Dynamic Optimization - University of Houston · 2020. 9. 29. · Background Dynamic Optimization in Discrete Time Dynamic Optimization in Continuous Time An EITM Example Dynamic Optimization

BackgroundDynamic Optimization in Discrete Time

Dynamic Optimization in Continuous TimeAn EITM Example

Dynamic OptimizationAn Introduction

M. C. Sunny Wong

University of San Francisco

University of Houston, June 22, 2013

EITM SUMMER INSTITUTE 2013 Dynamic Optimization: An Introduction



Outline1 Background

What is Optimization?EITM: The Importance of Optimization

2 Dynamic Optimization in Discrete TimeA Simple Two-period Consumption ModelThe Bellman EquationCake Eating ProblemProfit Maximization

3 Dynamic Optimization in Continuous TimeThe Method of Hamiltonian MultiplierCake Eating Problem Revisited

4 An EITM ExampleDynamics in a Money-in-the-Utility ModelTM: Theoretical ModelEI: Empirical Implications





Outline1 Background









BackgroundWhat is Optimization?

In general, optimization is a technique which either maximizesor minimizes the value of an objective function bysystematically choosing values of inputs (or choice variables)from a feasible range.Optimization is one of the key ideas in the literature ofEconomics.






In general, optimization is a technique which either maximizesor minimizes the value of an objective function bysystematically choosing values of inputs (or choice variables)from a feasible range.Optimization is one of the key ideas in the literature ofEconomics.






Adam Smith (1776): “It is not from the benevolence of thebutcher, the brewer, or the baker that we expect our dinner,but from their regard to their own self-interest. We addressourselves not to their humanity but to their self-love, andnever talk to them of our own necessities, but of theiradvantages.” (The Wealth of Nations, Book I, Chapter II)

Key assumptions: (1) Rationality, and (2) Efficiency.If rationality is assumed, firms will maximize their profits(supply) and households will maximize their utility (demand).When their results are achieved, the market is in equilibrium

(efficiency).

This idea is first discovered by Wilfredo Pareto (ParetoOptimality).








(efficiency).









(efficiency).









(efficiency).









(efficiency).






BackgroundTypes of Optimization

There are two general methods of optimization:Analytical optimization

Solving the optimal solution(s) mathematically.

Numerical (or computational) optimizationSearching for the optimal solution(s) according to differentalgorithms (using computers).For example, simulations, calibrations, and maximumlikelihood estimations.





































BackgroundAnalytical Optimization

There are three general types of analytical optimization:Optimization without Constraints

First-order conditions (FOCs)

Optimization with ConstraintsThe method of Lagrangian multiplier

Dynamic Optimization (with/without Constraints)Discrete time: The Bellman EquationContinuous time: The method of Hamiltonian multiplier
































Outline1 Background









EITM: The Importance of OptimizationCausality, Assumptions and Models

Social scientists are interested in causal effects:

How do x ’s affect y?

Empirical studies (e.g., regression analysis) can show us, atmost, correlations among variables (not causality!) If thecoefficient on x is significant, it could imply that:

x causes y ; ory causes x ; orthere is another unobservable variable, called z , whichcontributes x and y to move simultaneously.

But, how do we know if x ’s really cause y?
























































But, how do we know if x ’s really cause y?NOBODY TRULY KNOWS!!

We need to use our logical thinking and reasoning to describewhy x 0s can cause y .

But, the world is just too complex!An easy way to do so is to build a theoretical model whichdescribes some aspect of the market (or the society) thatincludes only those features that are needed for the propose athand.It is necessary to impose assumptions to make a model simpler.

But how do we impose “appropriate” assumptions in a model?



























































EITM: The Importance of OptimizationCausality, Assumptions, and Models

A Famous Quote from Robert Solow (1956, page 65):“All theory depends on assumptions which are not quite true.That is what makes it theory. The art of successful theorizingis to make the inevitable simplifying assumptions in such a waythat the final results are not very sensitive.”“A "crucial" assumption is one on which the conclusions dodepend sensitively, and it is important that crucial assumptionsbe reasonably realistic. When the results of a theory seem toflow specifically from a special crucial assumption, then if theassumption is dubious, the results are suspect.”






A Famous Quote from Robert Solow (1956, page 65):“All theory depends on assumptions which are not quite true.That is what makes it theory. The art of successful theorizingis to make the inevitable simplifying assumptions in such a waythat the final results are not very sensitive.”“A "crucial" assumption is one on which the conclusions dodepend sensitively, and it is important that crucial assumptionsbe reasonably realistic. When the results of a theory seem toflow specifically from a special crucial assumption, then if theassumption is dubious, the results are suspect.”






As NOBODY TRULY KNOWS how the world works, thetheoretical model we build could be “wrong”. In other words,the predicted results in the model can be inconsistent withwhat we observed in the real world.If this is the case, probably the assumptions we make are toosensitive (too strong) to the final results.Removing those assumptions / imposing some more realisticassumptions would be necessary.Therefore, both theoretical modeling (TM) and empiricaltesting (EI) enhance our understanding of the relationshipbetween x and y .























EITM: The Importance of OptimizationMicrofoundation of Macroeconomics

In the literature of economics, we assume that people (oreconomic agents) are rational.This assumption helps us formulate human behavior in orderto predict outcomes in aggregate markets. This is called themicrofoundation of macroeconomics.Microfoundations refers to the microeconomic analysis of thebehavior of individual agents such as households or firms thatunderpins a macroeconomic theory. (Barro, 1993)In this lecture, we study how agents face a dynamicoptimization problem where actions taken in one period canaffect the optimization decisions faced in future periods.























This Presentation

In this lecture, we study two methods of dynamic optimization:(1) Discrete-time Optimization - the Bellman equations; and(2) Continuous-time Optimization - the method of Hamiltonian multiplier.

Examples:Discrete-time case:

1 Cake-eating problem2 Profit maximization

Continuous-time case:1 Cake-eating problem2 Ramsey Growth Model (see lecture notes!)

EITM: Dynamics in a Money-in-the-Utility (MIU) Model(Chari, Kehoe, and McGrattan, Econometrica 2000; Christiano,Eichenbaum, and Evans, JPE 2005)





This Presentation










This Presentation










This Presentation










This Presentation









A Simple Two-period Consumption ModelThe Bellman EquationCake Eating ProblemProfit Maximization

Outline1 Background









Two-period Consumption Model

Two periods in the modelPeriod 1: The present; and Period 2: The future

The two-period utility function can be written as:

U = u (c1

)+1

1+ru (c

2

) .

We call 1/(1+r) as the discount factor, where r is called thediscount rate (or the degree of impatience).If an agent is more impatient (r ") 1/(1+r) #), then shewould put less weight on the utility of future consumption.






Two periods in the modelPeriod 1: The present; and Period 2: The future

The two-period utility function can be written as:

U = u (c1

)+1

1+ru (c

2

) .

Assuming that the agent has a first-period budget constraint:

Y1

+(1+ r)A0

= c1

+A1

,

where Yt

= exogenous income at time t, At

= assets / debtsthat the individual accumulates at time t, and r = anexogenous interest rate.






The complete model is:

The two-period utility function:

U = u (c1

)+1

1+ru (c

2

) .

The first-period and second-period budget constraints:

Y1

= c1

+A1

(1st-period BC), andY

2

+(1+ r)A1

= c2

(2nd-period BC).

We assume that the individual does not have any inheritance/debtin period 1 (i.e., A

0

= 0) and does not leave any bequest/debt afterperiod 2 (i.e., A

2

= 0).EITM SUMMER INSTITUTE 2013 Dynamic Optimization: An Introduction





Lagrangian Multiplier

The system:

U = u (c1

)+1

1+ru (c

2

) .

and

Y2

+(1+ r)A1

Y1

= c1

+A1

, and Y2

+(1+ r)A1

= c2

.

To maximize the system of equations, we can apply the method ofLagrangian multiplier to solve the model:

L= u (c1

)+1

1+ru (c

2

)+l1

(Y1

� c1

�A1

)+l2

(Y2

+(1+ r)A1

� c2

) ,

where l1

and l2

are the Lagrangian multipliers.EITM SUMMER INSTITUTE 2013 Dynamic Optimization: An Introduction





We have 5 choice variables: c1

, c2

, A1

, l1

, and l2

. We can solvefor those variables based on the 5 first-order conditions:

∂L∂c

1

= 0 ) u0 (c1

)�l1

= 0 (1)

∂L∂c

2

= 0 ) 11+r

u0 (c2

)�l2

= 0 (2)

∂L∂A

1

= 0 )�l1

+(1+ r)l2

= 0 (3)

∂L∂l

1

= 0 ) Y1

= c1

+A1

(4)

∂L∂l

2

= 0 ) Y2

+(1+ r)A1

= c2

. (5)






From equations (1)–(3), we have:

l1

= u0 (c1

) , (6)

l2

=1

1+ru0 (c

2

) , and (7)

�l1

+(1+ r)l2

= 0. (8)






Now we can plug (6) and (7) into (8), we have the followingequation:

u0 (c1

) =1+ r

1+ru0 (c

2

) . (9)

Equation (9) is called the Euler equation. By combining equations(4) and (5), we have the lifetime budget constraint:

Y1

+Y

2

1+ r= c

1

+c2

1+ r. (10)

Finally, given a certain functional form of u (·), we can useequations (9) and (10) to obtain the optimal levels of c

1

and c2,

(i.e., c⇤1

and c⇤2

).EITM SUMMER INSTITUTE 2013 Dynamic Optimization: An Introduction





What does the Euler equation: u0 (c1

) = 1+r

1+r u0 (c

2

) tell us?

Suppose that an agent gives up $1 consumption today (thepresent), the utility cost to her will be �u0 (c

1

) . In return, shewill get $(1+ r) additional consumption tomorrow (the future)so that her utility gain for the tomorrow will be (1+ r)u0 (c

2

) .

However, since we assume that agents are impatient, thetotally gain from giving up today’s consumption for tomorrowwould be 1

1+r ⇥ (1+ r)u0 (c2

).

In equilibrium, the agent will not give up more or less today’sconsumption for tomorrow at the optimal level only ifu0 (c

1

) = [(1+ r)/(1+r)]u0 (c2

) .







) = 1+r

1+r u0 (c

2

) tell us?


1


2

) .


1+r ⇥ (1+ r)u0 (c2

).


1

) = [(1+ r)/(1+r)]u0 (c2

) .







) = 1+r

1+r u0 (c

2

) tell us?


1


2

) .


1+r ⇥ (1+ r)u0 (c2

).


1

) = [(1+ r)/(1+r)]u0 (c2

) .







) = 1+r

1+r u0 (c

2

) tell us?


1


2

) .


1+r ⇥ (1+ r)u0 (c2

).


1

) = [(1+ r)/(1+r)]u0 (c2

) .





Two-period Consumption Model: An Example

Let the utility function be the power function, u (c) = 1

a ca , where

a 2 (0,1), and a = 1/2, we have:

u0 (c) = c�1/2. (11)

We can plug equation (11) into equation (9), we have:

c�1/21

=1+ r

1+rc�1/22

) c2

=

✓1+ r

1+r

◆2

c1

. (12)





Two-period Consumption ModelAn Example

Plug eq.(12) into the lifetime budget constraint (eq.(10)), we have:

Y1

+Y

2

1+ r= c

1

+1+ r

(1+r)2c1

Y1

+Y

2

1+ r= c

1

1+

1+ r

(1+r)2

!

) c⇤1

=

1+

1+ r

(1+r)2

!�1✓Y

1

+Y

2

1+ r

◆. (13)

Now we plug equation (13) into equation (12), we have:

c⇤2

=

✓1+ r

1+r

◆ 1+

1+ r

(1+r)2

!�1✓Y

1

+Y

2

1+ r

◆. (14)





Two-period Consumption ModelAn Example

The optimized consumption levels in period 1 and period 2 are:

c⇤1

=⇣1+ 1+r

(1+r)2

⌘�1

⇣Y

1

+ Y21+r

⌘

c⇤2

=⇣

1+r

1+r

⌘⇣1+ 1+r

(1+r)2

⌘�1

⇣Y

1

+ Y21+r

⌘





Outline1 Background









The Bellman Equation

In the previous section, we can use the method of Lagrangianmultiplier for solving a simple dynamic optimization problem.However, such the method can be sometime tedious andinefficient.This alternative technique is based on a recursiverepresentation of a maximization problem, which is called theBellman equation.The Bellman equation represents a maximization decisionbased on the forward (or backward) solution procedure withthe property of time consistency.This time consistency property of the optimal solution is alsoknown as Bellman’s optimality principle.























Consumption Dynamics

Let’s consider the following maximization problem:

maxc

t

"U =

•

Ât=1

✓1

1+r

◆t�1

u (ct

)

#, (15)

subject to the following budget constraint:

At

= (1+ r)At�1

+Yt

� ct

. (16)

At

is the state variable in each period t, which represents thetotal amount of resources available to the consumer;ct

is the control variable, where the consumer is choosing tomaximize her utility. Note that c

t

affects the amount ofresources available for the next period, that is, A

t

.







maxc

t

"U =

•

Ât=1

✓1

1+r

◆t�1

u (ct

)

#, (15)


At

= (1+ r)At�1

+Yt

� ct

. (16)

At



t


t

.







maxc

t

"U =

•

Ât=1

✓1

1+r

◆t�1

u (ct

)

#, (15)


At

= (1+ r)At�1

+Yt

� ct

. (16)

At



t


t

.






For this technique of dynamic optimization, the maximumvalue of utility not only depends on the level of consumptionat time t, but also the resource left for future consumption(i.e., A

t

).In other word, given the existing level of asset A

t�1

, the levelof consumption chosen at time t (that is , c

t

) will affect thelevel of assets available at time t+1, (that is, A

t

).






For this technique of dynamic optimization, the maximumvalue of utility not only depends on the level of consumptionat time t, but also the resource left for future consumption(i.e., A

t

).In other word, given the existing level of asset A

t�1

, the levelof consumption chosen at time t (that is , c

t

) will affect thelevel of assets available at time t+1, (that is, A

t

).






Since the consumer would like to maximize her utility from time tonwards, we define the following value function V

1

(A0

) whichrepresents the maximized value of the objective function from timet = 1 to the last period of t = T :

V1

(A0

) = maxc1

T

Ât=1

✓1

1+r

◆t�1

u (ct

) (17)

V1

(A0

) = maxc1

u (c

1

)+

✓1

1+r

◆u (c

2

)+ · · ·+✓

11+r

◆T�1

u (cT

)

!

V1

(A0

) = maxc1

(u (c

1

)+

✓1

1+r

◆"T

Ât=2

✓1

1+r

◆t�2

u (ct

)

#)(18)

subject toAt

= (1+ r)At�1

+Yt

� ct

. (19)






From equation (17), we see that V1

(A0

) is the maximized value ofthe objective function at time t = 1 given an initial stock of assetsA

0

.After the maximization in the first period (t = 1) , the consumerrepeats the same procedure of utility maximization according to theobjective function in period t = 2, given an initial stock of assets inperiod 1:

V2

(A1

) = maxc2

T

Ât=2

✓1

1+r

◆t�2

u (ct

) , (20)

subject to equation (19).By plugging equation (20) into equation (18), we have:

V1

(A0

) = maxc1

u (c

1

)+

✓1

1+r

◆V

2

(A1

)

�.





Consumption DynamicsThe Bellman Equation

Assuming the consumer is maximizing her utility every period, werewrite the maximization problem recursively. Therefore, we canpresent the well-known Bellman equation as follows:

Vt

(At�1

) = maxc

t

u (c

t

)+

✓1

1+r

◆Vt+1

(At

)

�,

where At

= (1+ r)At�1

+Yt

� ct

.






How to solve the Bellman equation in the problem?

Please see the whiteboard!





Outline1 Background









Cake Eating Problem

Question

Suppose the size of a cake at time t is ⇧t

; and the utility functionis presented as u (c

t

) = 2c1/2t

. Given that ⇧0

= 1, ⇧T

= 0, and thediscount rate is r , what is the optimal path of consumption?





Cake Eating Problem

Answer

This optimization problem can be written as:

maxc1,c2,...,c

T

T

Ât=1

✓1

1+r

◆t�1

u (ct

) , (21)

subject to ⇧t

= ⇧t�1

� ct

, for t = 1,2, . . . ,T , and ⇧0

= 1 and⇧T

= 1.In this case, we see that the choice variable is c

t

and the statevariable ⇧

t

.





Cake Eating ProblemThe Bellman Equation

Now we can formulate the Bellman equation:

V (⇧t�1

) = maxc

t

u (c

t

)+1

1+rV (⇧

t

)

�, (22)

subject to⇧t

= ⇧t�1

� ct

. (23)

Since u (ct

) = 2c1/2t

, we can rewrite the Bellman equation as:

V (⇧t�1

) = maxc

t

2c1/2

t

+1

1+rV (⇧

t

)

�. (24)





Cake Eating ProblemThe Bellman Equation







Cake Eating ProblemThe Optimal Path of Cake Consumption





Outline1 Background









Profit Maximization

Assuming that a representative firm maximizes the present value ofall future profits by choosing the levels of investment I

t

and laborLt

for t = 1,2, . . . ,T . Therefore, we have the followingmaximization problem:

maxI1,I2,...I

T

,L1,L2,...LT

ÂT

t=1

⇣1

1+r

⌘t

pt

maxI1,I2,...I

T

,L1,L2,...LT

ÂT

t=1

⇣1

1+r

⌘t

(F (Kt

,Lt

)�wt

Lt

� I ) ,

subject toKt+1

= Kt

�dKt

+ It

for t = 1,2, . . . ,T , and K1

and KT

are given, pt

is the level of profitat time t, K

t

is the stock of capital at time t, F (Kt

,Lt

) is aproduction function, w

t

is the wage rate, r and d is the interestrate and depreciate rate in the market, respectively.





Profit Maximization

In this case,the choice variable is: I

t

and Lt

, andthe state variable is K

t

.

According to the above system, we can formulate the followingBellman equation:

Vt

(Kt

) = maxI

t

,Lt

F (K

t

,Lt

)�wt

Lt

� It

+1

1+ rVt+1

(Kt+1

)

�, (25)

whereKt+1

= (1�d )Kt

+ It

. (26)





Profit Maximization

In this optimization problem, we have two important conditions:1 ∂F (K

t

,Lt

)∂L

t

= wt

.

This result suggests that the optimal amount of labor satisfiesthe condition where marginal product of labor (MPL) equalsthe real wage rate in each period, that is MPL

t

= wt

.

2 ∂F∂K

t

= r +d .This result suggests that the firm must choose a level ofinvestment such that the marginal product of capital (MPK)equals the sum of interest rate and depreciation rate in eachperiod, that is MPK

t

= r +d .




The Method of Hamiltonian MultiplierCake Eating Problem Revisited

Outline1 Background









Dynamic Optimization in Continuous Time

In the previous sections, we study the dynamic optimizationbased the discrete-time dynamic model, where the change intime �t is positive and finite (for example, we assume that�t = 1 for all t � 0, such that t follows the sequence of{0,1,2,3, . . .} .In the section, we consider the method of dynamicoptimization in continuous time, where �t ! 0. Therefore, weassume that agents make optimizing choices at every instantin continuous time.The method is called the technique of Hamiltonianmultiplier.


















A general continuous-time maximization problem can be written asfollows:

maxx

t

ZT

0

e�rt f (xt

,At

)dt, (27)

subject to the constraint:

At

= g (xt

,At

) , (28)

where At

is a time derivative of At

defined as dAt

/dt, and r is thediscount rate in the model, which is equivalent to the r we use in thediscrete time models.

If r = 0, then e�rt = 1. It implies that the agent does notdiscount the activities in the future. On the other hand, if r ",then e�rt # . It implies that the agent becomes more impatientand discounts the value (or utility) of future activities more.





The Method of Hamiltonian Multiplier

We set up the following Hamiltonian function:

Ht

= e�rthf (x

t

,At

)+lt

At

i, (29)

where lt

= µt

e�rt is called the Hamiltonian multiplier.The three conditions for a solution:

1 The FOC with respect to the control variable (xt

):∂H

t

∂xt

= 0; (30)

2 The negative derivative of the Hamiltonian function w.r.t. At

:

�∂Ht

∂At

=d (l

t

e�rt)

dt; (31)

3 The transversality condition:

limt!•

lt

e�rtAt

= 0. (32)







Ht

= e�rthf (x

t

,At

)+lt

At

i, (29)

where lt

= µt



):∂H

t

∂xt

= 0; (30)


:

�∂Ht

∂At

=d (l

t

e�rt)

dt; (31)


limt!•

lt

e�rtAt

= 0. (32)







Ht

= e�rthf (x

t

,At

)+lt

At

i, (29)

where lt

= µt



):∂H

t

∂xt

= 0; (30)


:

�∂Ht

∂At

=d (l

t

e�rt)

dt; (31)


limt!•

lt

e�rtAt

= 0. (32)





Outline1 Background









Cake Eating Problem

Let us consider the same cake eating problem in continuous time:

maxc

t

ZT

0

e�rtu (ct

)dt, (33)

subject to⇧t

=�ct

, (34)

and ⇧0

and ⇧1

are given. Again, the choice variable is ct

, and thestate variable is ⇧

t

.





Cake Eating Problem

We set up the Hamiltonian as follows:

Ht

= e�rt⇣u (c

t

)+lt

⇧t

⌘, (35)

where lt

is the co-state variable. Now, we can plug equation (34)into equation (35), we have the final Hamiltonian equation:

Ht

= e�rt (u (ct

)�lt

ct

) (36)





Cake Eating Problem

We obtain the following conditions:1 The FOC w.r.t. (with respect to) c

t

:

∂Ht

∂ct

= e�rt �u0 (ct

)�lt

�= 0

) u0 (ct

) = lt

; (37)

2 The negative derivative w.r.t. ⇧t

equals the time derivative oflt

e�rt :

�∂Ht

∂⇧t

=d (l

t

e�rt)

dt; (38)

and3 The transversality condition:

limt!•

lt

e�rt⇧t

= 0. (39)





Cake Eating Problem


t

:

∂Ht

∂ct

= e�rt �u0 (ct

)�lt

�= 0

) u0 (ct

) = lt

; (37)



e�rt :

�∂Ht

∂⇧t

=d (l

t

e�rt)

dt; (38)


limt!•

lt

e�rt⇧t

= 0. (39)





Cake Eating Problem


t

:

∂Ht

∂ct

= e�rt �u0 (ct

)�lt

�= 0

) u0 (ct

) = lt

; (37)



e�rt :

�∂Ht

∂⇧t

=d (l

t

e�rt)

dt; (38)


limt!•

lt

e�rt⇧t

= 0. (39)





Cake Eating Problem

How to solve the Hamiltonian equation in the problem?





Dynamics in a Money-in-the-Utility ModelTM: Theoretical ModelEI: Empirical Implications

Outline1 Background









An MIU Model: An Introduction

Why do people want to hold money?Holding cash does not generate any returns!

Two possible arguments:1 People feel good and safe if they hold some cash in their

pocket (Sidrauski, 1967)2 Money can provide transaction services (transaction purpose!)

(Baumol, 1952, Tobin, 1956)

Here we study a basic neoclassical model where agents’ utilitydepends directly on their consumption of goods and theirholdings of money (money demand).

Assumption: Money yields direct utility.

This is called the Money-in-the-Utility model(Chari, Kehoe, and McGrattan, Econometrica 2000; Christiano,Eichenbaum, and Evans, JPE 2005)

























































































Outline1 Background









An MIU Model

Consider the following Money-in-the-Utility Model:

W =•

Ât=0

✓1

1+r

◆t

u (ct

,mt

) ,

where mt

=Mt

/(Pt

Nt

) = real money holding per capita in aneconomy. The budget constraint in the whole economy is:

GDPt

=Consumptiont

+ Investmentt

+GovtSpendingt

+

NewNationalDebtt

+NewMoneyt

.

We can translate as:

Yt

=Ct

+[Kt

� (1�d )Kt�1

]� tt

Nt

+

B

Pt

� (1+ it�1

)Bt�1

Pt

�+

M

t

Pt

�Mt�1

Pt

�.





An MIU Model

Yt

=Ct

+[Kt

� (1�d )Kt�1

]� tt

Nt

+

B

Pt

� (1+ it�1

)Bt�1

Pt

�+

M

t

Pt

�Mt�1

Pt

�.

where Yt

= aggregate output (GDP), tt

Nt

= total lump-sumtransfers (positive) or taxes (negative), i =interest rate, and N

t

=population.





An MIU Model

Finally, the complete system is:

W =•

Ât=0

✓1

1+r

◆t

u (ct

,mt

) ,

and the budget constraint can be presented as follows:

Yt

+tt

Nt

+(1�d )Kt�1

+(1+ it�1

)Bt�1

Pt

+M

t�1

Pt

=Ct

+Kt

+M

t

Pt

+Bt

Pt

.

We also define the production function as Yt

= F (Kt�1

,Nt

).The per capita income is:

yt

=1Nt

F (Kt�1

,Nt

) = F

✓Kt�1

Nt

,Nt

Nt

◆= F

✓Kt�1

(1+n)Nt�1

,1◆

= f

✓kt�1

1+n

◆, where n = population growth rate.





An MIU Model

Finally, the complete system is:

W =•

Ât=0

✓1

1+r

◆t

u (ct

,mt

) ,

and the budget constraint (per capita) is:

f

✓kt�1

1+n

◆+t

t

+1�d1+n

kt�1

+(1+ i

t�1

)bt�1

+mt

(1+pt

)(1+n)= c

t

+kt

+mt

+bt

,

where pt

is the inflation rate, such that, Pt

= (1+pt

)Pt�1

,bt

= Bt

/(Pt

Nt

) , and mt

=Mt

/(Pt

Nt

) .





An MIU Model

We can now formulate the model as the Bellman equation:

V (Wt

) = maxu (c

t

,mt

)+1

1+rV (W

t+1

)

�,

subject to

Wt+1

= f

✓kt

1+n

◆+ t

t+1

+

✓1�d1+n

◆kt

+(1+ i

t

)bt

+mt

(1+pt+1

)(1+n)

= kt+1

+ ct+1

+mt+1

+bt+1

.





An MIU Model







An MIU Model

Finally we have the following equilibrium:um

(ct

,mt

)

uc

(ct

,mt

)=

i

1+ i.

Now assume that the utility function is of the constant elasticity ofsubstitution (CES) form:

u (ct

,mt

) =hac1�b

t

+(1�a)m1�b

t

i1/(1�b)

.

The final solution based on the CES utility function is:

mt

=

✓a

1�a

◆�1/b✓ i

1+ i

◆�1/b

ct

, or

lnmt

=1b

ln✓

1�a

a

◆+ lnc

t

� 1b

lng,

where g = i/(1+ i) = the opportunity cost of holding money.EITM SUMMER INSTITUTE 2013 Dynamic Optimization: An Introduction




Outline1 Background









Empirical Estimation of Money Demand

According to the theoretical solution:

lnmt

=1b

ln✓

1�a

a

◆+ lnc

t

� 1b

lng, (40)

where g = i/(1+ i) which can be called the opportunity cost ofholding money.Therefore, we empirical model can be written as:

lnmt

= a0

+a1

lnct

+a2

lng + et

. (41)

According to equation (41), we expect that he coefficient onlnc

t

is a1

= 1. It implies that consumption (income) elasticityof money demand is equal to 1.The coefficient on ln i

1+i

is a2

=�1/b. For simplicity, we callit as the interest elasticity of money demand.





Empirical Estimation of Money Demand

The empirical result of the money demand function for the UnitedStates based on quarterly data from the period of 1984:1 – 2007:2.

Source: Walsh (2010: 51)

lnm = the log of real money balance (M1); ct

= real peopleconsumption expenditures; and lng = ln i

1+i

= opportunity cost ofholding money = return on M1 minus the 3-month T-Bill rate.





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