PDV) of lifetime (i.e., two-
period) profits
 Dynamic profit function
 (specified in nominal terms – could specify in real terms…)
 Two-period model: k3 = 0 (no machines needed in “period 3”)
2 32 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2
( , )
( , )
1 1 1 1
P kP f k n P k Pw n
P f k n Pk Pwn Pk
i i i i
      
   
Period-1 profits
Total revenue in
period 1 (price x
output)
Value of pre-
existing
capital (an
asset for
firms)
Total labor
cost in
period 1
Total cost of
buying capital
for period 2
(time to build 
must purchase
period-2 capital
in period 1)
(PDV of) period-2 profits
Total revenue in
period 2 (price x
output)
Value of pre-
existing
capital (an
asset for
firms)
Total labor
cost in
period 2
Total cost of
buying capital
for period 3
(time to build 
must purchase
period-3 capital
in period 2)
= 0
12
FIRM PROFIT MAXIMIZATION
Model Structure
 FOCs with respect to n1, n2, k2
with respect to n1:
with respect to n2:
with respect to k2:
2 32 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2
( , )
( , )
1 1 1 1
P kP f k n P k Pw n
P f k n Pk Pwn Pk
i i i i
      
   
= 0
1 1 1 1 1( , ) 0nP f k n Pw 
2 2 2 2 2
( , )
0
1 1
nP f k n Pw
i i
 
 
2 2 2 2
1
( , )
0
1 1
kP f k n PP
i i
   
 
Identical
except for
time
subscripts
Equation 1
Equation 2
Equation 3
13
FIRM PROFIT MAXIMIZATION
Model Structure
 Re-express equation 3
2 2 2 2
1
( , )
0
1 1
kP f k n PP
i i
   
 
Divide by P1
2 2 2 2
1 1
( , )
1
(1 ) (1 )
kP f k n P
P i P i
 
 
Group terms
informatively
2 2
2
1
2
1
1 1
( , ) 1
1 1
k
P P
f k n
P i P i
      
       
       
P2/P1 = 1 + π2
2 2
2 21 1( , ) 1
1 1
kf k n
i i
     
    
    
Fisher equation
2 2( , ) 1 1
1 1
kf k n
r r
 
 
Multiply by 1+r
2 2( , ) 1 1kf k n r  
2 2( , )kf k n r
Equivalent/alternative
representation of firm
profit-maximizing condition
for capital
14
FIRM PROFIT MAXIMIZATION
Model Structure
 FOCs with respect to n1, n2, k2
with respect to n1:
with respect to n2:
with respect to k2:
 Profit-maximizing labor hiring implies
 Profit-maximizing capital purchases (for the future...) implies
2 32 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2
( , )
( , )
1 1 1 1
P kP f k n P k Pw n
P f k n Pk Pwn Pk
i i i i
      
   
= 0
1 1 1 1 1( , ) 0nP f k n Pw 
2 2 2 2 2
( , )
0
1 1
nP f k n Pw
i i
 
 
2 2 2 2
1
( , )
0
1 1
kP f k n PP
i i
   
 
Identical
except for
time
subscripts
Equation 1
Equation 2
Equation 3
2 2( , )k kf n r
equivalent
1 1 1( , )nf k n w
2 2( , )kf k n r
2 2 2( , )nf k n wAND
15
FIRM PROFIT MAXIMIZATION
Model Structure
 FOCs with respect to n1, n2, k2
with respect to n1:
with respect to n2:
with respect to k2:
 Marginal product of labor
 fn(kt,nt)
 Sometimes denote by mpnt
 Marginal product of capital
 fk(kt,nt)
 Sometimes denote by mpkt
2 32 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2
( , )
( , )
1 1 1 1
P kP f k n P k Pw n
P f k n Pk Pwn Pk
i i i i
      
   
= 0
1 1 1 1 1( , ) 0nP f k n Pw 
2 2 2 2 2
( , )
0
1 1
nP f k n Pw
i i
 
 
2 2 2 2
1
( , )
0
1 1
kP f k n PP
i i
   
 
Identical
except for
time
subscripts
Equation 1
Equation 2
Equation 3
2 2( , )k kf n r
equivalent
These FOCs are foundation for:
1. Labor Demand
2. Capital/Investment Demand
16
COBB-DOUGLAS PRODUCTION FUNCTION
Macro Fundamentals
 A commonly-used functional form in modern quantitative
macroeconomic analysis
 Describes the empirical relationship between aggregate GDP,
aggregate capital, and aggregate labor quite well
 measures capital’s share of output
 Hence measures labor’s share of output
 Interpretation
 The relative importance of (either) capital (or labor) in the production
process
 Estimates for U.S. economy:
1( , )t t t tf k n k n
 
(0,1)
(1 ) (0,1) 
0.3 
17
COBB-DOUGLAS PRODUCTION FUNCTION
Macro Fundamentals
 A commonly-used functional form in modern quantitative
macroeconomic analysis
 Describes the empirical relationship between aggregate GDP,
aggregate capital, and aggregate labor quite well
 measures capital’s share of output
 Hence measures labor’s share of output
 Interpretation
 The relative importance of (either) capital (or labor) in the production
process
 Estimates for U.S. economy:
 Estimates for Chinese economy: (not (yet) a very capital-rich
economy)
 Cobb-Douglas form useful for illustrating factor demands


1( , )t t t tf k n k n
  (saw Cobb-Douglas utility function
on Practice Problem Set 1)
(0,1)
(1 ) (0,1) 
0.3 
0.15 
( , ) (1 )t n t t t tmpn f k n k n
    
1 1( , )t k t t t tmpk f k n k n
    
18
MICRO-LEVEL LABOR DEMAND
Labor Demand in the Micro
 Firm-level demand for labor defined by the relation
(1 ) ( )t t t tw k n mpn
     for both t = 1 and t = 2
(1 ) tt
t
k
w
n


 
   
 
INVERSE RELATIONSHIP
BETWEEN wt and nt
labor
real
wage
D
Because exponent (-α) is a negative
number, can move to denominator
Follows from Equation 1 and Equation 2
19
LABOR DEMAND
Labor Demand in the Micro and the Macro
 Firm-level demand for labor defined by the relation
 Completes picture of the aggregate labor market
(1 ) ( )t t t tw k n mpn
     for both t = 1 and t = 2
(1 ) tt
t
k
w
n


 
   
 
Because exponent (-α) is a negative
number, can move to denominator
labor
real
wage
D
Sum over all firms
Firm-level labor demand function Aggregate-level labor demand function
(No tension between the
micro and macro as there
is for labor supply)
labor
real
wage
D
Follows from Equation 1 and Equation 2
INVERSE RELATIONSHIP
BETWEEN wt and nt
20
MICRO-LEVEL CAPITAL DEMAND
Capital Demand in the Micro
 Firm-level demand for capital defined by the relation
1 1 ( )t t t tr k n mpk
    
1
t
t
t
n
r
k



 
  
 
Because exponent (α – 1) is a negative
number, can move to denominator
k
r
capital demand function
Follows from Equation 3
INVERSE RELATIONSHIP
BETWEEN rt and kt
21
CAPITAL DEMAND
Capital Demand in the Micro and the Macro
 Firm-level demand for capital defined by the relation
 (Almost…) completes picture of the aggregate capital market
1 1 ( )t t t tr k n mpk
    
1
t
t
t
n
r
k



 
  
 
Because exponent (α – 1) is a negative
number, can move to denominator
k
r
capital demand function
Sum over all firms
Firm-level capital demand function Aggregate-level capital demand function
(No tension between the
micro and macro)
k
r
capital demand function
Follows from Equation 3
INVERSE RELATIONSHIP
BETWEEN rt and kt
22
FROM CAPITAL DEMAND TO INVESTMENT DEMAND
Investment Demand
 Capital is a stock variable
k
r
capital demand function
Want investment (a flow) to show up
here, not capital (a stock)
Investment is change in capital stock
between consecutive periods
23
FROM CAPITAL DEMAND TO INVESTMENT DEMAND
Investment Demand
 Capital is a stock variable
 Investment is a flow variable
k
r
capital demand function
Want investment (a flow) to show up
here, not capital (a stock)
Investment is change in capital stock
between consecutive periods
investment
r
investment demand function
inv1 = k2 – k1
At start of period 1, k1 cannot be
changed. Thus any rise in demand
for k2 is reflected one-for-one in a
rise in inv1.
 Capital demand and investment
demand functions have same shape
24
THE THREE MACRO (AGGREGATE) MARKETS
The Three Macro Markets
 Goods Markets
 Demand derived from C-L framework
(For S, have to consider how aggregate
NOMINAL P is determined…Chapter 15)
 Labor Markets
 Supply derived from C-L framework
 Demand derived from firm theory
in C-S framework
 Capital/Savings/Funds/Asset Markets
(aka Financial Markets)
 Supply derived from C-S framework
Demand derived from firm theory
in C-S framework
c
P
labor
wage
savings/in
vestment
real
interest
rate
S
D
S
D
D
S

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