The Romer Model - Pitt
Transcript of The Romer Model - Pitt
The Romer Model
David N. DeJongUniversity of Pittsburgh
Econ. 1540, Spring 2010
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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).
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The R&D Sector, cont.
Combining equations, the law of motion for new ideas is given by
�At = δAφLA, δ > 0, 0 < φ < 1.
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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.
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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 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.
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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
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.
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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
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.
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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
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.
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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
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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