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Doctoral Program and Advanced Degree in Sustainable Energy Systems
Doctoral Program in Mechanical Engineering
Ecological Economics
Week 6
Tiago Domingos
Assistant Professor
Environment and Energy Section
Department of Mechanical Engineering
Assignments
• Cost minimization
• a)
C1  57  M 1  0.5M 12
C2  93  2 M 2  M 22
CT  C1  C2  0.5M 12  M 1  M 22  2 M 2  150

M 1  1  1  2 M 22  2M 2  150  CT

M 1  M 2  100
• b)
min 0.5M 12  M 1  M 22  2M 2  150
M1 ,M 2
s.t. M 1  M 2  100
M 1 , M 2 ,    0.5M 12  M 1  M 22  2M 2  150   M 1  M 2  100
M 1  66ton
M 2  34ton
Assignments
• Equilibrium
• a)
AVC y   1 ay 2  aby  ab 2  c
3
3
MC  y   AVC y   y  b
2
3

2


a
y

b

c
if
y

b

Supply curve
2
0 otherwise
• b)
n
S  p    Si  p 
p  a y  b   c  y 
2
i 1
pc
b
a
For 2 firms :
p
S1+S2
S1
S2
y
2
 pc
  pc

 pc

Y


Y  
 b   
 b   2  
 b   p  a  b   c
a
a
a
2


 



Assignments
• Equilibrium
• c)

dq q p dq Ay  b 1



dp p q dp
y
A
• d) e e)
• Consumer surplus
q*
 MPB  p  dq
0
• Producer surplus
q*
  p  MPC  dq
0
Assignments
• Pollution economics
• Market equilibrium
MPC  MPB
Q  20
p  20
Consumer surplus  100
Pr oducer surplus  100
External cos t  200
• Social Optimum
MSC  MPB
Q  10
p  25
Consumer surplus  25
Pr oducer surplus  25
External cos t  100
Economic Growth and Dynamic
Optimization
Rui Mota – [email protected] Tel. 21 841 9442. Ext. - 3442
April 2009
Economic growth
Economic growth: Stylized facts (Check with IMF Data Mapper)
• Big differences in output per capita across countries.
• Growth rates vary substantially across countries.
• Convergence: In the long run, all countries will converge to
the same growth rate and level of income per worker.
– Income per worker converges to the same level across
countries conditional on the countries being structurally alike.
• Conditional on structural differences across countries, a
lower level of initial output tends to be associated with a
higher rate of growth.
• Growth rates of GDP per capita have been relatively
constant around 1.5-2 percent in Western Europe and North
America for at least 130 years.
Economic growth: Stylized facts (Check with IMF Data Mapper)
Economic growth: Kaldor (1961) Stylized facts
• Per capita output Y/L grows over time.
• Physical capital per worker K/L grows over time
• Rate of return to capital r nearly constant (profit on capital).
• Ratio of capital to output K/Y nearly constant
• Shares of labor wL/Y and physical capital rK/Y in national
income nearly constant.
Solow Model
• Solow (1956) – Main theoretical tool for economic growth until the
1980’s.
• 1987 Nobel prize in Economics for his contribution to the theory of
economic growth.
•
(1924- )
• In spite of being very limited and largely inappropriate to account for the
growth dynamics of modern economies, in particular the disparities of
economic growth across time and space, the Solow model is the starting
point for almost all analysis of economic growth. – Benchmark Model
Solow Model – Rationale
• Is it possible for an economy to enjoy positive growth rates
forever by simply saving and investing in its capital stock?
• Starting point: Try to relate the growth rate with the
willingness to save and invest.
• Simple model where the only source of growth is
accumulation of physical capital (durable physical inputs –
machinery, buildings, pencils ...).
• General growth model:
– Households own assets and inputs to production, and choose
fractions of their income to consume and save.
– Firms hire inputs (e.g., L, K) and use them with technology to
produce goods that they sell to households or other firms
– Markets exist for goods and inputs in production.
Solow Model – Assumptions
• Closed economy with no government;
• Single composite good is produced and transacted (this means that if we
have more than one good the relative prices are constant);
• the good can be used for consumption and investment;
• the good is produced, using capital and labor;
• investment allows for capital accumulation, therefore physical capital is a
reproducible input;
• population grows at an exogenous rate and all factors of production are
fully employed.
• all markets, i.e., factors, product and financial markets are perfectly
competitive;
Solow Model – Assumptions
• Can capital accumulation explain observed growth?
• How does the capital accumulation behaves along time and what are
the explanatory variables?
• Consumers:
– Receive income Y(t) from labour supply and ownership of firms
S (t )  sY (t ), 0  s  1
– consume a constant proportion of income
C (t )  (1  s )Y (t )
Solow Model – Assumptions
• Labour augmenting production function: Y (t )  F ( K (t ), A(t ) L(t ))
• Constant returns to scale
F ( K ,  AL )   F ( K , AL )  y(t )  f ( k ); x 
X
AL
• Positive and diminishing returns to inputs:
f (0)  0, f '( k )  0, f ''( k )  0
• Inada (1964) conditions:
lim f ' (k )  , lim f ' ( k )  0
k 0
k 
– Ensures the existence of equilibrium.
• Example of a neoclassical production function:
1
– Cobb-Douglas:
F ( K , AL)  K a  AL 
– Intensive form:
f (k )  k a
Solow Model – Dynamics
• Labour and knowledge (exogenous):
L
n
L
• Dynamics of man-made Capital
dK
 K  sY (t )   K (t )
dt
• Dynamics per unit of effective labor
k (t )  sf  k (t )    n  g    k (t )
• sf  k (t )  - actual investment per unit of effective labour
•
 n  g    k (t ) - break-even investment.
A
g
A
Solow Model – Balanced Growth Path
k
 lim k (t )  k *
k0  0 t 
How do the variables of the
model behave in the steady
state?
K*
AL

n

g

K*
AL
k0
k0
Y*
 n g
*
Y
K * L*
Y * L*
g * *
*
*
K L
Y L
t
Solow Model – Dynamics
• On the Balanced Growth Path (BGP)
– Each variable is growing at a constant rate.
– Growth of output per worker is determined
solely by the technological progress
• Stylized facts (Kaldor, 1961):
K*
AL

n

g

K*
AL
– Growth rates of labor, capital and output are
roughly constant;
Y*
 n g
*
Y
– Capital/output ratio roughly constant;
K * L*
Y * L*
g * *
*
*
K L
Y L
– Output per worker and capital per worker are
rising.
Solow Model – Central questions of growth theory
• Only changes in technological progress have growth effects on per capita
variables.
• Convergence occurs because savings allow for net capital accumulation,
but the presence of decreasing marginal returns imply that the this effect
decreases with increases in the level of capital.
• Two possible sources of variation of Y/L:
– Changes in K/L;
– Changes in g.
• Variations in accumulation of capital do not explain a significant part of:
– Worldwide economic growth differences;
– Cross-country income differences.
• Identified source of growth is exogenous (assumed growth).
Growth accounting: Short-run sources of growth
• Breakdown observed growth in GDP, Y Y  gY into components
associated to changes in factors of production.
Y (t )  F  K (t ), A(t ), L(t ) 
• Output growth only happens due to growth in productive inputs,
including technology.
• Tehcnological progress is measured by indirectly, i.e., growth not
attributed to changes in observable inputs.
• Solow refered to the residual as Total Factor Productivity (TFP)
K (t )
L(t )
gY (t )   YK (t )
  YL (t )
 R(t )
K (t )
L(t )
 YX 
F () X
X Y
R(t )   YA
A
A
Growth accounting: Short-run sources of growth
•Solow model explains more than ½ of output growth.
•An inportant part of growth is attributed to exogenous
“inputs”. What is Technological progress? (residual)
–Knowledge, institutions (property rights), education, culture, ...
Ramsey Model: Endogenous saving
• Frank Ramsey (1903-1930) - “A mathematical theory of saving”, 1928, Economic Journal.
How much of its income should a nation save?
• Solow vs Ramsey
– Solow: agents in the economy (or the dictator) follow a simplistic linear rule for
consumption and investment
– Ramsey: agents (or the dictator) choose consumption and investment optimally so as to
maximize their individual utility (or social welfare).
• Establishes the benchmark model for modern dynamic macroeconomics and
optimal intertemporal allocation of resources.
Ramsey Model: Rationale and households
• Does more savings today imply more consumption tomorrow? We must
contrast the cost to postpone our consumption today with the benefit of
enjoying it tomorrow. => Preferences for consumption at different dates
• Milton Friedman’s (1912 - 2006) permanent income and life cycle
hypothesis: Consumption/saving patterns determined not by current real
disposable income but by their longer-term income expectations (i.e.,
individual's real wealth).
• Representative Household wealth:

W (t )   u(c( s ))e
t
   s t 
ds
u ''(c )  0  u '(c )
•Most economic growth models assume infinite planning horizon.
–Isomorphic to a model with finite-lives and random death.
–Intergenerational altruism: individual not only derives utility from his
consumption but also from the bequest he leaves to his offspring. Each
individual internalizes the utility of all future members of the
“dynasty”. => decision makers act as if they have an infinite planning
horizon.
Ramsey Model: Representative Firm
• Representative Firm
– With no externalities and competitive markets, our focus on the aggregate
production possibilities set of the economy or on the representative firm is
without loss of any generality.
• Ouput is produced using capital and labor (Assume a constant
population normalized to 1). Capital does not depreciate. There is
no technological progress.
• Firms’ decisions: how to allocate capital and labour inputs to
production? How to expand activity?
• The output is either consumed or invested, i.e., added to the
capital stock (as in Solow’s model)
f (k )  c  k
• In a closed economy, savings equal investment.
Ramsey Model: Optimal saving
• Optimal growth with:
– Closed economy
– No population or technology growth
– Single composite (investment/consumption) good
– Competitive markets for inputs and outputs.
• The social planner (benevolent dictator) chooses how much the
representative household should consume/invest (add to capital to
provide consumption in future)

max  u(c)e t dt
c
0
s.t.
k  f ( k )  c , k ( 0)  k 0  0
•Discount factor: €1 in T periods from now, is worth exp(−rT ) today.
Same applies to utility.
Ramsey Model: Optimal saving
*
*
• Any solution ( k (t ), c (t ))

0
must obey:
c*
1
u '' (c )
'
*
*

f ( k )   ,  (c )  '
c
*
*
c
 (c )
u (c )


*
k  f ( k )  c , k ( 0)  k 0  0
*
*
lim u ' (c* (t ))k * (t )e  t  0
Instantaneous elasticity of
substitution between
consumption in two dates
t 
• Ramsey-Keynes rule: The higher the marginal product of capital,
relative to rate of time preference, the more it pays to depress the current
level of consumption in order to enjoy higher consumption later.
Interpret  !
• As in Solow model capital and ouput converge to a steady state and
growth will cease in the long run (without technological progress).
Ramsey Model: Phase Portrait
u c 
1
c
c
  c    0
1 
f (k )  k  , 0    1
k
Ramsey Model: Conclusions
• There is no long term growth. Per capita variables converge to a
stationary equilibrium.
• Ramsey models does not add any fundamental explanation of growth
other than the Solow model. The central implications of the Solow model
are not based on the assumption of constant savings.
• Solow + Ramsey: Significant differences in Y/L are explained by
differences in K/L only if differences in K/L and in rates of return to
capital are enormous. Not observed in the data.
• The identified source of growth is exogenous – Technological progress
• What is Technological progress? (residual)
– Knowledge, institutions (property rights), education, culture, ...
• Maybe capital is undervalued (using the private rate of return):
– Human capital, natural capital,
– Positive externalities of capital.
Dynamic Optimization: Infinite Horizon
• Optimal control: Pontryagin’s maximum principle
• Find a control vector u(t ) 
continuous r-vector such as to :
r
for some class of piece-wise

max  f 0 ( x(t ), u(t ), t )dt
u (t )
0
s .t .
x  f ( x(t ), u (t ), t ), x( 0)  x0
• Control variables are instruments whose value can be choosen by the
decision-maker to steer the evolution of the state-variables.
• Most economic growth models consider a problem of the above form.
Pontryagin’s Maximum Principle – Usual Procedure
• Step 1 – Construct Hamiltonian
H ( x, u , p , t )  f 0 ( )   x
• Step 2 – Maximize the Hamiltonian in w.r.t the controls
H
0
u

• Step 3 – Write the Euler equations
• Step 4 – Transversality condition
H
x
lim  (t ) x(t )  0
t 
Pontryagin’s Maximum Principle – With discount

max  f1 ( x(t ), u(t ), t )e   t dt
u (t )
0
s .t .
x  f ( x(t ), u (t ), t ), x( 0)  x0
• Step 1 – Construct the current value Hamiltonian
H c ( x, u , p , t )  f1 ( )e   t   c x
• Step 2 – Maximize the Hamiltonian in w.r.t the controls
H c
0
u
H c ( )
• Step 3 – Write the Euler equations  (t )   (t ) 
x(t )
c
• Step 4 – Transversality condition
c
lim  c (t )e t x(t )  0
t 
Dynamic Optimization: Cake-Eating Economy
• What is the optimal path for an economy “eating” a cake?

max  u(c )e  t dt
C
0
subject to
S (t )  c(t ), S (0)  S0
c1
u c 
  c    0
1 
• Optimal System:
S *  t   c*  t 
c*  t 
c*  t 
• Transversality condition:



lim  c (t )e t S (t )  0
t 
Dynamic Optimization: Cake-Eating Economy
C
S
lim  c (t )e t s(t )  0
t 
Dynamic Optimization: Cake-Eating Economy
• Explicit Solution:

 t
c*

– From the dynamics of consumption *    c* (t )  c0e 
c

– Resource stock constraint:
• The remaining stock of cake is the sum of all future consumption of
cake, i.e.,

 



S (t )   c ( )d   c e
*
*
t
*
0
t
 

d 


*
0
ce
 

t
 *  t
 c0 e

• In the planning horizon, all the cake is to be consumed, i.e,


0
0
S0   c(t )dt   c0*e

 t

c * (t )  S 0 e 

S (t )  S 0 e
*


 t


 t
dt  S0 
 *
c0


The optimal strategy is to consume
a fixed portion of the cake
 *
c (t )  S ( t )

*