The Romer Model

David N. DeJongUniversity of Pittsburgh

Econ. 1540, Spring 2010

DND () Romer Model Econ. 1540, Spring 2010 1 / 19

Overview of the Model

The model extends the Solow model by endogenizing thecharacterization of technological progress.

In Romer�s model, technological progress is represented as the generationof ideas, or blueprints, useful for producing new intermediate goods(goods used to produce �nal goods). The Solow model assumed a singleintermediate good; the Romer model features At intermediate goods,which evolves over time.

In both the Solow and Romer models, we assume a single �nal good (thus,we abstract from the innovation of consumption goods, and focusexclusively on the innovation of new intermediate goods).

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Overview of the Model, cont.

The model features four sectors:

Final-goods sector

Intermediate-goods sector

R&D sector

Labor market

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Overview of the Model, cont.

The model is represented in diagram form in the following slide. Thearrows represent �ows of resources between the sectors. These will bedescribed in detail below.

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R&D Sector

Produces blueprints for constructing intermediate goods

Workers

Supply labor to the R&D and Final-Goods sectors

Intermediate-Goods Sector

Group of monopoly producers of

intermediate goods

Final-Goods Sector

Produces the single final good using labor and existing

intermediate goods

The R&D Sector

In this sector, blueprints for producing new intermediate goods aregenerated by workers who spend their time exclusively towards this goal.

Physical capital is not required to produce blueprints; however, existingideas are useful for generating new ideas.

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The R&D Sector, cont.

Let the stock and �ow of ideas be given by:

Stock: At

Flow:�At

Also, let LA = sRL denote the proportion of the workforce devoted to theR&D sector.

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The R&D Sector, cont.

With this notation, the law of motion for new ideas is given by

�At = δLA,

whereδ = δAφ, δ > 0, 0 < φ < 1.

Note: individuals take δ as given, but their actions have an impact on δ(i.e., the search for new ideas generates an externality).

DND () Romer Model Econ. 1540, Spring 2010 7 / 19

The R&D Sector, cont.

Combining equations, the law of motion for new ideas is given by

�At = δAφLA, δ > 0, 0 < φ < 1.

DND () Romer Model Econ. 1540, Spring 2010 8 / 19

The R&D Sector, cont.

When a new blueprint is generated, it is sold to a �rm in theintermediate-goods sector at price PA.

Revenue generated by the sales of new blueprints are paid fully to workers.The wage rate in this sector is given by wR .

The �ow of resources into and out of the R&D sector is depticted in thefollowing diagram.

DND () Romer Model Econ. 1540, Spring 2010 9 / 19

R&D Sector

Produces blueprints for constructing intermediate goods

Workers

Supply labor to the R&D and Final-Goods sectors

Intermediate-Goods Sector

Group of monopoly producers of

intermediate goods

Final-Goods Sector

Produces the single final good using labor and existing

intermediate goods

The Intermediate Goods Sector

This consists of a large number of �rms, each of which has obtained themonopoly power to produce a unique intermediate good from the R&Dsector at price PA.

The j th �rm produces the j th unique good xj at the marginal resource costr , which represents the rental rate of physical capital.

The production of xj does not require a labor input, just the sacri�ce of xjunits of the �nal good (foregone consumption, as in the Solow model).

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Int. Goods Sector, cont.

The demand curve for xj stems from the �nal-goods sector; it is given by

pj = pj (xj ) , p0j (xj ) < 0.

The pro�t function of the j th �rm is given by

πj = pj (xj ) xj � rxj .

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Int. Goods Sector, cont.

The total quantity of intermediate goods supplied to the �nal-goods sectoris the sum (or integral) of the quantities supplied by the individualmonopolists:

At

∑j=1xj ,

AtZ0

xjdj .

In turn, the total resources paid to the individual monopolists is given by

At

∑j=1pj (xj ) xj ,

AtZ0

pj (xj ) xjdj .

The �ow of resources into and out of this sector is indicated in thefollowing diagram.

DND () Romer Model Econ. 1540, Spring 2010 12 / 19

R&D Sector

Produces blueprints for constructing intermediate goods

Workers

Supply labor to the R&D and Final-Goods sectors

Intermediate-Goods Sector

Group of monopoly producers of

intermediate goods

Final-Goods Sector

Produces the single final good using labor and existing

intermediate goods

Final Goods Sector

A single consumption good is produced in a perfectly competitive industry.(I.E., prices are taken as given.)

Here, labor LY = (1� sR ) L and intermediate goods combine to produceY , according to

Y = L1�αY

At

∑j=1xαj

or

Y = L1�αY

AtZ0

xαj dj .

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Final Goods Sector, cont.

The objective in this sector is once again to maximize pro�ts:

maxLY ,xj

L1�αY

AtZ0

xαj dj � wY LY �

AtZ0

pj (xj ) xjdj .

The �ow of resources into and out of this sector is indicated in thefollowing diagram.

DND () Romer Model Econ. 1540, Spring 2010 14 / 19

R&D Sector

Produces blueprints for constructing intermediate goods

Workers

Supply labor to the R&D and Final-Goods sectors

Intermediate-Goods Sector

Group of monopoly producers of

intermediate goods

Final-Goods Sector

Produces the single final good using labor and existing

intermediate goods

Labor Market

Workers are equally able and indi¤erent in supplying labor in the R&D andFinal Goods sectors.

Therefore, they choose sR , the fraction of labor devoted to the R&Dsector, such that wages are equated across sectors:

wY = wR = w .

The �ow of resources into and out of this sector is indicated in thefollowing diagram.

DND () Romer Model Econ. 1540, Spring 2010 15 / 19

R&D Sector

Produces blueprints for constructing intermediate goods

Workers

Supply labor to the R&D and Final-Goods sectors

Intermediate-Goods Sector

Group of monopoly producers of

intermediate goods

Final-Goods Sector

Produces the single final good using labor and existing

intermediate goods

Aggregating the Model

Yt = L1�αYt

AtZ0

xαj dj

Kt =

AtZ0

xαj dj

�K t = It � dKtYt = Ct + ItCt = (1� s)Yt�At = δAφ

t LAtLAt = sRLt , LYt = (1� sR )Lt�LtLt

= n

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Aggregating the Model, cont.

Endogenous prices:

wR ,wY : determined as the marginal product of labor in the R&Dand �nal-goods sectors, constrained to be equal

p(xj ) : determined as the marginal product of xj in the �nal-goodssector

PA : determined by the pro�ts a new blueprint can generate for anintermediate-goods monopolist

r : determined as the marginal product of K in producing Y

Also endogenous:sR

The aggregated �ow of resources is indicated in the following diagram.

DND () Romer Model Econ. 1540, Spring 2010 17 / 19

R&D Sector

Produces blueprints for constructing intermediate goods

Workers

Supply labor to the R&D and Final-Goods sectors

Intermediate-Goods Sector

Group of monopoly producers of

intermediate goods

Final-Goods Sector

Produces the single final good using labor and existing

intermediate goods

Solving and Implementing the Model

To solve the model, we must solve for (wR ,wY , p(xj ),PA, r , sR ) , thenusing these solutions, draw inferences for aggregate behavior.

This approach is tedious and complicated, so we�ll work in reverse: we�lltake as granted these soltuions, and move directly the to study ofaggregate behavior. Time permitting, we�ll then return to the solution of(wR ,wY , p(xj ),PA, r , sR ) .

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Solving and Implementing the Model, cont.

Further model analysis will be pursued interactively during class.

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