Neoclassical Growth Model

Michael Bar�

April 4, 2018

Contents

1 Introduction 2

2 Deterministic NGM, No Government 2

3 Solving the model 43.1 Solving the household�s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Solving the �rm�s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Solving for Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Quantitative Analysis 74.1 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Optimal saving rate at steady state . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Balanced growth path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5 Optimal saving rate on a BGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 Using the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 NGM with Taxes 145.1 Tax Distorted Competitive Equilibrium (TDCE) . . . . . . . . . . . . . . . . . . . 155.2 Solving for TDCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 E¤ects of taxes on equilibrium allocations and prices 176.1 Distorting taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 Temporary and permanent e¤ects of �scal policies . . . . . . . . . . . . . . . . . . 186.3 Quantitative Analysis of Taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.3.1 Example: anticipated once-and-for-all increase in labor tax . . . . . . . . . 21

�San Francisco State University, department of economics.

1

7 Risk Aversion 237.1 Examples of choice under uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.1.1 Demand for insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.1.2 Investment in risky asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.2 Elasticity of substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Final Thoughts 30

9 Appendix: Analytical Solution of Steady State 30

1 Introduction

In these notes we describe one of the workhorses of modern macroeconomics - the NeoclassicalGrowth Model (NGM). This model is sometimes referred to as the Ramsey Model of OptimalGrowth. The Solow model gave us some basic intuition about what factors are important forgrowth, but the Solow model lacks micro-foundations, in that consumers are assumed to use arule of thumb for dividing income into consumption and saving, and everybody works full time.Thus, the Solow model does not have a role for consumers�choices. In particular, we want to seehow consumers respond to changes in economic environment, government policies, etc. We wantto see how consumers change their saving behavior and their labor supply decisions as a responseto changes in economic conditions. The neoclassical growth model will allow us to do exactly that.Modern macroeconomic research on business cycles uses the neoclassical growth model, aug-

mented with stochastic shocks to productivity - Stochastic NGM. In order to study the e¤ectsof monetary policy, macroeconomists introduce money into the NGM, and then the prices of allgoods are nominal - in units of money. In order to allow monetary policy to have an e¤ect on realvariables in the economy, macroeconomists introduce price rigidities or wage rigidities in the model(i.e. relaxing the assumption that prices and wages always adjust immediately to equilibrium).Such models, with nominal rigidities, are called New Keynesian models. All these extensions ofthe NGM studied in these notes, are simulated and are used as a laboratory - a computationalexperiment, which gives quantitative results to well de�ned questions. For example, what causedthe great depression? What would have happened if the Federal Reserve had reacted to stock pricemisalignments prior to crashes and major �nancial crises such as the Great Recession? The ideaof the general methodology is described in Kydland, Finn E., and Edward C. Prescott 1996, "TheComputational Experiment: An Econometric Tool", Journal of Economic Perspectives, 10(1):69-85.

2 Deterministic NGM, No Government

1. Representative Household. There is a single household that lives forever, and enjoysconsumption and leisure. The period utility function is u (ct; lt), and the lifetime utility isU (fct; ltg1t=0) =

P1t=0 �

tu (ct; lt). We assume that u is strictly increasing and strictly concave.

2

The calculus conditions for these assumptions (provided that u is twice-di¤erentiable), are1

(a) : u1 (c; l) > 0, u2 (c; l) > 0

(b) : u11 (c; l) < 0; u11 (c; l)u22 (c; l)� u12 (c; l)2 > 0

Here � 2 (0; 1) is a discount factor, which the household uses to discount future consump-tion2. Households are endowed with one unit of time, which they can allocate between work,ht, and leisure lt. The household owns the capital stock kt and the �rm, to be describedbelow. In this version, the household makes the saving or investment decisions. Let xt be theamount of saving or investment at time t. Thus, the household�s income comes from work,renting the capital stock, and the dividends from the �rm, �t. The household�s problem istherefore:

maxfct;lt;ht;xt;kt+1g1t=0

1Xt=0

�tu (ct; lt) (1)

s:t:

[Time constraint] : ht + lt = 1 8t[Budget constraint] : ct + xt = wtht + rtkt + �t 8t

[Law of motion of capital] : kt+1 = (1� �) kt + xt 8t[Initial capital] : k0 > 0 is given

[Non-negativity] : ct; lt; ht; xt; kt+1 � 0 8t

This is a generic constrained optimization problem. The function to be maximized,P1

t=0 �tu (ct; lt)

is called the objective function, the variables under the word "max" are called the choicevariables, the abbreviation s:t: means subject to or such that, and it is followed by theconstraints. The left hand side of the budget constraint is the spending on consumptionand investment, and the right hand side is the income from labor, capital and dividends.Consumption is the numéraire, and its price is normalized to 1. The wages wt are thereforeexpressed in real terms, i.e. units of consumption per unit of time3, and rt is the rental rateof capital4.

2. Representative Firm. The total output is produced with the neoclassical productionfunction

Yt = F (Kt; Lt)

This means that F satis�es all the properties of the neoclassical production function that

1Condition (b) means that the hessian matix H =

�u11 u12u21 u22

�is negative de�nite. A matrix is negative

de�nite if and only if its leading principle minors alternate signs.2You can think of � = 1= (1 + �), where � is some discount rate, which works like interest rate in calculating

present values. Thus, the household wants to maximize present value of the steram of utilities.3Another option would be to introduce money in this model. Then the price of consumption could be expresses

in terms of money.4If rt = 0:1 for example, then the household is paid 0.1 units of consumption for each unit of capital it rents to

the �rm over a time period.

3

we discussed earlier. The �rm�s problem is, for all t

maxKt;Lt

�t = Yt � rtKt � wtLt (2)

s:t:

Yt = F (Kt; Lt)

The pro�t or the dividend is given to the household, but since the production function isCRS, we must have �t = 0 8t in equilibrium5.

3. Competitive Equilibrium. A competitive equilibrium consists of prices fwt; rtg1t=0 andallocations to the household fct; lt; ht; xt; kt+1g1t=0 and to the �rm fKt; Lt; Ytg1t=0, such that:

(a) Given the prices fwt; rtg1t=0 the allocation to the representative householdfct; lt; ht; xt; kt+1g1t=0 solves the household�s problem (1),

(b) Given the prices fwt; rtg1t=0 the allocation to the representative �rm fKt; Lt; Ytg1t=0solves the �rm�s problem (2),

(c) Markets are cleared

[Goods market] : Yt = ct + xt[Labor market] : ht = Lt[Capital market] : kt = Kt

In other words, supply = demand in all the markets.

3 Solving the model

The main di¢ culty in solving this model is with the household�s problem. Note that solving thisproblem entails �nding in�nitely many unknowns, fct; lt; ht; xt; kt+1g1t=0, that maximize the objec-tive function subject to in�nitely many constraints. The �rm�s problem is a simple static problem.The reason for that is the assumption that the household owns all the factors of production (laborand capital), and makes the investment decisions. It is a nice exercise to describe the environmentand de�ne competitive equilibrium in the case where the �rm owns the capital and makes theinvestment decisions. This alternative setting turns out to be exactly equivalent to the originalone, in the sense that equilibrium prices and equilibrium quantities are exactly the same.

3.1 Solving the household�s problem

We start with the household problem. First, since �t = 0 8t, we can omit this variable fromthe household�s problem. Next, substitute the time constraint into the objective function andsubstitute the law of motion of capital into the budget constraint. The simpli�ed problem becomes

maxfct;ht;kt+1g1t=0

1Xt=0

�tu (ct; 1� ht)

s:t:

ct + kt+1 = wtht + rtkt + (1� �) kt 8t5This result is a special case of Euler�s theorem.

4

For simplicity of notation, we stop writing that k0 > 0 is given and that the consumption, labor,capital must all be non-negative. But we bear in mind that these conditions are needed for solvingthe model numerically. The Lagrange function corresponding to the household�s problem is

L (fct; ht; kt+1; �tg1t=0) =1Xt=0

�tu (ct; 1� ht)�1Xt=0

�t [ct + kt+1 � wtht � rtkt � (1� �) kt]

where f�tg1t=0 is the sequence of Lagrange multipliers. The �rst order conditions with respect tofct; ht; kt+1g1t=0

[ct] : �tu1 (ct; 1� ht)� �t = 0 8t (3)

[ht] : ��tu2 (ct; 1� ht) + �twt = 0 8t (4)

[kt+1] : ��t + �t+1 (rt+1 + 1� �) = 0 8t (5)

Thus, we have in�nitely many �rst order conditions. Dividing (4) by (3) gives

u2 (ct; 1� ht)u1 (ct; 1� ht)

= wt 8t (6)

This condition is the familiar condition for optimal leisure, which says that the marginal rateof substitution between leisure and consumption equals the price ratio. Recall that the price ofconsumption is normalized to 1.Using (3) for period t and t+ 1 and (5) gives

�tu1 (ct; 1� ht)�t+1u1 (ct+1; 1� ht+1)

=�t�t+1

= rt+1 + 1� �

u1 (ct; 1� ht)u1 (ct+1; 1� ht+1)

= � (rt+1 + 1� �) 8t (7)

The last equation is called Euler equation. The Euler equation contains an important intuitionabout the optimal saving (or investment) condition. The Euler equation can be written as

u1 (ct; 1� ht) = �u1 (ct+1; 1� ht+1) (rt+1 + 1� �)

The left hand side is the "pain" from giving up one unit of consumption as the household investsone extra unit in future capital, i.e. increasing kt+1. The right hand side is the "gain" from thatinvestment. In particular, the future return on capital is rt+1 + 1� �, which is the rate of returnrt+1 plus the non-depreciated unit of capital. Multiplying this return by u1 translates the returninto units of utility. Finally, since the return on investment is collected in the next period, theright hand side is multiplied by � - the discount factor.

3.2 Solving the �rm�s problem

The �rm�s problem is fairly simple static problem. Substituting the technology constraint into theobjective, gives the unconstrained problem

maxKt;Lt

�t = F (Kt; Lt)� rtKt � wtLt

5

The �rst order conditions are

F1 (Kt; Lt) = rt

F2 (Kt; Lt) = wt

These are the familiar conditions that state that competitive �rms pay each input its marginalproduct.

3.3 Solving for Equilibrium

We need to combine the market clearing conditions with the above conditions for utility and pro�tmaximization. Thus, the factor prices are

F1 (kt; ht) = rt

F2 (kt; ht) = wt

Next, the equilibrium in the �nal goods market can be written as

ct + xt = F (kt; ht)

ct + kt+1 = F (kt; ht) + (1� �) kt 8t (8)

To summarize, the equilibrium sequences of consumption, labor, and capital, must satisfy 8t =0; 1; 2; :::

u2 (ct; 1� ht)u1 (ct; 1� ht)

= F2 (kt; ht) (9)

u1 (ct; 1� ht)u1 (ct+1; 1� ht+1)

= � [F1 (kt+1; ht+1) + 1� �] (10)

ct + kt+1 = F (kt; ht) + (1� �) kt (11)

In words, these are the condition for optimal time allocation (9), the Euler equation (10) andfeasibility constraint (11). The unknowns are fct; ht; kt+1g1t=0. Observe that saving and investmentin this economy is

st = xt = kt+1 � (1� �) ktand saving and investment rate is

stF (kt; ht)

=xt

F (kt; ht)=kt+1 � (1� �) kt

F (kt; ht)

Notice that unlike the Solow model, the saving (and investment) rate in NGM may change overtime, depending on productivity of the production function and the current level of capital.Since it is not possible to solve for in�nitely many unknowns, for numerical solutions we

approximate the solution by solving a truncated problem for t = 0; 1; :::; T , instead of t = 0; 1; 2; :::In other words, we truncate the time horizon to some large number, say T = 400, instead of in�nite

6

horizon. The above system of �rst order and feasibility constraints (9)-(11) becomes a system of3� (T + 1) equations. Lets now count the unknowns:

c0; c1; :::; cT ; cT+1 (T + 2 unknowns)

h0; h1; :::; hT ; hT+1 (T + 2 unknowns)

k0; k1; :::; kT ; kT+1 (T + 2 unknowns)

This makes the number of unknowns 3 � (T + 2). Does this mean that we are doomed, sincewe have 3 extra unknowns without equations? Well, not exactly. Recall that k0 is given, so thatmakes the number of unknowns to be 3 � (T + 1) + 2. We still need two more equations. If themodel converges to a steady state (css; hss; kss) and if T is big enough, then we can add two moreequations

cT+1 = css

hT+1 = hss

Alternatively, if we did not compute the steady state, the two extra equations can be

cT+1 = cT

hT+1 = hT

It turns out that without technological progress, and under some restrictions on the utility function,the NGM converges to a steady state, starting from any initial capital k0 > 0. This is similar toproposition 1 in the notes on Solow model.

4 Quantitative Analysis

Let us solve the NGM for a special case of Cobb-Douglas utility function and production function,i.e.

u (ct; 1� ht) = � ln (ct) + (1� �) ln (1� ht)F (kt; ht) = Atk

�t h1��t

The �rst order and feasibility conditions become 8t = 0; 1; 2; :::�1� ��

�ct

1� ht= (1� �)Atk�t h��t (12)

ct+1ct

= ���At+1k

��1t+1 h

1��t+1 + 1� �

�(13)

ct + kt+1 = Atk�t h1��t + (1� �) kt (14)

7

4.1 Steady state

If At = A 8t, then the above system converges to a steady state. This statement is not provedhere. Using (ct; ht; kt) = (css; hss; kss) 8t, gives a system of 3 equations with 3 unknowns�

1� ��

�css

1� hss= (1� �)Ak�ssh��ss (15)

1 = ���Ak��1ss h

1��ss + 1� �

�(16)

css = Ak�ssh1��ss � �kss (17)

This system can be solved analytically, but usually a numerical solution is applied. See theappendix for the analytical solution. The steady state output, wages and rate of return to capitalare given by

yss = Ak�ssh1��ss (18)

wss = (1� �)Ak�ssh��ss = (1� �)ysshss

(19)

rss = �Ak��1ss h1��ss = �

ysskss

(20)

Also notice that saving and investment at steady state is

sss = xss = �kss

and saving or investemet rate is

s =sssyss

=xssyss

= �kssyss

4.2 Optimal saving rate at steady state

Recall that in the Solow model the optimal saving rate is the one which maximizes the steady statelevel of consumption per worker, and is called the golden rule saving rate. In the Solow modelwith Cobb-Douglas production function we found that sGR = �. In the Neoclassical GrowthModel the equilibrium saving rate is optimal by construction of the model - the household choosesconsumption optimally in order to maximize the lifetime utility6. Let us compare the steady statesaving rate in the NGM with the golden rule saving rate in the Solow model. The steady statesaving (and investment) rate from the feasibility constraint in equation (17) is

s =�kssyss

) ysskss

=�

s

6Recall that in the Solow model consumers use a rule of thumb when choosing how much to save and how muchto consume.

8

Using this and � = 1= (1 + �) in equation (16) gives

1 = �

��ysskss

+ 1� ��

1 + � = ��

s+ 1� �

�+ � = ��

s

s =

��

�+ �

��

First notice that if � = 0, then the optimal saving rate in the NGM is the same as in theSolow model: s = sGR = �. Recall that � is a discount rate that the household uses to discountfuture utility. Higher values of � mean that the household is more impatient or cares about thefuture less (lives the day). Therefore, the optimal saving (and investment) rate is lower since thehousehold doesn�t want to invest that much in future capital. Consider a case of an economy withcapital share � = 0:35, depreciation rate of � = 0:05 and discount rate of � = 0:04. Based on theSolow growth model, the optimal saving rate in this economy is sGR = 35%. However, based onthe NGM, the optimal saving rate is much lower.

s =

�0:05

0:04 + 0:05

�0:35 = 19:44%

In fact, the maximal possible steady state saving rate in NGM is �. Suppose that in China, peoplecare about the future more than in the U.S., so the Chinese time discount rate is � = 0:02. Thesaving rate in China will be

sChina =

�0:05

0:02 + 0:05

�0:35 = 25%

We can also see that the optimal saving rate is increasing in depreciation rate �. This makesintuitive sense. If depreciation rate is � = 0, then capital never depreciates and we don�t need tosave or invest to maintain the steady state capital. With higher depreciation rate, maintaining thesteady state capital requires higher investment rate. If we change the depreciation rate to � = 0:1in the above example, the steady state saving rate becomes:

s =

�0:1

0:02 + 0:1

�0:35 = 25%

4.3 Balanced growth path

Recall that we proved that in the Solow model, when productivity grows at constant rate, thereexists a unique Balanced Growth Path, such that all endogenous variables grow at constant rate.The BGP is consistent with certain facts about modern growth experience in developed countries.Similarly, in the Neoclassical Growth Model, there exists a unique BGP, when productivity growsat constant rate. The next proposition characterizes the Balanced Growth Path in the NeoclassicalGrowth Model.

9

Proposition 1 Consider the Neoclassical Growth Model with Cobb-Douglass utility and produc-tion. Suppose that TFP grows at constant rate A, i.e. At = A0 (1 + A)

t. Then on a BGP,consumption, output, capital and wage, grow at constant rate , given below:

yt+1yt

=ct+1ct

=kt+1kt

=wt+1wt

= 1 + = (1 + A)1

1��

rt = r, ht = h

Also, the labor and rate of return on capital along the BGP are �xed.

Proof. The proof uses equilibrium equations (12)-(14). Let the BGP growth rates of ct, yt and ktbe c, y and k respectively. Using the Euler equation, (13), we can see that output and capitalgrow at the same rate:

ct+1ct

= ���At+1k

��1t+1 h

1��t+1 + 1� �

�1 + c = �

��yt+1kt+1

+ 1� ��

Since the left hand side is constant, we have that the right hand side must be constant for all t,and yt+1=kt+1 = �yk (constant) implies that y = k = . This also implies that rt = ��yk = r(constant) on a BGP.Using the feasibility constraint, equation (14), and dividing by kt and rearranging, gives

ct + kt+1 = Atk�t h1��t + (1� �) kt

ctkt+kt+1kt

=Atk

�t h1��t

kt+ (1� �)

ctkt+ 1 + =

ytkt+ 1� �

ctkt+ = �yk � �

Thus, ct=kt = �ck (constant) on a BGP, implying that c = y = k = .Dividing equation (12) by ct and multiplying by ht gives:�

1� ��

�ct

1� ht= (1� �)Atk�t h��t�

1� ��

�ht

1� ht= (1� �) Atk

�t h1��t

ct�1� ��

�ht

1� ht= (1� �) yt

ct= (1� �) �yc

where �yc = yt=ct (constant). The last step follows since we already proved that consumption andoutput grow at the same rate, so their ratio must be constant. The last equation implies thatht = h (constant) on a BGP.The wage wt is given by

wt = (1� �)Atk�t h��t =

�1� �h

�Atk

�t h1�� =

�1� �h

�yt

10

The second step uses the proven constancy of ht on a BGP. Thus, wt is proportional to outputand must grow at the same rate as yt.Finally, by constancy of rate of return on capital on BGP, we have:

rt+1rt

=�At+1k

��1t+1 h

1��t+1

�Atk��1t h1��t

=At+1k

��1t+1

Atk��1t

= (1 + A) (1 + )��1 = 1

) 1 + = (1 + A)1

1�� ;

which is the same as in the Solow model.The above proposition allows us to write the BGP equilibrium equations in terms of constant

ratios of variables which grow at the same rate:

BGP equations:�1� ��

�h

1� h = (1� �) �yc (21)

1 + = ����yk + 1� �

�(22)

�ck + = �yk � � (23)

4.4 Calibration

We tipically calibrate the model to match the growth trends of developed economies. Sucheconomies typically exhibit balanced growth, with business cycles. We need to calibrate theparameters �; �; (or A); � and �. The parameter can be calibrated directly from the data,as the average growth rate of real consuption per capita or real GDP per capita. As we discussedbefore, the capital share in the data is approximately 35%, so this pins down � = 0:35. Theremaining 3 parameters can be calibrated from the BGP equations (21)-(22), if indeed the ratiosytkt; ctkt; ytctare more or less constant in the data. Using the data averages for yt

kt; ctktand yt

ct, we obtain

values for �yk; �ck and �yc. Equation (23) pins down � (since we already found ). Alternatively,� can be measured directly, as depreciation rate of �xed assets (physical capital). Equation (22)allows us to calibrate �. Finally, equation (21) can be used to calibrate �. Suppose that thereare 100 hours of discretionary time per week (7 � (24� 8hours of sleep � 2personal care) = 98 � 100). Aworkweek of 40 hours means that h = 0:4. Thus,

h

1� h =0:4

0:6=2

3

Then, using equation (21), gives �1� ��

�2

3= (1� 0:35) �yc

with the only unknown �.

Example 2 From the U.S. data, we get �yk = 0:3, �yc = 1:5, = 3%. This implies that

�ck =ctkt=ctyt

ytkt=

1

1:5� 0:3 = 0:2

11

From (23) we get

�ck + = �yk � �� = �yk � �ck � � = �yk|{z}

0:3

� �ck|{z}0:2

� |{z}0:03

= 0:07

Alternatively, the annual depreciation rate in the data is about � = 6%. This is preferred estimate,since it is coming directly from the data, while the depreciation rate estimated based on (23) isvery sensitive to other estimates (�yk; �ck). From (22) we get

1 + = ����yk + 1� �

�1 + 0:03 = � [0:35 � 0:3 + 1� 0:06]

� = 0:98565

Finally, using �yc = 1:5 and h = 0:4 in equation (21), gives�1� ��

�h

1� h = (1� �) �yc� � 0:4

4.5 Optimal saving rate on a BGP

We would like to solve for the saving rate on the BGP in terms of the model�s parameters. Thelaw of motion of capital is:

kt+1 = (1� �) kt + xtktyt(1 + ) = (1� �) kt

yt+xtyt

s = + �

�yk

where s = xt=yt is the saving and investment rate. Notice that on a BGP, the saving rate isconstant. We want to express the saving rate in terms of the deep parameters of the model.Solving for �yk from (22) gives:

1 + = ����yk + 1� �

�(1 + ) (1 + �) = ��yk + 1� �1 + + �+ � = ��yk + 1� �

�yk =� + + �+ �

�

Plugging into the expression for saving rate, gives:

s = + �

�yk=

+ �

� + + �+ ��

12

First observe that when = 0, we have the steady state saving rate as a special case:

s =0 + �

� + 0 + �+ 0� =

�

� + ��

In general, we can see that the BGP saving rate increases in :

@s

@ =

� + + �+ �� (1 + �) (� + )(� + + �+ �)2

�

=� + + �+ �� � � � �� � �

(� + + �+ �)2�

=� (1� �)

(� + + �+ �)2� > 0

Intuitively, faster productivity growth A, and therefore faster , means that capital becomesincreasingly more productive and the return to investment (=saving) is higher. Therefore, theoptimal investment rate (=saving rate) is higher. This o¤ers another explanation for why Chinahas higher saving and investment rate than the U.S. To summarize, in the Neoclassical GrowthModel, the saving rate is determined endogenously, and depends on preferences and technology,while in the Solow model the saving rate is exogenous.

Example 3 The high Chinese saving rate puzzle. Suppose that U:S: = 0:02 and China =0:1, and for both countries the depreciation rate � = 0:06, � = 0:02, � = 0:35. Then, the savingrate in the two countries is:

sU:S: = + �

� + + �+ �� =

0:02 + 0:06

0:06 + 0:02 + 0:02 + 0:02 � 0:020:35 = 28%

sChina = + �

� + + �+ �� =

0:1 + 0:06

0:06 + 0:1 + 0:02 + 0:1 � 0:020:35 = 30:8%

From the above example we see that even big di¤erences in growth rates, such as 2% v.s. 10%,account for anly small di¤erences in saving or investment rate. In 2015, the saving rates of Chinaand U.S. were: 47.4% and 18.20% respectively. Notice that the NGM is unable to account forsuch high saving rate in China, if we assume that � = 0:35 in both countries. Even if the discountfactor in China is � = 0, i.e. the future is as important as the present, the maximal saving rateon a BGP in this model is � = 35%. Therefore we conclude that either (i) the capital share �in China is much higher, or (ii) China does not experience balanced growth (so our equation forBGP saving rate does not apply to China), or (iii) some componnents which are not included inour model (e.g. taxes) can account for the high saving rate.

4.6 Using the model

This model is not particularly interesting. The only question that we can ask is what is the e¤ect ofTFP and its growth on endogenous variables. The e¤ect of TFP level on steady state is discussedin appendix. In order to analyze the e¤ect of TFP on the time path of endogenous variables, weneed to solve numerically (using Matlab) the system (12)-(14). The next section discusses a modelwith various taxes, and is much more interesting.

13

5 NGM with Taxes

We now introduce a government into the model in order to study the e¤ects of various �scalpolicies on equilibrium outcomes. In this chapter we treat government policies as exogenous, anddo not attempt to model the government�s objective. The government imposes the following taxes:

1. � ct on ct

2. �xt on xt

3. �wt on wtht (i.e. on labor income)

4. � kt on rtkt (i.e. on capital income)

The government uses the taxes to �nance its expenditures gt and lump-sum transfers � t. Thus,in each period, the government�s income is � ctct + �xtxt + �wtwtht + � ktrtkt and its outlays aregt + � t. For simplicity, we will assume that the government always balances its budget:

gt + � t = � ctct + �xtxt + �wtwtht + � ktrtkt 8t

The household�s problem in (1) is now modi�ed to include taxes

maxfct;ht;xt;kt+1g1t=0

1Xt=0

�tu (ct; 1� ht)

s:t:

(1 + � ct) ct + (1 + �xt)xt = (1� �wt)wtht + (1� � kt) rtkt + � tkt+1 = (1� �) kt + xt

Notice that � ct and �xt are sales taxes, so the prices of consumption and investment increase asa result of taxes. The income taxes �wt and � kt on the other hand, reduce the income availablefor spending. Plugging the law of motion of capital into the budget constraint, simpli�es thehousehold�s problem to

maxfct;ht;kt+1g1t=0

1Xt=0

�tu (ct; 1� ht) (24)

s:t:

(1 + � ct) ct + (1 + �xt) kt+1 = (1� �wt)wtht + (1� � kt) rtkt + (1 + �xt) (1� �) kt + � t

Notice that if this household receives an extra unit of capital at time t, his income would increaseby (1� � kt) rt + (1 + �xt) (1� �). This magnitude is called the value of unit of capital, and wewill denote it by pkt:

pkt � (1� � kt) rt + (1 + �xt) (1� �)The �rm�s problem remains unchanged, and is not repeated here.

14

5.1 Tax Distorted Competitive Equilibrium (TDCE)

ATDCE, for given sequence of �scal policies fgt; � t; � ct; �xt; �wt; � ktg1t=0, consists of prices fwt; rtg1t=0

and allocations to the household fct; lt; ht; xt; kt+1g1t=0 and to the �rm fKt; Lt; Ytg1t=0, such that:

1. Given the prices fwt; rtg1t=0 and �scal policies fgt; � t; � ct; �xt; �wt; � ktg1t=0, the allocation to

the representative household fct; lt; ht; xt; kt+1g1t=0 solves the household�s problem (24),

2. Given the prices fwt; rtg1t=0 and �scal policies fgt; � t; � ct; �xt; �wt; � ktg1t=0, the allocation to

the representative �rm fKt; Lt; Ytg1t=0 solves the �rm�s problem (2),

3. Markets are cleared

[Goods market] : Yt = ct + xt + gt[Labor market] : ht = Lt[Capital market] : kt = Kt

4. Government budget is balanced

gt + � t = � ctct + �xtxt + �wtwtht + � ktrtkt

Remark. It is easy to show that household�s budget constraint, together with the feasibilityconstraint, imply that the government budget is balanced. Rearranging the household�s budgetconstraint:

(1 + � ct) ct + (1 + �xt)xt = (1� �wt)wtht + (1� � kt) rtkt + � tct + xt = wtht + rtkt � [� ctct + �xtxt + �wtwtht + � ktrtkt] + � tct + xt = Yt � [� ctct + �xtxt + �wtwtht + � ktrtkt] + � tct + xt = ct + xt + gt � [� ctct + �xtxt + �wtwtht + � ktrtkt] + � tgt + � t = � ctct + �xtxt + �wtwtht + � ktrtkt

Thus, the lump-sum transfer � t is always set in such a way that the government budget is balanced.For example, if gt = 10 and government�s revenue from all the taxes is 7, then � t = �3, i.e. thegovernment must collect extra lump-sum taxes in order to �nance its spending. Intuitively, this isa closed economy, so the government cannot borrow from abroad. We also do not have governmentbonds in this model, so the government cannot borrow from the consumer by selling bonds. Ourmain goal in this chapter is to study the e¤ects of taxes on consumer�s choices, while debt andde�cit issues are outside of our scope.

5.2 Solving for TDCE

As before, we start by deriving the optimality conditions for the household�s problem.

maxfct;ht;kt+1g1t=0

1Xt=0

�tu (ct; 1� ht)

s:t:

(1 + � ct) ct + (1 + �xt) kt+1 = (1� �wt)wtht + (1� � kt) rtkt + (1 + �xt) (1� �) kt + � t

15

The lagrange function is

L =1Xt=0

�tu (ct; 1� ht)�1Xt=0

�t [(1 + � ct) ct + (1 + �xt) kt+1

� (1� �wt)wtht � (1� � kt) rtkt � (1 + �xt) (1� �) kt � � t]

The �rst order necessary conditions with respect to fct; ht; kt+1g1t=0 are

[ct] : �tu1 (ct; 1� ht)� �t (1 + � ct) = 0

[ht] : ��tu2 (ct; 1� ht) + �t(1� �wt)wt = 0[kt+1] : ��t (1 + �xt) + �t+1 [(1� � kt+1) rt+1 + (1 + �xt+1) (1� �)] = 0

Rearranging these conditions yields the intra-temporal and inter-temporal optimality conditions

u2 (ct; 1� ht)u1 (ct; 1� ht)

=(1� �wt)(1 + � ct)

wt (25)

u1 (ct; 1� ht)(1 + �xt)

(1 + � ct)= �

u1 (ct+1; 1� ht+1)(1 + � ct+1)

[(1� � kt+1) rt+1 + (1 + �xt+1) (1� �)] (26)

These conditions are analogous to (6) and (7), but now the prices are distorted with taxes. Equa-tion (25) governs the optimal time allocation. The left hand side is the marginal rate of substitutionbetween leisure and consumption, while the right hand side is the relative price of leisure. Noticethat the price of leisure is the after tax wage (1� �wt)wt and the price of consumption is (1 + � ct).Equation (26) is the familiar Euler equation. The left hand side is the "pain" from increasinginvestment by 1 unit. The household has to pay now (1 + �xt) for one unit of investment and hegives up (1 + �xt) = (1 + � ct) units of consumption as a result. The "pain" is the loss of utilityfrom that extra unit of investment - u1 (ct; 1� ht) (1 + �xt) = (1 + � ct). On the right hand side wehave the future gain from the extra investment. Notice that taxes distort the return on capital.The rate of return is (1� � kt+1) rt+1, but on the bright side, the non-depreciated unit of capital isworth more because the price of capital next period is (1 + �xt+1).It is convenient to write the Euler equation in a more compact way:

u1 (ct; 1� ht) = �u1 (ct+1; 1� ht+1)Rt+1

where Rt+1 =(1 + � ct)

(1 + � ct+1)

��1� � kt+11 + �xt

�rt+1 +

�1 + �xt+11 + �xt

�(1� �)

�(27)

The term Rt+1 is the after-tax gross real interest rate between period t and t + 1. Recall thetwo-period consumption and saving model from undergraduate intermediate micro and macrocourses:

maxc1;c2

u (c1) + �u (c2)

s:t: c1 +c21 + r

= y1 +y21 + r

The optimality condition in this model is

u0 (c1) = �u0 (c2) (1 + r)

16

where r is the real interest rate. The exact same condition holds in our multi-period model, withR replacing 1 + r. Recall that 1 + r is the relative price of current consumption (c1) in therms offuture consumption (c2). Indeed, when we consume one more unit of c1, we give up 1 + r units ofc2 (i.e., the opportunity cost of 1 unit of current consumption is 1+r units of future consumption).

Exercise 1 Show that Rt+1 from (27) is the relative price ct in terms of ct+1. Hint: increase ctby 1 unit, and show that the household must give up Rt+1 units of ct+1.

Solution 1 Omitted.

We will investigate the e¤ects of di¤erent tax policies on the equilibrium, using the sameapproach we used in Solow and NGM models to investigate exogenous changes in parameters. Asbefore, if taxes are �xed and there is no technological progress, the model converges to a steadystate. We can solve the model for t = 0; 1; :::; T , where T is some large number of periods thatare enough for the model to converge close enough to the steady state. After combining withthe �rm�s optimality rt = F1 (k t; ht), wt = F2 (kt; ht), the equilibrium path fct; ht; kt+1g1t=0 has tosatisfy the following conditions 8t:

[Optimal labor] :u2 (ct; 1� ht)u1 (ct; 1� ht)

=(1� �wt)(1 + � ct)

F2 (kt; ht) (28)

[EE] : u1 (ct; 1� ht) = �u1 (ct+1; 1� ht+1)Rt+1 (29)

where Rt+1 =(1 + � ct)

(1 + � ct+1)

��1� � kt+11 + �xt

�rt+1 +

�1 + �xt+11 + �xt

�(1� �)

�[Feas] : ct + kt+1 + gt = F (kt; ht) + (1� �) kt (30)

Suppose that �scal policy is �xed over time: fgt; � t; � ct; �xt; �wt; � ktg1t=0 = (g; � ; � c; �x; �w; � k) 8tand suppose that all variables are at their steady state. Then the above system becomes

[Optimal labor] :u2 (c; 1� h)u1 (c; 1� h)

=(1� �w)(1 + � c)

F2 (k; h)

[EE] : 1 = ���1� � k1 + �x

�F1 (k; h) + 1� �

�[Feas] : c+ �k + g = F (k; h)

6 E¤ects of taxes on equilibrium allocations and prices

6.1 Distorting taxes

De�nition 4 Distorting taxes prevent the competitive equilibrium allocation from solving thesocial planner�s problem

maxfct;ht;kt+1g1t=0

1Xt=0

�tu (ct; 1� ht)

s:t:

ct + kt+1 + gt = F (kt; ht) + (1� �) kt

17

The solution to the social planner�s problem is a sequence of endogenous variables fct; ht; ktg1t=0,which satis�es the following conditions:

u2 (ct; 1� ht)u1 (ct; 1� ht)

= F2 (kt; ht) (31)

u1 (ct; 1� ht)u1 (ct+1; 1� ht+1)

= � [F1 (kt+1; ht+1) + 1� �] (32)

ct + kt+1 + gt = F (kt; ht) + (1� �) kt (33)

These conditions are exactly the same as equilibrium conditions in a model without taxes, (9)-(11). Thus, the competitive equilibrium in the model without government and without taxes isPareto-Optimal - coincides with solution to social planner�s problem. Also notice that equilibriumwith government and only lump-sum taxes is also Pareto-Optimal; set all the taxes to zero inTDCE conditions (28)-(30), and compare these conditions with the social planner�s (9)-(11).Thelump-sum transfer � t does not appear in the �rst order conditions, so if it was the only tax, theTDCE conditions would coincide with the �rst order conditions for the above social planner�sproblem, and in this case, TDCE is Pareto E¢ cient. This is one of the major results in public�nance - the most e¢ cient way to collect given revenues for the government is through lump-sumtaxes. The problem however is that from practical point of view, lump-sum taxes are di¢ cult toimplement.

6.2 Temporary and permanent e¤ects of �scal policies

By temporary e¤ects we mean e¤ects on the time path of equilibrium variables without changingthe steady state, while permanent e¤ect is such that the steady state (limiting behavior) of themodel changes. Some tax policies will have only temporary e¤ect, while others will alter the steadystate. We can only perform limited analysis with pencil and paper, but fortunately our Matlabcodes enable us to investigate the e¤ects of a host of �scal policies. Before we resort to Matlabhowever, lets try to develop some intuition about the possible e¤ects of �scal policies.Once again the su¢ cient conditions for equilibrium path fct; ht; ktg1t=0, are:

[Optimal labor] :u2 (ct; 1� ht)u1 (ct; 1� ht)

=(1� �wt)(1 + � ct)

F2 (kt; ht)

[EE] : u1 (ct; 1� ht) = �u1 (ct+1; 1� ht+1)Rt+1[Feas] : ct + kt+1 + gt = F (kt; ht) + (1� �) kt

where Rt+1 =(1 + � ct)

(1 + � ct+1)

��1� � kt+11 + �xt

�F1 (kt+1; ht+1) +

�1 + �xt+11 + �xt

�(1� �)

�The willingness of households to substitute between consumption and leisure is a¤ected by �w, � cand wages. Higher taxes on labor (�w ") reduce the after tax wage of the household, so the leisurebecomes cheaper (substitution e¤ect). This e¤ect alone would make the household work less. Thisis not the only way the taxes on labor income a¤ect the hours worked. Recall the lump-sumtransfers have to adjust to balance the government budget. Since we did not say anything aboutchanges in government spending, the entire increase in tax revenues will be given back to thehousehold in the form of lump-sum transfers. This means a positive income e¤ect, which on its

18

own leads to an increase in leisure (a normal good). On top of that, there is a general equilibriume¤ect since the equilibrium wage (marginal product of labor) will change. Hence without solvinga parametrized version of the model numerically, we can never be sure of the e¤ect of labor taxeson equilibrium labor ht.Observe that higher consumption tax, (� c ") has similar e¤ect on the intra-temporal substi-

tution. Higher tax on consumption means that with the same wage, the household can purchaseless consumption. Leisure then becomes relatively cheaper than consumption, and the same sub-stitution e¤ect as described above occurs. Notice however that taxes on consumption also a¤ectRt+1, which is the inter-temporal rate of substitution. In particular, observe that if � ct+1 > � ct,then Rt+1 #. Lower real after-tax gross interest rate means that current consumption becomescheaper relative to future consumption7. Thus, lower (1 + � ct) = (1 + � ct+1) makes the householdwant to consume more and save less. On top of that, as always, we have the general equilibriume¤ect which works through changes in equilibrium prices rt. Again, without solving numericallya parametrized version of the model, we cannot be sure about the e¤ects of � c on any of theequilibrium variables.The same story is with � k and �x. Higher taxes on capital income, reduce the real interest rate

Rt+1 and thus make the current consumption cheaper. This force will, ceteris paribus, tend toincrease consumption and reduce investment. But in general equilibrium, prices will be a¤ected aswell, so again we cannot predict exactly what would happen in this model economy. The importantlesson here is that it is not easy to analyze the e¤ects of �scal policies even in a simple model, letalone in the real world. If someone tells you that they are con�dent about a certain tax reform,you should be very skeptical about it. In this model, we can subject the arti�cial economy toparticular �scal policy, and attempt to analyze the numerical results. We will be able to concludewhich of the many e¤ects dominated, and how the parameters of the model a¤ect its response tothe policies.We can say a bit more about the permanent e¤ects of �scal policies, i.e., the e¤ects of policies

on the steady state. At the steady state, productivity and �scal policy must be �xed over time:fgt; � t; � ct; �xt; �wt; � ktg1t=0 = (g; � ; � c; �x; �w; � k). The steady state conditions are rewritten againfor convenience.

[Optimal labor] :u2 (c; 1� h)u1 (c; 1� h)

=(1� �w)(1 + � c)

F2 (k; h)

[EE] : 1 = ���1� � k1 + �x

�F1 (k; h) + 1� �

�[Feas] : c+ �k + g = F (k; h)

Notice that a �xed consumption tax � c does not a¤ect the Euler equation. It does have e¤ecton the itra-temporal substitution in the �rst optimality equation. Consumption becomes moreexpensive relative to leisure, so the substitution e¤ect is to reduce consumption and to increaseleisure (work less). Again, there is also income and general equilibrium e¤ect. Higher consumptiontax is accompanied by higher lump-sum transfers, which is a positive income e¤ect. If householdswork less, then this is yet another general equilibrium e¤ect through wages.

7Reacall that real interest rate represents the price of current consumption in terms of future consumption. Ifthe real interest rate is 5%, then consuming extra unit today, instead of saving it, means that you give up 1.05units of future consumption.

19

A bit more can be said about the e¤ects of �scal policies in a simpli�ed version of this modelwith inelastic labor supply. In this version of the model, the equilibrium is characterized by onlytwo equations

[EE] : u0 (ct) = �u0 (ct+1)Rt+1

where Rt+1 =(1 + � ct)

(1 + � ct+1)

��1� � kt+11 + �xt

�f 0(kt+1) +

�1 + �xt+11 + �xt

�(1� �)

�[Feas] : ct + kt+1 + gt = f (kt) + (1� �) kt

where f (kt) � F (kt; 1). So in the absence of the intra-temporal condition, taxes on labor incomeare not distorting and work like lump-sum taxes. This is intuitive since the labor supply is always1, so labor income is an exogenous constant from the household�s point of view. Consumptiontax, when �xed, is not distorting as well. Notice that if � ct = � ct+1 8t, then regardless of the levelof this tax, the steady state equations are una¤ected by it.

[EE] : 1 = ���1� � k1 + �x

�f 0 (k) + 1� �

�[Feas] : c+ �k + g = f (k)

So the e¤ect of changes in consumption tax is temporary, and constant consumption tax has noe¤ect on the steady state.

6.3 Quantitative Analysis of Taxes

Let us solve the NGMwith taxes for a special case of Cobb-Douglas utility function and productionfunction, i.e.

u (ct; 1� ht) = � ln (ct) + (1� �) ln (1� ht)F (kt; ht) = Atk

�t h1��t

In this case, the �rst order and feasibility equations become:

[Optimal labor] :�1� ��

�ct

1� ht=(1� �wt)(1 + � ct)

(1� �)Atk�t h��t

[EE] :ct+1ct

(1 + �xt)

(1 + � ct)= �

�(1� � kt+1) �Atk��1t+1 h

1��t+1 + (1 + �xt+1) (1� �)

�(1 + � ct+1)

[Feas] : ct + kt+1 + gt = Atk�t h1��t + (1� �) kt

And the steady state equations become:

[Optimal labor] :�1� ��

�c

1� h =(1� �w)(1 + � c)

(1� �)Ak�h��

[EE] : (1 + �x) = ��(1� � k) �Ak��1h1�� + (1 + �x) (1� �)

�[Feas] : c+ �k + g = Ak�h1��

20

The Matlab codes that solve this model are: (1) TDCEsseq.m, (2) TDCEeq.m, and (3) TDCEsim-ulations.m. The �rst function contains the steady state equations, i.e., the above 3 equations withthe unknowns (c; h; k). The second function contains the equations for equilibrium time paths offct; ht; ktgTt=0. Remember that we approximate 1 with large enough time horizon T , such thatthe model converges close enough to the steady state. The main script that the user needs tomodify for performing di¤erent experiments is TDCEsimulations.m. This program solves for thesteady state �rst, in order to determine the �nal point to which the model converges. Then theuser de�nes which experiments she wants to perform, and runs the code. The program solves themodel for the user provided �scal policies.Fiscal policy instruments are sequences of the form fgt; � ct; �xt; �wt; � ktg1t=0, where the lump-

sum transfer is not one of them because we proved above that without borrowing or lending inthis model, the government must balance its budget, so � t adjusts accordingly. There are in�nitelymany policies that we can asses, but economists are usually interested in conducting particulartypes of experiments:

1. Anticipated once-and-for-all change, when some policy instrument, say consumptiontax, changes once-and-for-all at a given date. For example, a sequence � ct = 0 8t = 0; 1; :::; 8and � ct = 20% 8t = 9; 10; ::: is such experiment. Households are assumed to know thisentire time path at period 0. Usually, we analyze the e¤ect of a speci�c change in one policyinstrument, so we can isolate the impact of particular �scal policy.

2. Anticipated one-time change, when the policy instrument deviates from its steady statevalue only during one particular period, and the households know about this change inadvance. For example, � ct = 0 for all t 6= 9, and � c9 = 20%.

3. Unanticipated once-and-for-all change, when the policy instrument changes at period 0from its steady state level, and stays there for ever. To conduct this experiment, we need tosolve for steady state for some original level of �scal policy instruments, and then make thedesired change at period 0. For example, �nd the steady state for fgt; � ct; �xt; �wt; � ktg1t=0 =(0; 0; 0; 0; 0) and then solve for the time path where � ct = 20% 8t = 0; 1; 2:::

4. Unanticipated one-time change, when the policy instrument changes from its originallevel at time 0 only, and goes back to its original level after that. For example, start theeconomy at steady state that corresponds to fgt; � ct; �xt; �wt; � ktg1t=0 = (0; 0; 0; 0; 0), and thensolve for the time path where � c0 = 20%, and � ct = 0 8t = 1; 2; ::: The resulting time pathsare called impulse response functions.

6.3.1 Example: anticipated once-and-for-all increase in labor tax

Suppose that the model parameters are � = 0:45, � = 0:97, � = 0:05, � = 0:35, A = 1. The�scal policy on the original steady state is given by g = 0:2 and � c = �x = �w = � k = 0. Theexperiment that we want to perform is a once-and-for-all increase in the tax on labor from the10th period and on, i.e. �wt = 0 8t = 0; 1; 2; :::; 9 and �wt = 20% 8t = 10; 11; ::: Using the Matlabprogram TDCEsimulations.m, we generate the next �gure:

21

Time0 20 40

ct

0.5

0.6

0.7Time path of c t

Time0 20 40

ht

0.4

0.45

0.5Time path of h t

Time0 20 40

kt

4

4.5

5Time path of k t

Time0 20 40

Rt

1.02

1.03

1.04Time path of R t

Time0 20 40

wt

1.4

1.5

1.6Time path of w t

Time0 20 40

pkt

1.02

1.03

1.04Time path of p kt

Time0 20 40

yt

0.9

1

1.1Time path of y t

Time0 20 40

It

0.1

0.2

0.3Time path of I t

Time0 20 40

g t

­2

0

2Time path of g t

Time0 20 40

rt

0.07

0.08

0.09Time path of r t

Time0 20 40

it

0.1

0.2

0.3Time path of i t

Time0 20 40

t

­0.2

­0.1

0Time path of t

Observe that since the change in policy is anticipated, the household changes its behavioreven prior to the actual change in policy. The horizontal lines correspond to the original steadystate with zero taxes. Notice that in anticipation of the labor tax increase the household reducesconsumption in the very beginning, and accumulates capital. Also notice that the household worksmore than the steady state, in anticipation of the coming increase in labor taxes. The reason forthis behavior is an attempt to smooth consumption. People with concave utility are risk averseand do not like abrupt changes from one period to the next. The way this household can smoothconsumption is by investing more and accumulating capital prior to the date when he is hit by thelabor taxes. Notice that after the taxes were imposed, the household gradually reduces his stockof capital. This smoothing of consumption is observed in the real world when students take loans,when workers save for retirement, and when people buy insurance. Risk aversion is the key factorthat determines how much households will respond to di¤erent shocks. Degree of risk aversion isparticularly important when we study people�s behavior during business cycles.

Exercise 2 Notice that following the once-and-for-all increase in �w, the prices rt and wt returnto their original steady state in the long run. Prove that with CRS production function, once-and-for-all changes in �w or � c, do not have a permanent e¤ect on rt and wt. That is, rt and wtconverge to their original steady state.

22

Solution 2 Based on Euler�s Theorem of homogeneous functions, if the production function ish.d.1. (i.e. CRS), the partial derivatives (marginal products) are h.d.0. and depend only on theratio of inputs. In particular,

r = F1 (k; h) = F1

�k

h; 1

�w = F2 (k; h) = F2

�k

h; 1

�The Euler Equation determines the steady state F1 (k; h),

[EE] : 1 = ���1� � k1 + �x

�F1 (k; h) + 1� �

�The steady state F1 (k; h) = F1

�kh; 1�determines the ratio k=h, which in turn determines r and

w. Note that this ratio at steady state is a¤ected only by � k and �x. Thus, �w or � c have no e¤ecton steady state k=h, r and w.

Exercise 3 Notice that following the once-and-for-all increase in �w, the level of capital stockpermanently declines, and converges to a lower steady state level. Prove that with CRS productionfunction, if higher �w increases steady state leisure, then the steady state capital must decrease.

Solution 3 We proved in the previous exercise that an increase in �w does not change the k=h.Therefore, if leisure goes up in steady state, it means that h decreases, and therefore k mustdecrease by the same factor as h, to keep the ratio k=h unchanged.

7 Risk Aversion

Most of the important decisions we face involve making choices under uncertainty. Decisions inthe presence of uncertainty is a well developed �eld of economics, from which �nance evolved. Ican�t do justice to this �eld by writing a few pages in these notes, so I encourage people to learnmore by reading a chapter about uncertainty in any graduate-level micro textbook. I will writehere the very minimum that we need for this course.

De�nition 5 A lottery is a probability distribution over some prizes.

For example

L =

�$1000$500

w.p. 0:5w.p. 0:5

is a lottery that pays $1000 or $500 with equal probabilities. More general example would beL = (�;x) such that � = �1; �2; :::; �N is a vector of probabilities with �i > 0 and

PNi=1 �i = 1,

and x = x1; x2; :::; xN are the corresponding prizes so that you get a prize xi with probability �i.Whenever an individual needs to make a choice between uncertain alternatives, she chooses

between di¤erent lotteries. Buying a car, pursuing an academic degree, adopting a pet or gettingmarried, are all lotteries. Therefore, we need to de�ne preferences over lotteries. Under someassumptions, consumer�s preferences over lotteries can be represented with expected utility.

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De�nition 6 Expected utility from a lottery with random prizes x is E [u (x)], where E is themathematical expectation of u (x).

For example, suppose that utility of certain outcomes is u (xi) and outcome i occurs withprobability �i. Then the expected utility from the lottery is

E [u (x)] =NXi=1

�iu (xi)

As another example, consider the lottery that pays $1000 or $500 with equal probabilities. Theexpected utility from that lottery is

E [u (x)] = 0:5 � u (1000) + 0:5 � u (500)

De�nition 7 An individual is risk averse if he prefers the expected value of any lottery tothe lottery itself. If preferences are represented with expected utility, then risk aversion meansu [E (x)] > E [u (x)], utility from expected value of the lottery is greater than the expected utilityfrom the lottery.

For example, consider the lottery that pays $1000 or $500 with equal probabilities. A riskaverse person prefers the expected value of this lottery, $750 with certainty, to participating inthe lottery, i.e.

u (750) > 0:5u (1000) + 0:5u (500)

Mathematically, the de�nition of risk aversion is equivalent to the function u being strictly concave.Intuitively, a function is concave if a cord connecting two points on the graph of the function liesbelow the graph of the function. The right hand side of the last inequality is a point on a cordconnecting two points on the graph of u, while the left hand side is a point on the graph of u.Intuitively, the more concave the utility function is, the more risk averse the individual is, so weexpect that any measure of risk aversion should contain the term u00 (x).

De�nition 8 Arrow-Pratt measure of relative risk aversion:

RRA = �u00 (x)

u0 (x)x

This function measures the extent to which the individual is willing to risk a certain fraction ofhis wealth, i.e. relative risk. For example, suppose that an investor is o¤ered to invest a fractionhis wealth in a an asset with risky return. It turns out that if the investor has higher RRA, hewill invest a smaller fraction of wealth in any risky asset.Since the utility is concave, u00 will always be negative, so the minus in front of the formula

converts the measure to positive numbers. Thus, bigger RRA means greater risk aversion. Thedivision by u0 (x) makes the measure invariant to multiplication of u by constant. For example,suppose one consumer has utility u and another has v = 5u. These consumers are equally riskaverse and will make the same choices between lotteries, but v00 will be 5 times greater than u00. Sothe division by the �rst derivative of u makes the measure invariant with respect to multiplicationby constant. The multiplication by x delivers a measure of relative risk aversion.

24

Example: Calculate the A-P measure of relative risk aversion for

u (x) =

�x1� �11� > 0, 6= 1ln (x) = 1

Solution:

u0 (x) = x� , u00 (x) = � x� �1

RRA = �u00 (x)

u0 (x)x = �� x

� �1

x� x =

Notice, that this utility function exhibits Constant Relative Risk Aversion (CRRA). This meansthat regardless of how big the wealth is, for given the individual has the same tendency torisk given fractions of his wealth. So for example, if Kim has = 0:5 and Ambrose has = 1,then Ambrose has higher relative risk aversion. In other words, he is less willing to risk a higherpercentage of his wealth than Kim. The next �gure shows the graphs of CRRA utility functionswith di¤erent .

Notice, higher makes the graph more curved ("more concave").

7.1 Examples of choice under uncertainty

In this section we present several examples in which agents need to make choices while facinguncertainty.

7.1.1 Demand for insurance

Suppose that Arnold has wealth w and faces a risk that with probability � his wealth will sustaindamage d 2 [0; w], and with probability 1�� his wealth remains intact. Arnold can buy insurancewith premium p per unit of wealth insured, and he needs to choose the amount of coverage q topurchase, where 0 � q � d. Assuming that the utility over certain outcomes is represented byu (�), which is strictly increasing and strictly concave, Arnold�s expected utility is

E [u] = �u (w � d� pq + q) + (1� �)u (w � pq)

25

Notice that in the case of damage occurring, he still needs to pay the premium, but he receives acompensation at the amount of his coverage q. His problem is therefore

max0�q�d

�u (w � d� pq + q) + (1� �)u (w � pq)

The �rst order condition for interior solution is:

�u0 (w � d+ q� (1� p)) (1� p)� (1� �)u0 (w � pq�) p = 0

The second order condition is

�u00 (w � d+ q� (1� p)) (1� p)2 + (1� �)u00 (w � pq�) p2 < 0

Thus, since Arnold is risk averse (utility is strictly concave i.e. u00 < 0), the second order conditionis satis�ed. For interior solution, we must have

�u0 (w � d+ q� (1� p)) (1� p) = (1� �)u0 (w � pq�) pu0 (w � d+ q� (1� p))

u0 (w � pq�) =(1� �)�

p

(1� p)

An important general result is that if insurance is actuarially fair, then a risk averse person willbuy full coverage, i.e. q� = d. An actuarially fair insurance requires that p = �, in which case theinsurance company makes zero expected pro�t: pq � �q = 0. Thus

u0 (w � d+ q� (1� p))u0 (w � pq�) = 1

w � d+ q� (1� p) = w � pq�

q� = d

If p > �, the insurance company makes positive pro�t and risk averse individuals will no buyfull coverage, and possible will not buy any insurance at all if the premium is too high8. In orderto �nd exactly how much insurance is purchased, we need to know the function u (�).

7.1.2 Investment in risky asset

Suppose that Barney has a wealth w and can invest an amount x 2 [0; w] in a risky asset. Withprobability � the net return on the asset is +r and with probability 1 � � the net return is �r,where r > 0. Barney�s problem is therefore

max0�x�w

�u (w + rx) + (1� �)u (w � rx)

The �rst order condition for interior optimum is

�u0 (w + rx�) r � (1� �)u0 (w � rx�) r = 08This might explain why many Americans do not purchase health insurance.

26

The second order condition for interior optimum:

�u00 (w + rx�) r2 + (1� �)u00 (w � rx�) r2 < 0which always holds for a risk averse person.Will Barney invest any amount in the risky asset? Mathematically, having x� > 0 is equivalent

to having positive slope of expected utility at x = 0, as shown in the next �gure.

x* x

E[u]

Thus, the condition for x� > 0 is

�u0 (w + r0) r � (1� �)u0 (w � r0) r > 0

�u0 (w) r � (1� �)u0 (w) r > 0

�r � (1� �) r > 0

� >1

2

This is another important and general result that any risk averse person will invest a positiveamount in a risky asset as long as the expected return on that asset is positive (expected returnis �r � (1� �) r > 0).Suppose that Barney�s utility from certain outcomes is of the CRRA form. We will show that

the higher is his risk aversion, the less he will invest in the risky asset. At the interior optimumwe have

� (w + rx�)� = (1� �) (w � rx�)�

��1= (w + rx�) = (1� �)�1= (w � rx�)w + rx� = a (w � rx�)

where a =

��

1� �

�1= Solving for x�

w + rx� = aw � rax�

rax� + rx� = aw � wx� =

aw � wr (a+ 1)

x�

w=

�a� 1a+ 1

�1

r(34)

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The left hand side is the fraction of wealth invested in the risky asset. We can show that thisfraction is decreasing in the Arrow-Pratt coe¢ cient of relative risk aversion, :

@ (x�=w)

@ =

@ (x�=w)

@a

@a

@ =

�(a+ 1)� (a� 1)

(a+ 1)2

�log

��

1� �

���

1� �

�1= �� �2

�=

�2

(1 + a)2

�log

��

1� �

�| {z }

>0

��

1� �

�1= �� �2

�< 0

Notice that since a risk averse person will invest in the rsiky asset only if � > 1 � �, we havelog�

�1���> 0.

Notice also that the fraction invested in the risky asset is decreasing in r, according to (34).The reason is that higher r means higher return in the good case, but also lower return in the badcase. Risk averse investor has concave utility, which is impacted more by losses than by gains ofthe same magnitude.

7.2 Elasticity of substitution

Risk aversion means that people prefer average over extremes, or in other words would like tosmooth consumption. Our intuition tells us that in a dynamic model such as the NGM, risk aversehouseholds would prefer more-or-less equal consumption over time, as opposed to low consumptionin one period and high consumption in the next period. Thus, intuitively, households willingnessto substitute consumption in period t and t0 should be related to their risk aversion. Formally,the willingness of households to substitute one good for another is measured by the elasticity ofsubstitution between those two goods.

De�nition 9 Let the utility function be u (x; y). The elasticity of substitution (in utility)between goods x and y is:

ESx;y =%�(y=x)

%�MRSx;y=

d (y=x)

dMRSx;y� MRSx;y

y=x

In words, the elasticity of substitution measures how much the ratio of the two goods changes asthe slope of the indi¤erence curves changes. In the case of production function, the elasticity ofsubstitution between two inputs measures how the ratio of the inputs changes as the slope of theisoquants changes. In that case, we replace MRS with TRS.

Example: suppose that the household derives utility from consumption in two periods ac-cording to

u (c1; c2) =c1� 1

1� + �c1� 2

1� with > 0, where it is understood that = 1 is the case of log utility. Find the elasticity of(inter-temporal) substitution between c1 and c2. First, �nd the marginal rate of (inter-temporal)substitution

u1 (c1; c2) = c� 1 , u2 (c1; c2) = �c� 2

) MRSc1;c2 =c� 1�c� 2

=1

�

�c2c1

� 28

Next, it is convenient to �nd 1=ESc1;c2:

1

ESc1;c2=dMRSc1;c2d (c2=c1)

� c2=c1MRSc1;c2

=1

�

�c2c1

� �1� c2=c11�

�c2c1

� = The elasticity of substitution is therefore

ESc1;c2 =1

This example illustrates that when the period utility is of the constant relative risk aversion(CRRA) form, the inter-temporal elasticity of substitution is also constant (CES), and is theinverse of the A-P measure of relative risk aversion. The next �gure illustrates indi¤erence curveswith di¤erent parameter .

Notice that when is large (ES is low), the household is more risk averse and at the sametime, the household is less willing to substitute inter-temporally. In the extreme when !1, the indi¤erence curves approach the perfect complements right angle curves. In that case,the household wants to preserve a constant ratio of c1=c2 and not willing to substitute inter-temporally (ES ! 0). In homework exercises you will experiment with di¤erent CES parametersto see how the household changes its response to �scal policies, and to business cycles. The CESparameter is also the key to �nding optimal policies for combating global warming. Any policyto reduce emissions will reduce current consumption but also may prevent a sharp drop in futureconsumption due to climatic disasters. Thus, if our is high, the people are more risk averse andwant to smooth consumption more. In that case, a policy that would prevent disastrous drops

29

in future consumption would be optimal. If on the other hand is low, people are more toleranttowards risk, and don�t care about smoothing consumption. In that case, we don�t have to doanything about global warming, and wait for the disaster to hit. Our former faculty, YanchunZhang, is currently working on this topic.

8 Final Thoughts

Analyzing the e¤ects of �scal policies on the economy is not an easy task. We have made a stepin the right direction by constructing a laboratory that helps us conduct controlled experimentswith �scal policies - a luxury that we don�t have with actual economies.

9 Appendix: Analytical Solution of Steady State

In this appendix, we solve the system of equations (15)-(17). Substituting (17) into (15) gives.�1� ��

�Ak�ssh

1��ss � �kss1� hss

= (1� �)Ak�ssh��ss (35)

Rearranging equation (16) gives the relationship between hss and kss

1 = ���Ak��1ss h

1��ss + 1� �

�1

�� 1 + � = �Ak��1ss h

1��ss

1�� 1 + ��A

!k1��ss = h1��ss

hss = �kss

where � =

1�� 1 + ��A

! 11��

Using hss = �kss in equation (35) gives�1� ��

�Ak�ss (�kss)

1�� � �kss1� �kss

= (1� �)Ak�ss (�kss)���

1� ��

�A�1��kss � �kss

1� �kss= (1� �)A���

kss

�1� ��

��A�1�� � �

�= (1� �)A��� (1� �kss)

kss

�1� ��

��A�1�� � �

�= (1� �)A��� � (1� �)A�1��kss

kss

�1� ��

��A�1�� � �

�+ (1� �)A�1��kss = (1� �)A���

kss

��1� ��

��A�1�� � �

�+ (1� �)A�1��

�= (1� �)A���

30

Thus, the steady state equations are:

kss =(1� �)A�����

1���

�(A�1�� � �) + (1� �)A�1��

� (36)

hss = �kss =(1� �)A�1����

1���

�(A�1�� � �) + (1� �)A�1��

� (37)

css = Ak�ssh1��ss � �kss = A�1��kss � �kss =

�A�1�� � �

�kss (38)

where � =

1�� 1 + ��A

! 11��

The following proposition is left as an exercise.

Proposition 10 The steady state labor supply and saving rate (or investment rate) are indepen-dent of the TFP.

Proof. I just provide a guideline for the proof. You need to work out the details in the homework.First, to show that hss is independent of A, notice that A always appears multiplied by �1��. Usingthe de�nition of �, you need to show that A cancels out in the product A�1��. The Steady statesaving (or investment) is sss = �kss, from equation (17). The saving rate is therefore

s =�kss

Ak�ssh1��ss

Using hss = �kss gives

s =�

A�1��

Which does not depend on A.The intuition is as follows. First, note that the labor supply depends on the demand for

leisure. When A ", the steady state wage wss ". This has two e¤ects on leisure - income e¤ect andsubstitution e¤ect. With higher income, consumers want to increase the consumption of normalgoods, including leisure. But on the other hand, the price of leisure goes up when the wage ishigher. It turns out that with Cobb-Douglas preferences, income and substitution e¤ect canceleach other. Second, note that higher income makes consumers want to consume more now, aswell as in the future. So it is not necessary that consumers will change the fraction of incomeconsumed and saved. It turns out that with Cobb-Douglas preferences, consumers will not changetheir steady state saving rate.

31