004: Macroeconomic Theory

004: Macroeconomic Theory

Lecture 17

Mausumi Das

Lecture Notes, DSE

October 30, 2014

Growth of Per Capita Income without Technical Progress?

In the last class we have argued that we can have long run growth of per capita income in the Solow model even without technological progress only if we are willing to relax some of the Neoclassical properties of the production function.

One way to get this result is to relax one of the Inada conditions, namely that lim

k!∞f⁰(k) =0.

Examples:

Jones-Manuelli Production Function:

Yt =K_t^αN_t^{1 α}+βKt;0<α<1; β>0.

In this case,f(kt) =k_t^α+βkt; and lim

k!∞f⁰(k) =β6=0.

CES Production Function:

Y_t =^hαK_t^ρ+ (1 α)N_t^ρi¹

ρ;0<α<1; ρ>0.

In this case,f(kt) = h

αk_t^ρ+ (1 α) i¹

ρ;and lim

k!∞f⁰(k) =α

1 ρ 6=0.

Growth of Per Capita Income without Technical Progress (Contd.):

While there is no natural justi…cation for the Inada conditions, (and most well-known production functions, except the Cobb-Douglas, typically violate one of these), relaxing the Inada conditions may not still generate a balanced growth path. (Recall our obsession with balanced growth!!)

Another way to generate long run growth of per capita income in the Solow model without technological progress is to relax the assumption of diminishing returns (i.e., f⁰⁰(k) 0).

An example of non-dinimishing returns but CRS production function is the linear technology case:

Y_t =AK_t+BN_t; A,B >0.

A special case of linear technology - AK production function:

Yt =AKt; A>0.

Growth of Per Capita Income without Technical Progress (Contd.):

The AK production technology indeed generates a balanced growth path.

Rate of growth of per capita output (in long as well as short run):

sA (n+δ) 1+n .

Notice that now the government can directly a¤ect the growth rate of the economy by in‡uencing the savings ratio!

There are several interesting economic justi…cations for this kind of AK production technology. (And, no... having and AK production technology does notnecessarily mean that labour does not play any role in the production process!)

We shall examine some of these justi…cations at the very end of this course.

Neoclassical Growth with Optimizing Agents:

Let us now extend the Solow model to allow for optimizing agents.

There are two frameworks which allow for optimizing consumption/savings behaviour by households:

1 The Ramsey-Cass-Koopmans Ini…nite Horizon Framework (henceforth R-C-K);

2 The Samuelson-Diamond Overlapping Generations Framework (henceforth OLG).

The basic di¤erence between the two is that in the R-C-K model agents optimize over in…nite horizon; while in the OLG model, agents optimize over a …nite time horizon (usually 2 periods).

As we shall see later, this apparently innocuous di¤erence in terms of time horizon spells out very di¤erent growth trajectories for the two models.

Neoclassical Growth with Optimizing Agents: The R-C-K Model

We start with the R-C-K model. This model is still Neoclassical - beacuse it retains allthe assumptions of the Neoclassical production function (including the diminishing returns property and the Inada conditions.)

In fact the production side story is exactly identical to Solow.

As before, the economy starts with a given stock of capital (Kt) and a given level of population (L_t) at time t. (We are ignoring

technological progress for now).

Since the production side story is identical to Solow, we know that the …rm-speci…c production functions can be aggregated to generate an aggregate production function:Y_t =F(K_t,N_t).

And at every point of time the market clearing wage rate and the rental rate of capital are given by:

wt =F_N(Kt,Nt); rt =F_K(Kt,Nt).

The R-C-K Model: The Household Side Story

There are H identical households indexed by h.

Each household consists of a singlein…nitely lived member to begin with (at t=0). However population within a household increases over time at a constant raten. (And each newly born member is in…nitely lived too!) This implies that total population also increases at the raten.

At any point of timet,the total capital stock and the total labour force in the economy are equally distributed across all the households, which they o¤er inelastically to the market at the market wage rate w_t and the market rental rater_t.

Thus total earning of a household at time t: w_tN_t^h+r_tK_t^h. Corresponding per member earning:y_t^h =wt+rtk_t^h,

wherek_t^h is the per member capital stockin household h, which is also theper capita capital stock (or the capital-labour ratio, kt) in the economy.

The Household Side Story (Contd.):

In every time period, the instantaneous utility of the household depends on itsper member consumption:

u_t =u c_t^h ; u⁰ >0; u⁰⁰ <0; lim

c^h!⁰

u⁰(c^h) =_∞; lim

c^h!∞u⁰(c^h) =0.

The household at time 0 chooses its entire consumption pro…le c_t^{h ∞}_t=0 so as to maximise the discounted sum of its life-time utility:

U₀^h =

∑

∞ t=0

β^tu c_t^h

subject to its period by period budget constraint.

Notice once again that identical households implied thatper

member consumption (c_t^h) of any household is also equal to the per capita consumption (c_t) in the economy at timet.

The R-C-K Model: Centralized Version (Optimal Growth)

There are two version of the R-C-K model:

A centralized version - which analyses the problem from the perspective of a social planner.

A decentralized version - which analyses the problem from the perspective of a perfectly competitive market economy where

‘atomistic’households and …rms take optimal decisions in their respective individual spheres.

The centralized version was developed by Ramsey (way back in 1928) and is oftem referred to as the ‘optimal growth’problem.

It is assumed that there exists an omniscient,omnipotent, benevolent social planner who wants to maximise citizens’welfare.

Since all households are identical, the objective function of the social planner is identical to that of the households:

Max.U₀=

∑

∞ t=0

β^tu(c_t). (1)

The R-C-K Model: Centralized Version (Contd.)

The social planner maximises (1) subject to its period by period budget constraint.

Notice that in a centrally planned economy there are no markets (hence no market wage rate or market rental rate), and there is no private ownership of assets (capital) and no personalized income.

The social planner employs the existing capital stock in the economy (either collectively owned or owned by the government) and the existing labour force to produce the …nal output -using the aggregate production technology.

After production it distributes a part of the total output among its citizens for consumption puoposes and invests the rest.

Thus the budget constraint faced by the planner in periodt is nothing but the aggregate resource constraint:

Ct+It =Yt =F(Kt,Nt).

The R-C-K Model: Centralized Version (Contd.)

Investment augments next period’s capital stock:

K_t+1 =I_t + (1 δ)K_t.

Thus the budget constraint faced by the planner in periodt can be written as:

C_t +K_t+1 =F(K_t,N_t) + (1 δ)K_t. Writing in per capita terms:

c_t + (1+n)k_t+1 =f(k_t) + (1 δ)k_t.

Thus the dynamic optimization problem of the social planner is:

Max.

f^ctg^∞t=0,f^kt+1g^∞t=0

∑

∞ t=0

β^tu(c_t) subject to

(i)c_t 5 ^f(k_t)for all t =^0;

(ii)kt+1 = ^f(k_t) + (1 δ)k_t c_t

1+n ; kt =^{0 for allt} =^{0; k0 given.}

Dynamic Optimization in Discrete Time: Dynamic Programming

Consider the following canonical discrete time dynamic optimization problem:

Max.

f^xt+1g^∞t=0,f^ytg^∞t=0

∑

∞ t=0

β^tU˜ (t,xt,yt) subject to

(i)y_t 2 ^G˜(t,x_t)for all t =^0;

(ii)x_t+1 = f^˜(t,x_t,y_t); x_t 2^{X for all t} =^{0; x}0 given.

Hereyt is the control variable; xt is the state variable; U˜ represents the instantaneous payo¤ function.

(i) speci…es what values the control variable y_t is allowed to take, given the value ofxt at time t;

(ii) speci…es evolution of the state variable as a function of previous period’s state and control variables (state transition equation).

Dynamic Programming (Contd.)

It is often convenient to use the state transition equation given by (ii) to eliminate the control variable and write the dynamic programming problem in terms of the state variable alone:

Max.

f^xt+1g^∞t=0

∑

∞ t=0

β^tU(t,xt,xt+1) subject to

(i)xt+1 2^G(t,xt) for allt =^{0; x0 given.}

We are going to focus onstationary dynamic programming problems, where time (t) does not appear as an independent argument either in the objective function of in the constraint function:

Max.

f^xt+1g^∞t=0

∑

∞ t=0

β^tU(xt,xt+1) subject to

(i)xt+1 2^G(xt)for all t =^{0; x0 given.}

Stationary Dynamic Programming: Value Function &

Policy Function

Ideally we should be able to solve the above stationary dyanamic programming problem by employing the Lagrange method. Let

x_t+1 ^∞_t=0 denote such a solution.

We can then write the maximised value of the objective function as a function of the parameters alone, in particular as a function ofx0 :

V(x₀) Max.

f^xt+1g^∞t=0

∑

∞ t=0

β^tU(x_t,x_t+1); x_t+1 2^G(x_t) for all t=^0;

=

∑

∞ t=0

β^tU(x_t,x_t+1).

The maximized value of the objective function is called the value function.

The functionV(x0)represents the value function of the dynamic programming problem at time 0.

Value Function & Policy Function (Contd.)

Suppose we were to repeat this exercise again the next period i.,e. at t =1.

Now of course the time period t =1 will be counted as the initial point and the corresponding initial value of the state variable will be x₁.

Let τdenote the new time subscript which counts time fromt =1 to

∞.By construction then, τ t 1.

When we set the new optimization exercise (relevant for

t =1,2....,∞) in terms of τ it looks exactly similar. In particular, the new value function will be given by:

V(x₁) Max.

f^xτ+1g^∞τ=0

∑

∞ τ=0

β^τU(x_τ,x_τ+1); x_τ+1 2^G(x_τ) for all τ=^0;

=

∑

∞ τ=0

β^τU(x_τ,x_τ+1).

Value Function & Policy Function (Contd.)

Noting the relationship between t and τ, we can immediately see that the two value functions are related in the following way:

V(x₀) =

∑

∞ t=0

β^tU(x_t,x_t+1)

= U(x0,x₁) +β

∑

∞ t=1

β^{t 1}U(x_t,x_t+1)

= U(x₀,x₁) +β

∑

∞ τ=0

β^τU(x_τ,x_τ+1)

= U(x₀,x₁) +βV(x₁).

The above relationship is the basic functional equation in dynamic programming which relates two successive value functions recursively.

It is called the Bellman Equation. It breaks down the ini…nite horizon dynamic optimization problem into a two-stage problem:

what is optimal today (x₁);

what is the optimal continuation path (V(x₁)).

Value Function & Policy Function (Contd.)

Since the above functional relationship holds for any two successive values of the state variable,we can write the Bellman Equation more generally as:

V(x) = Max

˜

x2^G(x)[U(x,x˜) +βV(x˜)] for all x 2^{X. (2)} The maximizer of the right hand side of equation (2) is called a policy function:

˜

x =π(x), which solves the Bellman Equation above.

If we knew the value function V(.)and if it was di¤erentiable, we could have easily found the policy function by solving the following FONC (called theEuler Equation):

˜

x : ∂U(x,x˜)

∂x˜ +βV⁰(x˜) =0. (3)

Value Function & Policy Function (Contd.)

Unfortunately, the value function is not known.

In fact we do not even know whether it exists; if yes then whether it is unique, whether it is continuous, whether it is di¤erentiable etc.

In fact a lot of theorems in Dynamic Programming go into

establishing conditions under which a value exists; is unique and has all the nice properties (continuity, di¤erentibility and others).

Without going into the details, we shall simply assume that all these conditions are satis…ed for our problem.

In other words, we shall assume that for our problem the value function exists and is well-behaved (even though we do not know its precise form).

Once the existence of the value function is established, we can then solve the FONC (3) (the Euler Equation)to get the policy function.

But there is still one hurdle: what is the valueV⁰(x˜)? Here the Envelope Theorem comes to our rescue.

Value Function & Policy Function (Contd.)

Recall that V(x˜)is nothing but the value function for the next period wherex˜ is next period’s initial value of the state variable (which is given - from next period’s perspective).

Since the Bellman equation is de…ned for all x 2^X,we therefore get a similar relationship betweenx˜ and its subsequent state value (x):ˆ

V(x˜) = Max

ˆ

x2^G(x˜)[U(x,˜ xˆ) +βV(xˆ)]. Then applying Envelope Theorem:

V⁰(x˜) = ^∂U(x,˜ xˆ)

∂x˜ . (4)

Combining the Euler Equation (3) and the Envelope Condition (4), we get the following equation:

∂U(x,x˜)

∂x˜ +β∂U(x,˜ xˆ)

∂x˜ =0 for allx 2^X.

Value Function & Policy Function (Contd.)

Replacingx,x,˜ xˆ by their suitable time subscripts:

∂U(x_t,x_t+1)

∂xt+1

+β∂U(x_t+1,x_t+2)

∂xt+1

=0; x_t given. (5) Equation (5) is a di¤erence equation which we should be able to solve to derive the time path of the state variablext (and consequently that of the control variabley_t).

Since it is a di¤erence equation of order 2, apart from the initail condition, we need another boundary condition.

Typically such a boundary condition is provided by the following transversality condition:

tlim!∞β^t∂U(x_t,x_t+1)

∂x_t x_t =0.

References:

For Solow Model with Technological Progress & Growth of Per Capita Income without Technical Progress:

1 D. Acemoglu: Introduction to Modern Economic Growth; Pages 55-67 (sections 2.6 & 2.7)

For Dynamic Prgramming Technique:

1 Stokey, Lucas & Prescott: Recursive Methods in Economic Dynamics, Chapter 2.

2 D. Acemoglu: Introduction to Modern Economic Growth; Chapter 6 (sections 6.1, 6.2, 6.6.1).

(Scanned copy of the readings available at C:n^Coursen⁰⁰⁴⁾