1 Exogenous Growth Model 1.1 Constant Technology
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1
Exogenous Growth Model
(Based on Solow 1956)
1.1
Constant Technology
Discrete time: = 0 1 2 ∞
Two factors of production:
- Labor
- Capital
Produce one final good that can be used for consumption or as capital in the
production process.
Factor supply
Labor supply at + 1 :
+1 = (1 + )
where:
0 is given
−1
capital supply at + 1 :
+1 = + (1 − )
where:
0 is given
- aggregate saving
∈ [0 1]
A1:
+ 0
Production
output produced at time :
= ( )
A2:
1
( ) ( ) 0 ( ) ( ) 0 for all
0
lim→0 ( ) = ∞
lim→∞ ( ) = 0
(0 ) = 0
( ) = ( )
→
= ( ) = ( 1) ≡ ( )
where ≡
It follows from A2:
(0) = 0
( (0) = (0 ) = 0)
for all 0 :
0 ( ) = ( ) 0
( ( ) = ( ) = 0 ( ))
and
00 ( ) = 0
( = 0 ( ) = 00 ( ) )
lim →0 0 ( ) = ∞ lim →∞ 0 ( ) = 0
Moreover:
since: ( ) = ( )differentiating with respect to :
( ) = +
and dividing by :
( ) = 0 ( ) +
→
( ) − 0 ( ) = 0
Remark:
In a competitive environment:
the rate of return per unit of capital (rental rate):
= 0 ( )
2
the wage rate per unit of labor:
= ( ) − 0 ( )
Remark:
Since ( ) = + it follows from differentiating with respect to that
= + +
→
+ = 0
→
0
Consumption, Saving and Investment
=
where ∈ [0 1]
Capital Accumulation:
+1 = + (1 − )
= ( ) + (1 − )
→
→
+1 =
+1
( ) + (1 − )
=
+1
+1
+1 =
The Dynamical System
{ }∞
0 such that
( ) + (1 − )
≡ ( )
1+
+1 = ( ) ∀
where 0 is given
Let be output per worker
= = ( )
3
→
∞
{ }∞
0 uniquely determines { }0
Properties of ( ) :
(0) = 0
0 ( ) + (1 − )
0 ∀ 0
1+
00 ( )
00 ( ) =
0 ∀ 0
1+
lim 0 ( ) = ∞
0 ( ) =
→0
lim 0 ( ) =
→∞
1−
∈ [0 1)
1+
Remark:
The strict concavity of ( ) follows from:
1. the strict concavity of ( )
2. saving is a constant fraction of output
Steady states
̄ such that:
̄ = (̄) =
→
(̄) + (1 − )̄
1+
( + )̄ = (̄)
→ there exist 2 steady states:
̄ = 0 unstable
̄ 0 stable
Remark:
+ 0 (̄)
Comparative Statics
Proposition.
̄
0
̄
0
4
̄
=0
0
Proof.
Let
(̄ ) ≡ ( + )̄ − (̄) = 0
→
̄
̄
= −
=−
0
0 (̄)
+
−
̄
̄
(̄)
=
= −
0
0 (̄)
+
−
̄
initial condition do not matter since there exists a unique globally stable
steady state equilibrium
Comparative Dynamics
Let
≡
+1 −
Proposition.
0
0
0
Proof.
→
¸
( ) + (1 − )
−
=
1+
( ) − ( + )
=
(1 + )
( )
+
=
−
(1 + ) 1 +
∙
1−
( )
−
0
=−
2
(1 + ) (1 + )2
5
( )
=
0
(1 + )
=−
[( ) − 0 ( ) ] 0
(1 + )2
Conclusion: no growth in the long-run without technological progress
Testable Implications and Evidence
conditional convergence
convergence
convergence
1.2
Threshold Externalities
= ( ) = ( )
where ≡
and = ( )
( ) =
Dynamics:
+1 =
Testable Implications
club convergence
⎧
⎨ ̃
⎩
≤ ̃
⎧
⎨ ( ) ̃
⎩
( ) ≤ ̃
6
2
Growth Accounting
(Solow 1956)
Production
= ( )
→
+ ∆
+ ∆
∆ - the change in the variable between two periods, =
→
∆
∆ ∆ ∆
=
+
+
→
∆ ∆
∆
∆
=
−
−
- the share of capital = the elasticity of output with respect to capital
- the share of labor = the elasticity of output with respect to labor
If ( ) is characterized by CRS:
∆ = ∆
= ()
=
=
→
∆
∆ ∆
=
−
7
3
Overlapping Generations Model
(Based on Diamond 1965)
Production
= ( )
satisfies A2
→
= ( ) = ( 1) = ( )
where ≡
Wage per worker
= ( ) − 0 ( )
Return to capital (= 1 + between − 1 and )
= 0 ( ) + 1 −
Individuals
A generation of size is born every period and lives for two periods
Individuals:
supply labor inelastically, consume and save in their first life period
consume in the second
Utility of the working generation:
= ( +1 )
Budget constraint
→
+1 = +1 = +1 [ − ]
+
+1
=
+1
Optimization:
= ( +1 )
since consumption (in second period) is normal:
0
A3
≥0
+1
8
Remark: a sufficient assumption instead of A3 is that the absolute slope of
the supply of saving with respect to the return to saving is larger than the
slope of the demand for capital. In particular, since consumption in second
period is normal, substitution effect and income effect operate in the same
direction, implying that second period consumption is increasing with +1
and therefore the elasticity of saving with respect to +1 is larger than −1
Therefore, 0 ( ) 0 can replace A3.
The evolution of capital
+1 =
→
+1 = ( +1 ) = ( ( ) − 0 ( ) 0 (+1 ) + 1 − )
→ under A3,
+1 = ( )
Properties of ( ):
0 ( ) 0
(follows from the normality of consumption)
(0) = 0
( ) ≤ ( )
Comment:
Aggregate saving per capita in the economy
= +
where
= −(1 − )
= +1
→
= + = +1 − (1 − )
in the steady state:
= + =
9
3.1
Cobb-Douglas production and utility
Production
= ( ) = 1− =
≡
wage per worker
= (1 − )
return to capital
= −1 + 1 −
utility
= ( +1 ) = ln +
optimization:
=
1
ln +1
1+
1
2+
The evolution of capital
+1 = =
1
(1 − ) ≡ ( )
(2 + )
Properties of ( ) :
(0) = 0
0 ( ) 0 ∀ 0
00 ( ) 0 ∀ 0
lim 0 ( ) = ∞
→0
lim 0 ( ) = 0
→∞
steady state
̄ =
3.2
µ
(1 − )
2+
¶1(1−)
The general case
Multiple steady state equilibria are possible
10
4
4.1
Endogenous Growth
Ak Basic Model
(Based on Rebelo JPE 1991)
Production:
=
where ( + )
The Dynamical System:
→
=
( ) + (1 − )
1+
+ (1 − )
=
1+ ¸
∙
+ 1 −
≡ ( )
=
1+
+1 =
therefore:
(0) = 0
¸
+ 1 −
0
( ) =
1 ∀ 0
1+
00 ( ) = 0 ∀ 0
∙
growth rate:
¸
( ) + (1 − )
=
−
1+
− −
=
1+
+
−
=
1+ 1+
→
→
no conditional convergence
∙
0
0
=0
11
4.2
Human Capital Accumulation
(Related to Uzawa IER 1965, Barro JPE 1990)
Production:
= ( )
satisfies A2
=
where = = 1 and +1 = ( ) =
note that this implies that human capital fully depreciate at the end of each
period
output per capita.
(In Uzawa’s model labor’s productivity growth rate is an increasing function
of the fraction of workers in the education sector).
= = ( )
where = =
sum of investment in physical and human capital is
= = ( )
Assumption: physical capital fully depreciate at the end of each period ( =
1)
efficient allocation of investment:
max +1 = (+1 +1 )
+1 + =
+1 = +1 = ( )
it follows from the optimization that
(+1 +1 ) = (+1 +1 )0 ( )
→ the capital labor ratio that maximizes +1 ̄ satisfies:
0 (̄) = [ (̄) − 0 (̄)̄]
̄ is unique (and independent of ) since 0 (̄) decreases with ̄ and [ (̄)−
0 (̄)̄] increases with ̄
12
→ = ̄ for all
Remark: ̄ is strictly decreasing in (follows from implicit differentiation)
Let be the fraction of physical capital in investment, and therefore, 1 −
is the fraction of human capital investment:
+1 = ( )
+1 = (1 − ) ( )
→
+1 =
where is efficient
→
= such that
+1
=
+1
(1 − )
+1 = ̄ =
→
→
=
(1 − )
̄
1 + ̄
(̄)
1 + ̄
→ (multiplying both sides of the equation with (̄)):
+1 = (1 − ) ( ) =
+1 = +1 (̄) =
=
(̄)
(̄)
1 + ̄
(̄)
1 + ̄
+1 −
(̄)
=
−1
1 + ̄
→ for sufficiently high: productivity of education saving rate and pro(̄)
ductivity of final output ( )
− 1 0.
1+̄
The positive effect of on +1 (despite its negative effect on, and thus on
̄) follows from the envelop theorem noting that +1 is strictly increasing in
for any where = arg max +1 .
=
• no convergence
• no limit to human capital accumulation
13
4.3
Endogenous Technical Change
(Based on Frankel AER 1962, Romer JPE 1986, Lucas JME 1988)
The level of technology is:
= ( )
is external to the firm
(in Lucas 1988 () is a function of human capital)
Production:
= ( ) ( )
where ( ) is derived from a function ( ) that satisfies A2
Saving per worker is ∈ (0 1)
a. ( )( ) is linear in
example:
( ) = 1−
( ) =
→ model with constant factor shares.
inconsistence with conditional convergence.
b. lim →∞ ( )( ) is linear in
→
share of capital 9 1
conditional convergence
14
4.4
4.4.1
Endogenous R&D
Quality Ladder Model
(Related to Lucas JME 1988, Grossman Helpman 1991; Aghion Howitt
Econometrica 1992)
Production of the final good:
The final good produced by each worker in the final good sector is
=
where,
= −1 +
−1 is the non-excludable existing technology and is new knowledge (inventions) purchased by the worker.
Individuals
In each period a population of size joins the economy
Individuals are active one period in which they work in the final good sector
or in the R&D sector
The number of workers in the R&D sector is
The number of workers in the final good sector (production) is
+ =
Production of technology:
The number of non-rival inventions each worker in the R&D sector produces
in is:
−1
inventions are made at the beginning of the period and sold to producers
Equilibrium
In equilibrium all workers purchase all inventions:
= −1
→
= −1 + −1 = −1 (1 + )
15
The surplus generated by each invention used by each worker is 1
The surplus is divided between production workers and R&D workers: a
fraction ∈ (0 1] is allocated to the R&D worker and 1− to the production
worker.
income of each R&D worker in is
= −1
income of each production worker in is
= −1 + (1 − )−1
for 1 equilibrium in the labor market (individuals are indifferent
between the two occupations) implies:
=
−1 = −1 + (1 − )−1
= 1 + (1 − )
→
1 + (1 − )
1 + −
− =
1 +
=
=
→ if 1 for all :
= − 1
= (1 − ) + 1
if ≤ 1
= 0
=
16
Since = −1 + −1
= =
⎧
⎨ − 1 1
− −1
= =
⎩
−1
0
≤ 1
Conclusions
1. Growth is affected by:
scale
R&D productivity
patents property rights
2. Crucial elements:
technology is non-rival and excludable
linearity of technological progress with respect to the technological level
Comments
1. monopolistic competition
→ may generate over investment in R&D
2. externality to technology
→ may generate under investment in R&D
3. if investment in technology takes place before benefits from the technology
are exhausted
→ the interest rate/time preference have an effect on R&D investment
17
4.4.2
Criticism
(Jones 1995)
1. Economies of scale
2. Non decreasing productivity in R&D
inconsistent with empirical evidence from the 20th century
3. New technology is proportional to the stock of old technology
Define
∆
+1 −
≡
=
Suppose
+1 = + (&) = [1 + (&)]
where R&D is constant over time (it can be replaced by your favorite candidate, human capital, population, or anything else)
→
= = (&)
and
∆ = (&)
Suppose, in contrast
∆ = (&)
6= 1
∆
= (&)−1
6= 1
→ if 1 is growing over time converging to infinity
→ if 1 is declining over time converging to zero
=
18
4.4.3
Scale Effect in a Malthusian Economy
(based on Kremer 1993)
Production
= ( )
1−
=
µ
¶
where is the adult population in , is the constant land size, augmented
by a productivity coefficient,
→ income per adult individual is
¶
µ
= =
Individuals
live two periods: childhood and adulthood.
adults work, consume and raise children
Preferences:
= (1 − ) log + log ∈ (0 1)
- consumption in the household
- number of children
Budget constraint
¶
µ
+ =
is the cost of raising a child
Optimization
µ
=
¶
where =
The evolution of population
+1 = = = = ( ) 1−
→ for any given there exists a unique globally stable steady state ,
= 1
Suppose evolves sufficiently slow → the economy is at the proximity of
the Malthusian equilibrium:
19
= 1
Consider population dynamics under:
1. Technological progress is constant
+1 −
=
→
+1 = (1 + )
where 0 is given.
→
= (1 + ) 0
ln( ) = ln(0 ) + ln(1 + )
→ Prediction: log population evolves linearly over time.
2. Technological progress is increasing with population size
+1 −
= ( );
0 ( ) 0
→ Prediction: log population is a convex function of time.
Evidence (from million BC until the 20th century)
Consistent with #2
Interpretation:
A larger population generates more non-excludable inventions.
A growing population allows for increasing scope for division of labor.
20
5
5.1
Inequality and Growth
The credit market imperfection approach
(Galor and Zeira 1993)
Production of the final good:
= ( ) + ( )
where, is the number of unskilled workers producing in the agricultural
sector, is the number of skilled workers producing in the manufacturing
sector, + = 1 is the constant population size of each generation.
Production in the Agricultural sector is
( ) =
and the production in the Manufacturing sector, ( ) is CRS production function characterized by decreasing positive marginal products and
boundary conditions that assure an interior solution to the producers maximization problem.
Individuals
The population consists of overlapping generations
A generation of size 1 is born every period and lives for two periods
Each individual has one parent and one child
Individuals:
in their first life period: are endowed with a parental bequest, invest in
human capital
in their second life period: supply labor inelastically, consume and bequeath
Preferences of individual born in are defined by the utility function:
= (1 − ) log +1 + log +1
where ∈ (0 1).
Budget constraint
+1 + +1 = +1
Hence the optimal, non-negative, transfer of individual born in period is
given by,
+1 = (+1
) = +1
21
The production of human capital
there is an indivisible cost, invested in to become skilled in + 1
Capital markets
unrestricted international capital flows at the world interest rate
→ = for all such that:
0 () + 1 − = 1 + =
→
= = () − 0 ()
as follows from the production function
=
A1: is sufficiently small such that
−
the interest rate for borrowers for sake of investment in human capital is
where 1
A2: is sufficiently large such that:
−
Investment decisions and income
if ≥
+1
= + ( − )
if
+1
= max{ − ( − ) + }
where, as follows A1 and A2 there exists
̂ =
− +
∈ (0 )
( − 1)
such that
− ( − ̂) = + ̂
22
and individuals choose to invest in human capital if and only if ≥ ̂
alternative presentation: the cost of education, which is strictly decreasing in for is equal to the return, (− )+ = − (−1) =
−
The dynamical system
⎧
⎫
̂
[ + ]
⎨
⎬
≡ ( )
+1 =
[ − ( − )] ∈ [̂ )
⎩
⎭
[ + ( − )] ≥
A3: is sufficiently small and is thereby sufficiently large such that
1
A4: is sufficiently small such that
̂(1 − )
implying that
( + ̂) ̂
Note that: (1) this assumption can be expressed as an assumption on the
parameters: (1−)(− )[−1] (2) A4 implies that 1
Assumptions A1 - A4 assure that the dynamical system is characterized by
2 stable steady states:
=
1 −
i.e., = ( + )
=
( − )
1 −
i.e., = (( − ) + )
and a threshold unstable steady state
=
( − )
1 −
i.e., = (( − ) + )
23
5.1.1
Replacing the non-convexities of the technology
(Moav 2002)
Production
= ( )
where is efficiency units of human capital.
Individuals
as in the Galor-Zeira model, with the following utility function:
= (1 − ) log +1 + log( + +1 )
(1)
where ∈ (0 1) and 0 from maximization subject to the budget con
straint, +1
= +1 + +1 the optimal, non-negative, transfer of individual
born in period is,
⎧
+1
≤ ;
⎨ 0
+1 = (+1 ) =
⎩
(+1
− ) +1
;
where ≡ (1 − ).
Capital markets
unrestricted international capital flows at the world capital rate of return,
uniquely determine the wage per efficiency unit of human capital
individuals can not borrow for sake of investment in human capital.
The formation of human capital
the level of human capital of an individual +1 is an increasing concave
function of real resources invested in education,
⎧
⎨ 1 + ;
+1 = ( ) =
⎩
1 + ≥
It is assumed that the marginal return to human capital, for is larger
than the marginal return to physical capital:
assuring that individuals invest in human capital. Noting that is a decreasing function of Assumption A1 implies that is sufficiently low.
24
The evolution of income
second life period income, +1
is uniquely determined by first life period
bequest,
⎧
;
⎨ (1 + )
+1 = ( ) ≡
⎩
(1 + ) + ( − ) ≥
the evolution of income within a dynasty is uniquely determined:
⎧
( − ) 0;
⎪
⎪
⎪
⎪
⎨
(1 + ( − ))
( − ) ∈ [0 ];
+1 = ( ) =
⎪
⎪
⎪
⎪
⎩
(1 + ) + (( − ) − ) ( − )
where 0 is given. Note that, +1
= ( ) ≥ for all
Additional restrictions on the parameter values are required in order for the
dynamical system to generate multiple income level steady states:
[(1 + ) − ]
→
( − )
− 1
1
ˆ such that dynasties with income
→ there exists an income threshold,
ˆ converge to the poverty trap income level, and
below the threshold,
ˆ converge to the high
dynasties with income above the threshold, 0
income steady state. The threshold is,
( − 1)
ˆ =
− 1
the dynamical system, +1
= ( ) generates multiple steady states. A
poverty trap, = a high income steady state, , and a threshold
income ˆ ∈ ( ) where,
=
( + 1) − ( + )
1 −
25
5.1.2
Robustness and endogenous wages
(Related to Banerjee and Newman 1993)
Consider the Galor Zeira model with random shock to income and endogenous wages:
Random shocks
Suppose that:
a fraction 1 of individuals who did not invest in human capital become
skilled workers
a fraction 2 of individuals who did invest in human capital are unskilled
workers
1 and 2 are small and, for sake of simplicity, are ignored by individuals
when making investment decisions.
Assumption A3 assures that the positive shock places individuals in the basin
of attraction of the high income steady state and it is assumed that the
negative shock places individuals in the basin of attraction of the low income
steady state. (this assumption holds for sufficient low values of and
)
The fraction of skilled individuals in (once the range ̂ − is empty):
+1 = (1 − 2 ) + 1 (1 − ) = 1 + (1 − 1 − 2 )
Therefore, the steady state value of ,
=
1
1 + 2
is independent of the initial wealth distribution.
Endogenous wages
output in the agricultural sector,
= ( )
where, is the constant land size, is a CRS production function characterized by decreasing positive marginal products and boundary conditions
that assure an interior solution to the producers maximization problem.
therefore:
= ( )
( ) 0 and lim →0 ( ) = ∞
26
In addition it is assumed that (1) ̂ where the dynamical system
describing the evolution of is characterized by two steady states for all
̂ and by one steady state for all ̂
→ there exists ̂ such that (̂) = ̂
Assume that
2
≡ ̃ ̂
1 + 2
Therefore:
If 0 ̂ the economy will converge to the steady state:
= ̃ =
1
1 + 2
If 0 ̂ (and 2 is sufficiently small) the dynamical system governing
the evolution of is characterized by a unique high income steady state.
In equilibrium therefore a sufficient fraction of the population will have a
bequest and the net return to education declines to zero, uniquely
determining the number of uneducated workers in the steady state, ̄:
− (̄) =
Replacing in the dynamical system:
+1 = [ (̄) + ] = [ + ( − )]
which is linear in equilibrium and has a unique high income globally stable
steady state. In the steady state all individuals converge to the high income
steady state, but they are subject to negative income shocks that place them
in a lower point along the dynamical path. In the steady state = 1 − ̄
Hence, endogenous wages in the Galor-Zeira model imply that it is robust to
random shocks - steady-state is affected by initial conditions.
27
5.2
The Political Economy Approach
Alesina-Rodrik 1994, Persson-Tabellini 1994, Benabou 2000
(the model is based on Benabou 2000)
One period endowment economy
Wealth distribution
Cumulative distribution of (pre-tax) wealth:
⎧
0
⎨ 0
() = [(1 + )] ∈ [0 (1 + )]
⎩
1
(1 + )
where ∈ (0 1]
The density function:
⎧
0
⎨ ¡ 0¢
−1
∈ [0 (1 + )]
1+
() =
⎩
0
(1 + )
Mean income:
̄ =
Z
(1+)
()
¶
Z (1+) µ
=
1
+
0
¯(1+)
¶1+
µ
¯
¯
1+ ¯
=1
=
¯
1+
0
0
Median income,
() = 12
Hence,
∙
() =
(1 + )
and therefore
=
¸
=
1+
≡ ()
21
28
1
2
Properties of :
→ 0
lim () = 0
()
=
(1+) log 2−
21 3
0 ∈ (0 1]
=1
= 1
1
0
0
1
Indices of equality:
1. Median/Mean ratio:
()
=
= ()
̄
→ the higher is the median, that is the higher is the lower is inequality.
2. Income variance:
2
Z
2
(1+)
() = ( ) − ̄ =
2 () − 1
0
¶
Z (1+) µ
=
1+
1
+
0
¯(1+)
¶
µ
¯
1
2+ ¯
¯
=
1+
2+
0
1
=
(2 + )
→ the variance of is decreasing in that is the higher is the lower is
inequality
Note that for = 1 () = 13 equal to the variance of a uniform
distribution with a range of 2 (the variance of the uniform distribution is
given by 2 12, where is the range), and for → 0 () → ∞
29
• Equality increases with
→ is a measure of equality.
Redistribution
Post-tax Income of individual , ̃ :
̃ = (1 − ) + ̄
- fraction of wealth taxed and redistributed equally among individuals
1 - distortionary taxation
1 - beneficial taxation
Redistribution is preferred by if
̃ ↔ ̄
Hence, if = 1 (i.e., taxation is neither distortionary nor beneficial) individual supports redistribution if and only if her income is below the mean.
Since ̄ = 1 redistribution is beneficial for if
̃ ↔
Hence, redistribution is supported by a fraction () of the population:
⎧
0
⎨ 0
() = [(1 + )] ∈ [0 (1 + )]
⎩
1
(1 + )
For ∈ (0 1)
⎧
→ 0
⎨ () = 1
() 0 ∈ (0 1]
⎩
() = 2
= 1
Note that:
For → 0 :
lim→0 = 0 and thus () = 1 since 0
For = 1 :
= ̄ and thus () 12 since 1
Hence for distortionary taxations:
30
• More equality reduces the pressure for redistribution.
For ∈ (1 2)
∀
() 12
() 0
() 0
(2) = 1
∈ (exp(ln 4 − 1)2 2]
=1
The non-monotonic impact of inequality captures two effects:
1. More inequality increases the proportion of less than average income
individuals who support redistribution.
2. More inequality increases the cost of redistribution for high income
individuals who object redistribution
Inequality and Growth
The theory predicts that inequality:
1. Has a negative effect on growth if taxation is distorting 1
2. Has a negative effect on growth if 1213 and inequality is sufficiently low (high ).
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6
Two Dimensional Dynamical System
An Example
= ;
0 0 + 1
+1 = ;
+1 =
0
;
0
It is further assumed that:
+ 1;
+ 1
→ the dynamical system:
+1 = ≡ ( )
≡ ( )
+1 = +
where ( ) ( ) 0 ( ) ( ) 0
The Locus
Let be the locus of all pairs ( ) such that is in a steady-state:
≡ {( ) : +1 = } As follows from the dynamical system there
exists a function
(1−)
̄( ) = ()1(1−)
such that if = ̄( ) then +1 = ( ) = That is, the Locus
consists of all the pairs (̄( ) ).
As follows from the properties of the dynamical system
The Locus
Let be the locus of all pairs ( ) such that is in a steady-state:
≡ {( ) : +1 = } As follows from the dynamical system there
exists a function
(1−−)
̄( ) = ()1(1−−)
such that if = ̄( ) then +1 = ( ) = That is, the Locus
consists of all the pairs (̄( ) ).
As follows from the properties of the dynamical system, the non-trivial
levels of ̄( ) and ̄( ) are unique and globally stable.
+ 1 → (1 − ) 1
→ the locus is increasing and concave with respect to
If + + 1 → (1 − − ) 1
→ the locus is increasing and concave with respect to
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7
Institutions
Endogenous Property Rights
(Based on Mayshar, Moav & Neeman 2013)
The principal-agent problem
• The principal designs the contract to maximize its expected income
• Agents are risk neutral and choose their effort level to maximize their
expected welfare
• The economy exists for two periods
Output (per agent in each period):
½
if = and =
=
otherwise
• ∈ { } - effort (unobserved by the principal)
• ∈ { } - state of nature (observed by the agent before exerting
effort)
• ∈ (0 1) - the probability that =
Information
∈ {̃ ̃} - a public signal about the state of nature
Signal accuracy ≥ 12
= Pr(̃|) = Pr(̃|)
1 − = Pr(̃|) = Pr(̃|)
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- observed after the effort decision
Interpretation of the signal
a. Observation of output in other plots provides information about the
state of nature at a specific plot depending on the correlation across plots.
Interpretation of the signal
b. An observable signal, such as the ‘Nilometer’ that measures the amount
of water in the Nile.
The cost of maintaining the agent
0 if effort is low ( = )
0 if effort is high ( = )
Assumptions:
≥
(low output is larger than the maintenance cost)
−
(effort is efficient)
Agent’s Income and Utility
- agent’s expected income
= − - agent’s periodic utility when exerting effort
- the agent’s discount factor
- the value of the agent’s employment in the next period
- agent’s value of unemployment
Incentive scheme - the carrot:
The principal pays the agent:
a bonus ≥ 0 if output is high ( = )
a basic wage ≥ regardless of output
Incentive scheme - the stick:
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∈ {0 1} - the probability the agent is dismissed after the first period if:
= and = ̃
(otherwise the agent is retained)
- the cost of replacing the agent
→ Two types of contracts are possible in the first period:
= 0 “Pure Carrot”
and
= 1 “Stick and Carrot”
The optimization implies that =
→ An employment contract is fully described by and
(a carrot and a stick)
Second period optimization
The principal’s objective function, 2 :
min +
≥0
subject to the agent’s incentive compatibility constraint, 2 , for =
+− ≥
→ 2 =
The agent’s welfare is independent of the state of nature in the second
period since the is binding
→ The value of employment at the second period
=
The principal’s objective function at the first period, 1 :
min1 + + (1 − )(1 − )
≥0
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subject to the agent’s incentive compatibility constraint, 1 for = :
1 + − +
≥ + (1 − ) + (1 − )
Solution
The IC is binding:
1 = (1 − )
Stick & Carrot: ( = 1)
= (1 − )
Pure Carrot: ( = 0)
=
If:
1 ( = 0) ≤ 1 ( = 1)
↔
↔
≤ + (1 − )(1 − )
≤ ̂ =
(1 − )
+ (1 − )
→ ‘Pure Carrot’
• ̂ 1
• For (1 − ) ̂ 12
→ For some set of parameters:
• ‘pure carrot’ contract is optimal for low
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• ‘stick & carrot’ contract is optimal for high
Intuition: a principal relying on a “stick” to incentivise the agent has to
incur the cost of dismissal with probability:
(1 − )(1 − )
→ The expected cost of using the “stick”:
(1 − )(1 − )
is decreasing with the quality of information
Expected Income in the first period
Pure Carrot
The expected income of the agent
= +
The expected income of the principal
= ( − ) + −
Efficient outcome:
+ = ( − ) +
Stick & Carrot
The expected income of the Agent
= + (1 − )
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is decreasing with
The intuition for the decline of with above ̂ :
Holding constant the bonus, , a higher implies a lower probability of
dismissal, increasing the value of employment. Therefore, as increases
has to decline to hold the incentive constraint binding.
The expected income of the principal
= ( − ) + −
−(1 − )(1 − )
is increasing with
inefficient outcome:
+ = ( − ) + − (1 − )(1 − )
inefficiency declines with within the ‘stick & carrot’
Conclusions:
• Opacity can lead to de facto property rights and increase the welfare
of subjects
• Ownership of land doesn’t increase incentives to exert effort - it is
granted by the elite because at low transparency it is too costly for the
elite to expropriate the subjects
• Transparency can lead to micro management by the elite and reduced
efficiency
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