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 6 .