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Ch3: Productivity, Output,
and Employment
Abel & Bernake: Macro Ch3
Varian: Ch10, Ch19
1
Chapter Outline
The Production Function
 The Demand for Labor
 The Supply of Labor
 Labor Market Equilibrium
 Unemployment
 Relating Output and Unemployment: Okun’s Law

2
The production function
describe relationship between inputs and output.
 Real Output (Y)
 Inputs: factors of production 生產要素
Y = AF(K, N)
(3.1)

K = capital: tools, machines, and structures
N = labor: physical and mental efforts of workers
F(．) reflects the economy’s level of technology
A= “total factor productivity”
(the effectiveness with which capital and labor are used)
3
Table 3.1 The Production
Function of the United
States, 1979-2007
Assumes constant returns to scale
Cobb-Douglas production function
works well for U.S. economy:
Y = A K0.3 N0.7
(3.2)
Productivity grew slowly in 1980s
and the first half of the 1990s, but
increased since the mid-1990s.
4
Returns to scale:
Initially Y1 = AF (K1 , N1 )
Scale all inputs by the same factor z:
K2 = zK1 and N2 = zN1
(e.g., if z = 1.25, then all inputs are increased by
25%)
What happens to output, Y2 = F (K2, N2 )?
If constant returns to scale, Y2 = zY1
If increasing returns to scale, Y2 > zY1
If decreasing returns to scale, Y2 < zY1
5
Examples
F (K , N )  K  N : CRS
F (K , N )  K 2  N 2 : IRS
F (K , N ) 
KN : CRS
F (K , N ) 
K  N : DRS
2
K
F (K , N ) 
: CRS
N
6
Diminishing marginal returns:
diminishing MPN

Marginal Product of Labor: MPN  Y  Y
N
N
Diminishing marginal returns: diminishing MPN
 Suppose N while holding K fixed
 fewer machines per worker
 lower worker productivity


Marginal Product of Capital:
Y Y
MPK 

K K
7
Fig 3.1 The Production Function Relating Output and Capital
Fig 3.2 The marginal product of capital
MPK 
Y Y

K K
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MPN 
Fig 3.3: MPN ( K fixed )
Diminishing marginal returns
Y Y

 AFN ( K , N )
N N
Y
output
Y  AF (K , LN)
1
MPN
MPN
As more labor is
added, MPN 
1
MPN
1
Slope of the production
function equals MPN
N
labor
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Eg, diminishing MPN

Which of these production functions have
diminishing marginal returns to labor?
a) F (K , N )  2K  15N
b) F (K , N ) 
KN
c) F (K , N )  2 K  15 N
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Supply shocks





Supply shock = productivity shock
= a shift in an economy’s production function (Fig. 3.4)
Supply shocks affect the amount of output that can be
produced for a given amount of inputs
Negative (adverse) shock: Usually slope of production
function decreases at each level of input
(eg, if shock causes parameter A to decline)
Positive shock: Usually slope of production function
increases at each level of output
(eg, if parameter A increases)
eg, weather, inventions and innovations, government
regulations, oil prices
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Fig3.4 An adverse supply shock that lowers the MPN
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Firm: Profit Optimization
Assume: Supply of each factor is fixed.
 Assume markets are competitive:
each firm takes W, Re, and P as given.
P = price of output, W = nominal wage, Re= nominal rental rate
W /P = real wage (measured in units of output), Re /P= real rental rate

Max   PF ( K , N )  Re K  WN
FOC wrt K: P  MPk  Re
FOC wrt N: P  MPN  W
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Demand for labor
W
P  MPN  W , MPN 
w
P
Profit Maximization ( FOC Y) : P  ?  MC
benefit = MPN, cost = real wage
A firm hires each unit of labor
if the cost does not exceed the benefit.

W
MPN 
P
W
MPN 
: Demand for labor
P
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Fig 3.5: MPN = Demand for labor
Units of
output
Each firm hires labor
up to the point where
MPN = W/P.
Real
wage
MPN,
Labor
demand
Units of labor, N
Quantity of labor
demanded
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Summary 2
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Labor Market: the equilibrium real wage
Units of
output
Labor
supply
equilibrium
real wage
N
L
MPN,
Labor
demand
Units of labor, N
17
A↑ or K↑→ ? → ?
Fig 3.6 The effect of a beneficial supply shock on
labor demand (here A↑)
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Summary 3
19
The Supply of Labor
Aggregate supply of labor is
the horizontal sum of individuals’ labor supply
 Labor supply of individuals depends on
consumption-leisure choice

20
Individual: Utility Optimization

The consumption-leisure trade-off
Max U(C, L)
St. time constraint: L + h = T
budget constraint: C ≦ wh + V
U: utility, C: consumption, L: leisure,
h: working hours, T: time endowment,
w: real wage rate, V: nonlabor income,
 w: price of leisure, opportunity cost of leisure
 Constraint combined: C ≦ w(T-L) + V
 Trade-off: more h, less L, but more income and more C
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Optimal consumption and leisure (參考)
h > 0: working, h=0: not in the LF
at point E: corner solution -- indifferent
Consumption ($)
$1200
Y
A
$1100
P
$500
U1
U*
E
$100
U0
0
110
70
40
110
Hours of
Leisure
0
Hours of
Work
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A pure income effect (IE): V↑
Winning a lottery : V↑
 A pure income effect:
Demand for normal goods increase: C↑, L↑
 Winning the lottery: no SE
because it doesn’t affect the reward for working

 L↑=>
h↓
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An increase in real wages: w↑
An increase in the real wage : w↑
 Substitution effect (SE):
w↑: price of leisure ↑
Use cheaper C to substitute more costly L
=> C↑,L↓ => h↑
 Income effect (IE):
w↑for same h => income ↑ => C↑,L ↑ => h ↓
 w↑total effect: has offsetting IE and SE
 h ↑ if SE > IE
h ↓ if SE < IE

24
Fig 3.7 labor supply curve of an individual worker
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Temporary vs. Permanent increase in w
Optimization over time (Ch4)
ISE: intertemporal substitution effect
 ISE between current C and future C’
ISE between current L and future L’
 If temporary w↑: strong ISE + weak IE
ISE > IE => L↓, h ↑
 If permanent w↑ : weak ISE + strong IE
ISE < IE => L ↑, h ↓
 Empirical evidence support the implication.

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Aggregate labor supply
 When
current real wage rises
 Some
people work more hours
 Other people enter labor force
Result: Aggregate labor supply curve slopes
upward
27
Fig 3.8 The effect on labor supply of
an increase in wealth
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Factors that shifts aggregate labor supply

Factors increasing labor supply
 Decrease in wealth
 Decrease in expected future real wage
 Increase in working-age population
(higher birth rate, immigration)
 Increase in labor force participation
(increased female labor participation,
elimination of mandatory retirement)
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Summary 4
30
Application:
comparing U.S. and European labor markets
Unemployment rates were similar in the U.S. and
Europe in 1970s and 1980s,
but are higher in Europe since then (Fig. 3.9)
 3 reasons for higher unemployment rates in Europe:
generous unemployment insurance systems,
high tax rates,
government policies that interfere with labor markets

31
Fig 3.9 Unemployment rates in the U.S. and Europe,
1982-2008
Source: OECD Factbook 2009, Harmonised Unemployment Rates.
32
Labor Market Equilibrium
Equilibrium: Labor supply equals labor demand
 Classical model of the labor market:
real wage adjusts quickly
 Determines full-employment level of employment
and market-clearing real wage
 Problem with classical model:
can’t study unemployment

33
Fig 3.10 Labor market equilibrium
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Full-employment output

Full-employment output = potential output
= level of output when labor market in equilibrium
Yf= AF(K, Nf)
(3.4)
Y  AF ( K , N )

An adverse supply shock: A↓
MPN =AFN ↓→ DN↓→ Nf↓ (Fig. 3.11)
Yf ↓ because both A↓and Nf ↓
35
Fig 3.11 Effects of a temporary adverse supply shock
on the labor market
Sources: Producer price index for fuels and related products and power from research.stlouisfed.org/fred2/series/PPIENG; GDP
deflator from research.stlouisfed.org/fred2/GDPDEF. Data were scaled so that the relative price of energy equals 100 in year
2000.
36
Application: output, employment, and the real wage
during oil price shocks
 Sharp
oil price increases
in 1973–1974, 1979–1980, 2003–2008 (Fig. 3.12)
 Adverse supply shock—lowers labor demand,
employment, the real wage, and the fullemployment level of output
 First two cases: U.S. economy entered recessions
 Research result: 10% increase in price of oil
reduces GDP by 0.4 percentage points
37
Fig 3.12 Relative price of energy,1960-2008
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Determination of factor prices （補充）
Varian: 19.7-19.9 and Appendix
Factor prices are determined by supply and demand in
factor markets.
 Assume: Supply of each factor is fixed.
 Assume markets are competitive:
each firm takes W, Re, and P as given.

Max   PF ( K , N )  Re K  WN
FOC wrt K : P  MPK  Re
FOC wrt N : P  MPN  W
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Why assuming CRS?
Eg, Cobb-Douglas Production Function



Y  AK N  AK N
1
A is exogenous, CRS: α+β=1  β=1-α
Each factor’s MP is proportional to its AP.
MPK   AK
 1
1
Y

K


N
MPN  (1   ) AK N
Y
 (1   )
N
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Neoclassical Theory of Distribution:
C-D production function in competitive markets
In the competitive market: ( ReP )  MPK , ( WP )  MPN
 C-D production function (CRS)
 constant factor shares:
Re
(
capital income≡ P ) K  MPK K  Y
labor income ≡ ( WP ) N  MPN N  (1   )Y
 = capital’s share of total income
1- = labor’s share of total income


Assumes CRS Cobb-Douglas production function works
well for U.S. economy: Y = A K0.3 N0.7
(3.2)
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The ratio of labor income to total income
in the U.S.
Labor’s
1
share of
total
0.8
income
0.6
Labor’s share of income
is approximately constant over time.
(Hence, capital’s share is, too.)
0.4
0.2
0
1960
1970
1980
1990
2000
42
Taiwan data: labor share
薪資報酬佔所得比例(%)
100
80
%
60
40
20
0
1997
1998
1999
2000
2002
2001
年度
2003
2004
2005
2006
薪資報酬佔所得比例
43
Neoclassical Theory of Distribution
Proof that
MPN  N  MPK    (WP )N  ( Re
)K  Y
P
Exhaustion of the product
 imply zero profits for competitive
firms in the LR.
 Since π=0 for all periods,
can ignore intertemporal analysis:
profit maximization over-time

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Ch3: 勞動力之分類
臺灣地區總人口
未滿十五歲人口
武裝勞動力
十五歲以上人口
監管人口
民間人口
（現役軍人）
（民間）勞動力
就業者
非勞動力
失業者
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Fig 3.13 Worker flow:
Changes in employment status in a typical month (June 2007)
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Duration of Unemployment 失業期間
Duration of unemployment
（length of unemployment spell）
 Most unemployment spells are of short duration
 Most unemployed people on a given date are
experiencing unemployment spells of long
duration
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3 types of unemployment

Frictional unemployment 摩擦性失業
Search activity of firms and workers due to heterogeneity.
Matching process takes time.

Structural unemployment結構性失業
Reallocation of workers (lack of new skill) out of shrinking
industries or depressed regions:
matching takes a long time

Cyclical unemployment景氣性失業
48
The natural rate of unemployment

The natural rate of unemployment (u )
when output and employment are at full-employment levels


u= frictional + structural unemployment
Cyclical unemployment:
difference between actual unemployment rate and natural
rate of unemployment, u  u
49
Okun’s Law:
Relating Output and Unemployment

Relationship between
output (relative to full-employment output) and
cyclical unemployment
(3.5)
Y Y
Y
 2(u  u )
Alternative formulation:
if average growth rate of full-employment output is 3%:
Y/Y = 3 – 2 u
(3.6)

50
Fig 3.14 Okun’s Law in the US: 1951-2008
Sources: Real GDP growth rate from Table 1.1.1 from Bureau of Economic Analysis Web site, www.bea.gov/bea/dn/nipaweb. Civilian unemployment rate for all civilian workers from Bureau of
Labor Statistics Web site, data.bls.gov.
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2010/10/03 工商時報
台灣歐肯係數 四小龍最小 u = a -0.1(Y/Y )

主計處研究報告:現階段我國的歐肯係數約在0.10～0.16之間，
即經濟成長每提升1%，只能降低失業率0.10%～0.16%。
亞洲四小龍最小，顯示台灣GDP成長對改善失業的效果，相對較低:
 金融海嘯前（97年第1季）台灣的歐肯係數為0.11，
係數低於美、德、英等14個先進國家。
 97年第1季新加坡的歐肯係數為0.17、香港0.23、南韓0.35。


台灣致力發展高科技產業，雖能創造GDP，但由於所能提供的就業機
會非常有限。
主計處表示，金融海嘯期間，台灣的實質GDP衰退幅度達10.1％，
台灣的歐肯係數較低，卻也使得台灣在金融海嘯期間失業率上升幅度
相對較小。
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