202: dynamic macroeconomic...

202: Dynamic Macroeconomic TheoryRomer Model: A Simplified Version

Mausumi Das

Lecture Notes, DSE

March 25, 2015

Das (Lecture Notes, DSE) Dynamic Macro March 25, 2015 1 / 21

Romer Model: A Simplified Version

Romer (1986) provides an outline of a model which analyses howknowledge accumulation alone can be an engine of growth.

It also highlights the crucial role of the government in influencing thegrowth path of an economy.

However the original construction of Romer only shows variouspossibilities - some of which are conjectures and not rigorous enough.

The conclusions of the model can be proved more rigorously if weassume simplify the model and add more structure.

Here we attempt to do precisely that.


Romer Model: A Simplified Version (Contd.)

We simplify the Romer model in the following way:

we assume specific function forms - which makes the analysis simplerand tractable;we replace the ‘knowledge production technology’in Romer by thestandard assumption that one unit of the final good (corn) can beconverted into one unit of knowledge - one to one.

We retain all other assumptions of the original Romer model.


Simplified Romer Model: Economic Structure

Economic structure is identical to the original model.

A single final commdity is produced - which can be either consumedor invested in R&D activities to produce ‘knowledge’.

The economy has S identical firms and H identical households.

Each household consists on a single infinitely lived member.There is no population growth, which implies that the size of labourforce in every period:N = H.

Also, S = H = N so that there is no difference between the percapita, per household and per firm value of a variable and these alsocoincide with the economy-wide average.


Simplified Romer Model: Production Side Story

As before, each firm has access to an identical technology representedby the following production function:

Yi = F (ki ,K , li ); F1,F2,F3 > 0,

where

ki ≡ amount of private knowledge in access of the firm;

K ≡ economy-wide aggregate stock of knowledge;

li ≡ labour input employed by the frim.

The firms takes the aggregate stock of knowledge (K ) as given anddecides on the optimal quantities of the firm specific inputs: ki and li .


Simplified Romer Model: Production Side Story (Contd.)

We shall assume a specific functional form:

F (ki ,K , li ) = (ki )α (Kli )

1−α , 0 < α < 1.

Notice that in accordance with the original Romer model, theproduction function F is concave and CRS in the firm specific inputs,ki and li and actually exhibits IRS in all its three factors, ki , K and li .

But since the firms treat K as exogenous, they do not internalise theincreasing returns; in their perception the production function is CRS.

This allows perfect competetion to prevail in the market economy.



Once again we shall assume that the total labour input (constant insupply) is equally distributed over all the firms so that for each firm:

li = l .

This assumption allows us to focus only on knowledge accumulationand its implication for growth.

Since there is a representive firm which engages with a representativehousehold and there is a single member in that household,(and notingthat ki = kj = k) per capita (as well as per firm) output in thismarket economy would be given by:

F (k,K , l) ≡ (l)1−α(k)α (K )1−α = f (k,K )



Notice that with identical firms, aggregate stock of knowledge in theeconomy is:

K = Ski = Sk .

Thus when each firm decided to increases its private knowledge inputki by λ proportion, K also goes up by the same proportion.

While the private firms do not recognise this link, the ‘omniscient’social planner social planner surely does.

Thus the relevant per capita production function for the socialplanner:

y = f (k, Sk) = (l)1−α(k)α (Sk)1−α = (Sl)1−α k ≡ φ(k).

While f (k,K ) is still concave in k (keeping K constant), φ(k) is nowlinear in k :

φ′(k) = (Sl)1−α.


Technology for Knowledge Production (R&D Technology):

New knowledge (k) is produced by investing the final good (corn) inR&D.

Unlike Romer, we assume that one unit of corn invested in knowledgeproduction generates one unit of knowledge:

k = I

where I ≡investment in research (in terms of final good).As before, we assume that knowledge accumulation is irreversible:one unit of final good once invested in knowledge creation cannot beconverted back into corn.


Simplified Romer Model: Household Preferences

Preferences of the single-member infinitely-lived representativehousehold is denoted by the following life-time utility function:

U0 =

∞∫t=0

log (ct ) exp−ρt dt; ρ > 0.

It is easy to very that the log specification of the utility functionsatisfies all the standard properties, namely,

u′(c) > 0; u′′(c) < 0; limc→0

u′(c) = ∞; limc→∞

u′(c) = 0.

In the market economy each household maximises the above utilityfunction subject to its bugdet constraint.

The social planner in benevolent; so he maximises the same utilityfunction, but his budget constarint would be different than that of thehousehold.


Romer Model: Social Planner’s Problem

The dynamic optimization problem of the social planner is given by:

∞∫t=0

log (ct ) exp−ρt dt (I)

subject to

k = (Sl)1−α kt − ct ; k(t) = 0, k0 given.


Romer Model: Problem of the Market Economy

The corresponding problem for a household operating in the marketeconomy is given by:

∞∫t=0

log (ct ) exp−ρt dt (II)

subject to

k = (l)1−α(kt )

α (Kt )1−α − ct ; k(t) = 0, k0 given.

Since the firms treat K as exogenous, market return to knowledgeaccumulation:

∂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:

Ht = u (ct ) + µt [φ(kt )− ct ]= log (ct ) + µt

[(Sl)1−α kt − ct

]Corresponding FONCs:

∂Ht∂c

= 0⇒ u′ (ct )− µt = 0

i.e., µt =1ct. (1)

∂Ht∂k

= −µ+ µρ

i.e.,µ

µ= ρ− φ′(kt ) = ρ− (Sl)1−α

. (2)


Solution to the Social Planner’s Problem: FONCs (Contd.)

∂Ht∂µ

= k

i.e.,k = (Sl)1−α kt − ct . (3)

TVC: limt→∞

µt exp−ρt kt = 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 dynamicequations:

cc= (Sl)1−α − ρ; (5)

kk= (Sl)1−α − ct

kt(6)

Equations (5) and (6) along with the TVC now characterise theoptimal path for the social planner.


Social Planner’s Problem: Characterization of the OptimalPath

Given the function forms, we can explicitly characterise the optimalpaths in this case.We shall focus on the balanced growth path: the path where allvariable in the economy grow at constant rates.As is clear from (5), consumption in this planned economy is alreadygrowing at a constant rate.From (6), knowledge will also grow at a constant rate if and only ifctktis a constant.

Butctktwill be a constant if and only if ct and kt grow at the same

rate.Hence for this planned economy, the balanced growth path ischaracterized by:

cc=kk= (Sl)1−α − ρ.


Social Planner’s Problem: Optimal Path (Contd.)

But, what about the initial consumtion level along this optimal path?(Recall that k0 is given, but c0 is not).

Since along the balanced growth path,kk= (Sl)1−α − ρ, plugging

this value in the LHS of (6), we get:

(Sl)1−α − ρ = (Sl)1−α − ctkt

i.e., ct = ρkt for all t. (7)

Thus given the initial k0, the social planner chooses the initialconsumption such that c0 = ρk0, and thereafter allows bothconsumption and knowledge stock to grow at a constant rate(Sl)1−α − ρ.

Fianlly, Is TVC satisfied along this balanced growth path? (Verify)


Corresponding Problem for the Competitive MarketEconomy:

Recall that the only difference between the social planner problemand the household’problem in the market economy is in terms of theper capita production function: The social planner knows that the percapita output is given φ(k) while the household/firm reads the percapita output as f (k,K ).

Accordingly, one can easily derive the FONCs of the Householdsproblems as follows:

µt =1ct

(1′)

µ

µ= ρ− α (l)1−α

(kt )α−1 (Kt )

1−α (2′)

k = (l)1−α(kt )

α (Kt )1−α − ct (3′)

TVC: limt→∞

µt exp−ρt kt = 0. (4′)


The Competitive Market Economy: Characterization ofthe Optimal Path

Once again from (1′), (2′) and (3′) one can get the dynamicequations representing the optimal paths for the competitive marketeconomy as:

cc= α (l)1−α

(kt )α−1 (Kt )

1−α − ρ; (8)

kk= (l)1−α

(kt )α−1 (Kt )

1−α − ctkt

(9)

Assuming perfect foresight on the part of the households, and therebysubstituting Kt = Skt in the equations above:

cc= α (Sl)1−α − ρ; (10)

kk= (Sl)1−α − ct

kt(11)


Competitive Market Economy: The Optimal Path(Contd.)

Equations (10) and (11) along with the TVC now characterise theoptimal path for the competitive market economy (under perfectforesight).

Once again one can find out the balanced growth path for thiseconomy.

Arguing as before, it can be shown that along the optimal balancedgrowth trajectory for this competitive market economy:

cc=kk= α (Sl)1−α − ρ.

Thus clearly the growth rate of per capita consumption/output islower in the competetive market economy than in the plannedeconomy.

What about the initial level of consumption?


Competitive Market Economy: The Optimal Path (Contd.)

Since along the balanced growth path,kk= α (Sl)1−α − ρ, plugging

this value in the LHS of (6), we get:

α (Sl)1−α − ρ = (Sl)1−α − ctkt

i.e., ct =[ρ+ (1− α) (Sl)1−α

]kt for all t. (12)

Hence the initial optimally chosen level of consumption for the marketeconomy is given by c0 =

[ρ+ (1− α) (Sl)1−α

]k0.

Thus given k0, the initial level of consumtion is higher in thecompetitive 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-timeutility along the balanced growth path for both the cases. Derivethese utility values and then compare them.)


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