romer’s model of expanding varieties (part 2)

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Romer’s Model of Expanding Varieties (Part 2)

Econ 4968: Economic Growth

Romer’s (1990) Model: MicrofoundationsRomer’s main contribution was to write a micro-founded model of endogenous growth.By this, we mean a model where the production function for TFP and the aggregate economy, i.e.,

are not assumed but rather derived from a detailed description of the micro-economy.Solid micro-foundations what makes a model part of modern macro.


A = δ LA

λ Aφ and Y = Kα ALY( )1−α

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Romer’s Model: Three componentsRomer’s model has three sectors:1. Final goods producing sector: produces goods from intermediate

good that are sold to consumers.2. Intermediate goods producing sector: Each firm in this sector

owns a patent to an intermediate good invented by the R&D sector. It produces it and sells it to sector 1.

3. R&D sector: Invents new varieties of intermediate goods by devoting funds to innovation. Sells production rights to sector 2.

In this economy all growth will come from the invention of new “varieties” by the R&D sector.But the incentives of these firms to innovate will depend on what happens in sectors 1 and 2 as well.


1. Final-Goods SectorLarge number of perfectly competitive firms (as in Solow)Using the following technology (unlike Solow):

Notice that this is like adding up a bunch of Cobb-Douglas production functions, each using one intermediate good:


1

1

where denotes an intermediate goodA

Y j jj

Y L x xa a-

=

æ ö= ç ÷

è øå

1 1 11 2 ...Y Y Y AY L x L x L xa a a a a a- - -= + + +

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1. Final-Goods Sector (cont’d)An intermediate good is also called a “differentiated good” because they are not substitutes for each other. For example, tires, engine, brakes, dashboard, etc, etc. are intermediate goods that are used in manufacturing a car (final good).We used A to denote the number of intermediate goods used in production. This is not a coincidence.The same A—the number of varieties—will also turn out to measure TFP growth in this economy. Finally, for a fixed A, the production function is CRS. Since final goods producing firms do not control A, they face CRS.


One difficulty with the formulation above is that it assumes A is an integer. In this case, the math becomes very messy. Instead, we switch to the this formulation, where A is continuous:

Firm’s profit maximization problem:


1. Final-Goods Sector (cont’d)

1

0

A

Y jY L x dja a-= ò

1

,0 0

max Y j

A A

Y j y j jL xL x dj wL p x dja a-é ù

- -ê úë û

ò ò

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FOC:

These conditions simply say that firm hires (both labor and intermediate goods) until price equal marginal product.Equation for is also the inverse demand condition used by sector 2.


1. Final-Goods Sector (cont’d)1

,0 0

max Y j

A A

Y j y j jL xL x dj wL p x dja a-é ù

- -ê úë û

ò ò

( ) ( )1 11 and for 0,j Y jY

Yw p L x j AL

a aa a - -= - = Î

x j


2. Intermediate-Goods Sector Each firm in this sector is a monopolist because they hold the patent to the design of an intermediate good.Each patent is bought from the R&D sector for a one-time fixed cost.Production is very simple: one unit of raw capital is converted one-for-one into one unit of the intermediate good. Profit maximization:

( )maxj

j j j j jxp x x rxp = -

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2. Intermediate-Goods Sector (cont’d)

maxx j

αLY1−α x j

α−1( )p j

x j − rx j =αLY

1−α x jα − rx j

FOC :

r =α 2LY1−α x j

α−1 ⇒ x j =α 2

r⎛⎝⎜

⎞⎠⎟

11−α

LY

Plug in into the expression for :

1j j

j

x p

p ra

=


Note that the price of intermediate good j is the same for all j. Therefore, all monopolists charge the same price and produce the same amount of the intermediate goods:Exercise: Show that the profit in this model is:

Since intermediate goods are produced from capital goods, the amount of capital in time t is:

jx x=

( )1 YA

p a a= -

0

. Since /A

j jx dj K x x x K A= = ® =ò


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This is the same Cobb-Douglas production function we assumed in the Solow model and the “simple” Romermodel.Here we derived it from a more general model.

( )

1

0

1

1

Substitute / :

!!

A

j j Y

Y

Y

x x x dj Ax Y AL x

x K AY AL K A

Y K AL

a a a a

a a a

aa

-

- -

-

= ® = ® =

=

=

=

ò


romer’s model of expanding varieties (part 2)

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