Romer's Model of Expanding Varieties (Part 2)

Romer’s Model of Expanding Varieties (Part 2)

Econ 4960: Economic Growth

Romer’s (1990) Model: Microfoundations Romer’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.,

1−α A = δ Lλ Aφ and Y = K α AL A ( Y ) 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. Econ 4960: Economic Growth

1 Romer’s Model: Three components Romer’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.

Econ 4960: Economic Growth

1. Final-Goods Sector Large number of perfectly competitive firms (as in Solow) Using the following technology (unlike Solow): ⎛⎞A YL1−αα x where x denotes an intermediate good = Yj⎜⎟∑ j ⎝⎠j=1 Notice that this is like adding up a bunch of Cobb- Douglas prod. Functions, each using one intermediate good: 11−−αα α α 1 − α α YL=+++YY x12 L x... L YA x

Econ 4960: Economic Growth

2 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. Econ 4960: Economic Growth

1. Final-Goods Sector (cont’d)

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: A YL= 1−αα xdj Yj∫ 0 Firm’s profit maximization problem: AA ⎡⎤1−αα max⎢⎥LxdjwLpxdjYj−− yjj LxYj, ∫∫ ⎣⎦00 Econ 4960: Economic Growth

3 1. Final-Goods Sector (cont’d) AA ⎡⎤1−αα max⎢⎥LxdjwLpxdjYj−− yjj LxYj, ∫∫ ⎣⎦00 FOC: Y 11−−αα wpLxjA=(1−αα) and jYj= for ∈( 0, ) LY These conditions simply say that firm hires (both labor and intermediate goods) until price equal marginal product.

Equation for x j is also the inverse demand condition used by sector 2. Econ 4960: Economic Growth

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:

maxπ jjjjj=pxx( ) − rx x j

Econ 4960: Economic Growth

4 2. Intermediate-Goods Sector (cont’d) max α L1−α xα −1 x − rx = α L1−α xα − rx x ( Y j ) j j Y j j j  p j FOC : 1 ⎛ α 2 ⎞ 1−α r = α 2 L1−α xα −1 ⇒ x = L Y j j ⎝⎜ r ⎠⎟ Y

Plug in xpjj into the expression for : 1 pr= j α Econ 4960: Economic Growth

2. Intermediate-Goods Sector (cont’d)

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: xxj = Exercise: Show that the profit in this model is: Y πα=(1− α) A Since intermediate goods are produced from capital goods, the amount of capital in time t is: A xdj== K. Since x x→ x= K / A ∫ jj 0 Econ 4960: Economic Growth

5 2. Intermediate-Goods Sector (cont’d) A x= x→ xαα dj= Ax→ Y= AL1− αα x jj∫ Y 0 Substitute xKA= / : 1−−αα α Y= ALY K A α 1−α Y= K( ALY ) !! This is the same Cobb-Douglas production function we assumed in the Solow model and the “simple” Romer model. Here we derived it from a more general model.

Econ 4960: Economic Growth