202: Dynamic Macroeconomic Theory
Romer Model: A Simpli…ed Version
Mausumi Das
Lecture Notes, DSE
March 25, 2015
Romer Model: A Simpli…ed Version
Romer (1986) provides an outline of a model which analyses how knowledge accumulation alone can be an engine of growth.
It also highlights the crucial role of the government in in‡uencing the growth path of an economy.
However the original construction of Romer only shows various possibilities - some of which are conjectures and not rigorous enough.
The conclusions of the model can be proved more rigorously if we assume simplify the model and add more structure.
Here we attempt to do precisely that.
Romer Model: A Simpli…ed Version (Contd.)
We simplify the Romer model in the following way:
we assume speci…c function forms - which makes the analysis simpler and tractable;
we replace the ‘knowledge production technology’in Romer by the standard assumption that one unit of the …nal good (corn) can be converted into one unit of knowledge - one to one.
We retain all other assumptions of the original Romer model.
Simpli…ed Romer Model: Economic Structure
Economic structure is identical to the original model.
A single …nal commdity is produced - which can be either consumed or invested in R&D activities to produce ‘knowledge’.
The economy has S identical …rms and H identical households.
Each household consists on a single in…nitely livedmember.
There is no population growth, which implies that the size of labour force in every period:N =H.
Also,S =H=N so that there is no di¤erence between the per capita, per household and per …rm value of a variable and these also coincide with the economy-wide average.
Simpli…ed Romer Model: Production Side Story
As before, each …rm has access to an identical technology represented by the following production function:
Yi =F(ki,K,li); F1,F2,F3 >0, where
ki amount of private knowledge in access of the …rm;
K economy-wide aggregate stock of knowledge;
li labour input employed by the frim.
The …rms takes the aggregate stock of knowledge (K) as given and decides on the optimal quantities of the …rm speci…c inputs: ki andli.
Simpli…ed Romer Model: Production Side Story (Contd.)
We shall assume a speci…c functional form:
F(ki,K,li) = (ki)α(Kli)1 α, 0<α<1.
Notice that in accordance with the original Romer model, the
production functionF is concave and CRS in the …rm speci…c inputs, ki and li and actually exhibits IRS in all its three factors, ki,K andli. But since the …rms treat K as exogenous, they do not internalise the increasing returns; in their perception the production function is CRS.
This allows perfect competetion to prevail in the market economy.
Simpli…ed Romer Model: Production Side Story (Contd.)
Once again we shall assume that the total labour input (constant in supply) is equally distributed over all the …rms so that for each …rm:
li =¯l.
This assumption allows us to focus only on knowledge accumulation and its implication for growth.
Since there is a representive …rm which engages with a representative household and there is a single member in that household,(and noting that ki =kj =k)per capita (as well as per …rm) output in this market economy would be given by:
F(k,K,¯l) (¯l)1 α(k)α(K)1 α =f(k,K)
Simpli…ed Romer Model: Production Side Story (Contd.)
Notice that with identical …rms, aggregate stock of knowledge in the economy is:
K =Ski =Sk.
Thus when each …rm decided to increases its private knowledge input ki by λproportion, K also goes up by the same proportion.
While the private …rms do not recognise this link, the ‘omniscient’
social planner social planner surely does.
Thus the relevant per capita production function for the social planner:
y =f(k,Sk) = (¯l)1 α(k)α(Sk)1 α = (S¯l)1 αk φ(k).
While f(k,K) is still concave ink (keeping K constant), φ(k) is now linear in k :
φ0(k) = (S¯l)1 α.
Technology for Knowledge Production (R&D Technology):
New knowledge (k) is produced by investing the …nal good (corn) in˙ R&D.
Unlike Romer, we assume that one unit of corn invested in knowledge production generates one unit of knowledge:
k˙ =I
whereI investment in research (in terms of …nal good).
As before, we assume that knowledge accumulation is irreversible:
one unit of …nal good once invested in knowledge creationcannot be converted back into corn.
Simpli…ed Romer Model: Household Preferences
Preferences of the single-member in…nitely-lived representative household is denoted by the following life-time utility function:
U0 =
Z∞ t=0
log(ct)exp ρtdt; ρ>0.
It is easy to very that the log speci…cation of the utility function satis…es all the standard properties, namely,
u0(c)>0; u00(c)<0; lim
c!0u0(c) =∞; lim
c!∞u0(c) =0.
In the market economy each household maximises the above utility function subject to its bugdet constraint.
The social planner in benevolent; so he maximises the same utility function, but his budget constarint would be di¤erent than that of the household.
Romer Model: Social Planner’s Problem
The dynamic optimization problem of the social planner is given by:
Z∞ t=0
log(ct)exp ρtdt (I)
subject to
k˙ = (S¯l)1 αkt ct; k(t)=0, k0 given.
Romer Model: Problem of the Market Economy
The corresponding problem for a household operating in the market economy is given by:
Z∞ t=0
log(ct)exp ρtdt (II)
subject to
k˙ = (¯l)1 α(kt)α(Kt)1 α ct; k(t)=0, k0 given.
Since the …rms treat K as exogenous, marketreturn to knowledge accumulation:
∂f(k,K)
∂k = α(¯l)1 α(kt)α 1(Kt)1 α.
Solution to the Social Planner’s Problem: FONCs
The Current-value Hamiltonian:
Hˆt = u(ct) +µt [φ(kt) ct]
= log(ct) +µt h
(S¯l)1 αkt cti Corresponding FONCs:
∂Hˆt
∂c = 0)u0(ct) µt =0 i.e.,µt = 1
ct. (1)
∂Hˆt
∂k = µ˙ +µρ
˙
µ 1
Solution to the Social Planner’s Problem: FONCs (Contd.)
∂Hˆt
∂µ = k˙
i.e.,k˙ = (S¯l)1 αkt ct. (3)
TVC: lim
t!∞µtexp ρtkt =0. (4)
Social Planner’s Problem: Solution
We shall now analyse the dynamics of the social planner’s problem.
Notice that from (1):
˙ µ µ = c˙
c.
Hence from (1), (2) and (3), we get the following two dynamic equations:
˙ c
c = (S¯l)1 α ρ; (5)
k˙
k = (S¯l)1 α ct kt
(6) Equations (5) and (6) along with the TVC now characterise the optimal path for the social planner.
Social Planner’s Problem: Characterization of the Optimal Path
Given the function forms, we can explicitly characterise the optimal paths in this case.
We shall focus on the balanced growth path: the path where all variable in the economy grow at constant rates.
As is clear from (5), consumption in this planned economy is already growing at a constant rate.
From (6), knowledge will also grow at a constant rate if and only if ct
kt
is a constant.
But ct kt
will be a constant if and only if ct and kt grow at the same rate.
Hence for this planned economy, the balanced growth path is characterized by:
˙ c c = k˙
k = (S¯l)1 α ρ.
Social Planner’s Problem: Optimal Path (Contd.)
But, what about the initial consumtion level along this optimal path?
(Recall thatk0 is given, butc0 is not).
Since along the balanced growth path, k˙
k = (S¯l)1 α ρ,plugging this value in the LHS of (6), we get:
(S¯l)1 α ρ = (S¯l)1 α ct kt
i.e., ct = ρkt for all t. (7) Thus given the initialk0,the social planner chooses the initial
consumption such that c0 =ρk0,and thereafter allows both consumption and knowledge stock to grow at a constant rate (S¯l)1 α ρ.
Corresponding Problem for the Competitive Market Economy:
Recall that the only di¤erence between the social planner problem and the household’problem in the market economy is in terms of the per capita production function: The social planner knows that the per capita output is givenφ(k)while the household/…rm reads the per capita output asf(k,K).
Accordingly, one can easily derive the FONCs of the Households problems as follows:
µt = 1 ct
(10)
˙ µ
µ =ρ α(¯l)1 α(kt)α 1(Kt)1 α (20) k˙ = (¯l)1 α(kt)α(Kt)1 α ct (30)
TVC: lim
t!∞µtexp ρtkt =0. (40)
The Competitive Market Economy: Characterization of the Optimal Path
Once again from (10),(20)and (30) one can get the dynamic
equations representing the optimal paths for the competitive market economy as:
˙ c
c =α(¯l)1 α(kt)α 1(Kt)1 α ρ; (8) k˙
k = (¯l)1 α(kt)α 1(Kt)1 α ct kt
(9) Assuming perfect foresight on the part of the households, and thereby substituting Kt =Skt in the equations above:
˙ c
c =α(S¯l)1 α ρ; (10)
k˙ c
Competitive Market Economy: The Optimal Path (Contd.)
Equations (10) and (11) along with the TVC now characterise the optimal path for the competitive market economy (under perfect foresight).
Once again one can …nd out the balanced growth path for this economy.
Arguing as before, it can be shown that along the optimal balanced growth trajectory for this competitive market economy:
˙ c c = k˙
k = α(S¯l)1 α ρ.
Thus clearly the growth rate of per capita consumption/output is lower in the competetive market economy than in the planned economy.
What about the initiallevel of consumption?
Competitive Market Economy: The Optimal Path (Contd.)
Since along the balanced growth path, k˙
k = α(S¯l)1 α ρ,plugging this value in the LHS of (6), we get:
α(S¯l)1 α ρ = (S¯l)1 α ct kt
i.e.,ct = hρ+ (1 α) (S¯l)1 αikt for all t. (12) Hence the initial optimally chosen level of consumption for the market economy is given by c0 =hρ+ (1 α) (S¯l)1 αik0.
Thus given k0,the initial level of consumtion is higher in the competitive market economy than the planned economy.
Finally, what about the welfare of the households?
(Notice that here you can calculate the precise value of the life-time utility along the balanced growth path for both the cases. Derive