004: Macroeconomic Theory

004: Macroeconomic Theory

Lecture 18

Mausumi Das

Lecture Notes, DSE

October 31, 2014

Das (Lecture Notes, DSE) Macro October 31, 2014 1 / 13

R-C-K Model: Centralized Version Revisited

:

In the last lecture we have analysed how to solve a dynamic

programming problem (using the Bellman equation) to arrive at the optimal paths for the control and state variable.

Today we are going to apply this technique to solve the social planner’s problem in a centralized economy.

Recall that the social planner’s problem is given by:

Max.

f^ctg^∞t=0,f^kt+1g^∞t=0

∑

∞ t=0

β^tu(ct) subject to

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

(ii)kt+1 = ^f(kt) + (1 δ)kt ct

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

Herect is the control variable; kt is the state variable, and the corresponding state space is given by <^+.

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

As before, we can use constraint (ii) to eliminate the control variable and write the dynamic programming problem in terms of the state variable alone:

Max.

f^kt+1g^∞t=0

∑

∞ t=0

β^tu(f(k_t) + (1 δ)k_t (1+n)k_t+1) subject to

(i) k_t+1 = (1 δ)k_t

(1+n) ^{for all t}=^{0; k0 given.}

This now looks exactly like the canonical stationary dynamic programming problem that we had seen earlier.

Hence we can write the corresponding Bellman equation relating the two value functions V(k₀)andV(k₁)as:

V(k₀) =Max

f^k1g [u(f^f(k₀) + (1 δ)k₀ (1+n)k₁g) +βV(k₁)].

Das (Lecture Notes, DSE) Macro October 31, 2014 3 / 13

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

More generally, we can write the Bellman equation for any two time periods t andt+1 as:

V(k_t) = Max

f^kt+1g[u(f^f(k_t) + (1 δ)k_t (1+n)k_t+1g) +βV(k_t+1)]. Maximising the RHS above with respect tokt+1, we get the FONC as:

u⁰(f^f(k_t) + (1 δ)k_t (1+n)k_t+1g) [1+n] =βV⁰(k_t+1) (1) Noting thatV(k_t+1)andV(k_t+2) would be related through a similar Bellman equation and applying Envelope Theorem on the latter:

V⁰(kt+1) = u⁰(f^f(kt+1) + (1 δ)kt+1 (1+n)kt+2g) f⁰(kt+1) + (1 δ) . (2) Combining (1) and (2):

u⁰(f^f(k_t) + (1 δ)k_t (1+n)k_t+1g) [1+n]

= βu⁰(f^f(k_t+1) + (1 δ)k_t+1 (1+n)k_t+2g) f⁰(k_t+1) + (1 δ) .

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

The above equation looks more complicated than it actually is! In fact the equation simpli…es the moment we recognise that the terms inside the u⁰(.) functions are nothing butct andct+1 respectively.

Thus bringing back the control variables (using the constraint (ii)), we get the FONC of the social planner’s optimization problem as:

u⁰(ct) [1+n] =βu⁰(ct+1) f⁰(kt+1) + (1 δ) . (3) (Interpretation?)

We also have the constraint function:

k_t+1= ^f(k_t) + (1 δ)k_t c_t

1+n ; k₀ given. (4)

Equations (3) and (4) represents a 2X2 system of di¤erence equations which implicitly de…nes the ‘optimal’trajectories of c_t and k_t.

We still need two boundary conditons to precisely characterise the solution paths for this 2X2 system.

Das (Lecture Notes, DSE) Macro October 31, 2014 5 / 13

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

One such boundary condition is of course the given initial condition:

k₀.

The other boundary condition is provided by the Transversality condition:

tlim!∞β^t∂u(f^f(k_t) + (1 δ)k_t (1+n)k_t+1g)

∂k_t k_t = 0

i.e., lim

t!∞β^t f⁰(kt) + (1 δ) .u⁰(ct)kt = 0 However, the dynamic equations (3) and (4) are very involved and it is not easy to analytically derive the solution paths, unless we ascribe speci…c functional forms to u(c)andf(k) (a route which we shall explore in a moment).

Nonetheless, one can easily charaterise the corresponding (non-trivial) steady state.

R-C-K Model (Centralized Version): Steady State

At steady state:

c_t =c_t+1 =c ; kt =kt+1 =k .

Using this steady state de…nition in equations (3) and (4), the steady state for this system is given by:

k : f⁰(k ) = ¹

β[1+n] (1 δ); (5)

c = f(k ) (n+δ)k (6)

Is this steady state dynamically e¢ cient? For that we have to characterize the corresponding ‘golden rule’.

Recall that ‘golden rule’de…nes a steady state point where the per capita consumption (at steady state) is maximised.

Also recall that in the Solow model the golden rule steady state was de…ned by:

k_g :f⁰(k_g) = (n+δ)

Das (Lecture Notes, DSE) Macro October 31, 2014 7 / 13

R-C-K Model (Centralized Version): Golden Rule &

Dynamic E¢ ciency

Is the above de…nition of ‘golden rule’still valid for this R-C-K model?

The answer is: yes. (Why?)

The next question is: is the above steady state k (de…ned by equation (5)) dynamically e¢ cient?

Once again, the answer is: yes. (Why?)

Thus in the centralized version of the R-C-K model, when the social planner decides on how much to save and how much to leave for households’consumption in order to maximise households’utility over in…nite horizon, it ensures that the corresponding steady state isnot dynamically ine¢ cient (as expected.)

Does this conclusion hold for the decentralized market economy as well? We’ll see.

But before that let us characterise the precise optimal paths of consumption and capital stock for this economy.

R-C-K Model (Centralized Version): Optimal Paths

For analytical convenience, we shall use speci…c functional forms for u(c)and f(k).

Let

u(ct) =logct; f(k_t) = (k_t)^α; 0<α<1.

Also let the rate of depreciation be 100%, i.e., δ=1.

Our dynamic system is then represented by the following two equations:

ct+1

c_t = β α(kt+1)^{α 1}

1+n ; (3⁰)

k_t+1 = (k_t)^α c_t

1+n . (4⁰)

The associated boundary conditions are:

k0 given; lim

t!∞β^tα(kt)^{α 1}.kt

c_t =0

Das (Lecture Notes, DSE) Macro October 31, 2014 9 / 13

Characterization of the Optimal Paths:

We are now all set to characterise the dynamic paths ofc_t and k_t as charted out by the above simpli…ed system.

Before that let us quickly characterize the steady state.

Using the steady state condition that ct =ct+1 =c and

k_t =k_t+1 =k in equations (30^{) and (4}0), the steady state values are given by:

k = ^αβ

1+n

1 1 α

; c = (k )^α (1+n)k .

Now let us chart out the opyimal paths of ct andkt - starting from any given initial value of k₀.(Note that the initial value of c₀ is not given - it is to be optimally determined).

We could have used the phase diagram method to qualitatively characterise the solution paths. But even with the speci…c functional forms the di¤erence equation remains too involved.

Characterization of the Optimal Paths (Contd.):

Instead, we shall use the direct method of ‘guess and verify’.

Let us make a conjecture that the optimal path of per capita stock looks as follows:

kt+1 =M(kt)^α for all t, (C) whereM is a yet unknown constant.

If (C) is indeed the solution path forkt+1 for all t, then (from (3⁰)) the corresponding solution path forc_t would be given by:

ct = (kt)^α (1+n)kt+1 = [1 M(1+n)] (kt)^α. Likewise,

ct+1 = [1 M(1+n)] (kt+1)^α.

Das (Lecture Notes, DSE) Macro October 31, 2014 11 / 13

Characterization of the Optimal Paths (Contd.):

Now from equation (4⁰): c_t+1

ct

= β α(k_t+1)^{α 1} 1+n i.e., (kt+1)^α

(k_t)^α = ^αβ(kt+1)^{α 1} 1+n i.e.,k_t+1 = ^αβ

1+n(k_t)^α ) ^M(k_t)^α = ^αβ

1+n(k_t)^α (given our conjecture).

Thus our conjecture would indeed be true (and satisfy all the relevant equations) i¤

M αβ

1+n.

Hence by the guess and verify method we have indeed identifed the optimal solution paths ofc_t and k_t for this simpli…ed problem.

Characterization of the Optimal Paths (Contd.):

These optimal paths are:

c_t = (1 αβ) (k_t)^α for all t=^0;

kt+1 = ^αβ

1+n(kt)^α for allt =^0.

It is easy to check that starting from anyk₀, thec_t andk_t values monotonically approach their respective steady state valuesc andk . So the strong stability result of Solow is preserved,but the steady state is now necessarily dynamically e¢ cient. (What is the economy’s savings ratio in this example?)

Growth predictions are exactly the same that of the Solow model:

Without technical progress, the per capita income does not grow in the long run;

The aggregate income in the long run grows at a constant exogenous raten.

Das (Lecture Notes, DSE) Macro October 31, 2014 13 / 13