marek kapicka lecture 2 basic intertemporal model
DESCRIPTION
Where are we? 1) A Basic Intertemporal Model A) Consumer Optimization B) Market Clearing C) Adding capital stock D) Welfare Theorems E) Infinite horizonTRANSCRIPT
Econ 208
Marek KapickaLecture 2
Basic Intertemporal Model
Where are we? 1) A Basic Intertemporal Model
A) Consumer Optimization B) Market Clearing C) Adding capital stock D) Welfare Theorems E) Infinite horizon
Consumer’s optimization Consumers maximize utility subject to
budget constraints
Lagrangean)1(
. )()(max
122
11121,, 121
rbyc
ybctscUcUbcc
))1((
) (
)()(max),,,,(
2122
1111
21,,21121121
crbybcy
cUcUbccLbcc
Consumer’s optimization First order conditions
Euler Equation)1(
)()(
21
22
11
rcUcU
)()1()( 21 cUrcU
A) Consumer’s optimization Log utility:
Solution:
)1(1
2 rcc
*11
*1
21
*1 1
1
cyb
ryy
c
Where are we? A Basic Intertemporal Model
A) Consumer Optimization B) Market Equilibrium C) Adding capital stock D) Welfare Theorems E) Infinite horizon
B) Market Equilibrium Suppose that there is N identical agents Market clearing condition is
Log utility:
0)( **1 rNb
1
2*
21
1
11
1
yyr
ryy
y
Where are we? A Basic Intertemporal Model
A) Consumer Optimization B) Market Clearing C) Adding capital stock D) Welfare Theorems E) Infinite horizon
C) Adding Capital Stock Shortcomings of the previous
model Production is not determined within
the model Solution: Introduce production
There is a firm producing output using capital stock it owns
Consumers own the firm, get the profits
C) Adding Capital StockFirm’s Problem Production function
Capital changes according to
Initial capital stock K1 given Capital stock K3 can be sold at the end of period 2
)()(
22
11
KFyKFy
23
112
)1()1(KK
IKK
C) Adding Capital StockFirm’s Problem Profits
Maximize the present value of profits
In the optimum:
rI
1
max 21
rKFK )( 2
322
111
KYIY
C) Adding Capital StockConsumer’s problem revisited Budget Constraints:
B1 are savings from period 1 to period 2
r is the interest rate
)1()()(
122
111
rBrCrBC
C) Adding Capital StockMarket Equilibrium Market Clearing
Properties of Equilibrium:322
111
)()(KKFC
KFIC
)(]1)([)()1()(
22
21
cUKFcUrcU
K
Where are we? A Basic Intertemporal Model
A) Consumer Optimization B) Market Clearing C) Adding capital stock D) Welfare Theorems E) Infinite horizon
D) Efficiency of EquilibriumPareto Efficiency Thought experiment: How to choose
consumption and investment if one doesn’t need to obey the markets
The only constraints are the resource constraints
This is the best one can possibly do! Will the solution coincide with the
market solution?
D) Efficiency of EquilibriumPareto Efficiency Pareto Efficient Allocation satisfies
Properties of Pareto Optimum:
)(]1)([)( 221 cUKFcU K
322
11121,,
)(
)( . )()(max121
KKFC
KFICtsCUCUICC
D) Efficiency of EquilibriumWelfare Theorems The allocation is the same as in the
competitive equilibrium The equilibrium allocation is (Pareto)
efficient Practical advantages of this result:
Solving for Pareto Optimum is easier How to figure out what the prices
must be?
Where are we? A Basic Intertemporal Model
A) Consumer Optimization B) Market Clearing C) Adding capital stock D) Welfare Theorems E) Infinite horizon
E) Infinite Horizon Shortcomings of the previous model:
2 periods are arbitrary Solution: Infinite number of periods Solve the Pareto Problem
given )1()(
..
)(max
0
1
0},{ 1
KKKFKC
ts
CU
tttt
tt
t
kc tt
E) Infinite HorizonEuler Equation again and Steady State Consumption satisfies:
Steady State:
)(]1)([)( 11 ttKt cUKFcU
11)(
]1)([1
)(]1)([)(
ssK
ssK
ssssK
ss
KF
KF
CUKFCU