14 06 Lecture Notes Intermediate Macroeconomics George Marios Angeletos MIT Department of Economics Spring 2004 Chapter 7 Endogenous Growth II R D and Technological Change 7 1 Expanding Product Variety The Romer Model There are three production sectors in the economy A final good sector an intermediate good sector and an R D sector The final good sector is perfectly competitive and thus makes zero profits Its output is used either for consumption or as input in each of the other two sector The intermediate good sector is monopolistic There is product di erentiation Each intermediate producer is a quasi monopolist with respect to his own product and thus enjoys positive profits To become an intermediate producer however you must first acquire a blueprint from the R D sector A blueprint is simply the technology or know how for transforming final goods to di erentiated intermediate inputs 129 George Marios Angeletos The R D sector is competitive Researchers produce blueprints that is the technology for producing an new variety of di erentiated intermediate goods Blueprints are protected by perpetual patents Innovators auction their blueprints to a large number of potential buyers thus absorbing all the profits of the intermediate good sector But there is free entry in the R D sector which drive net profits in that sector to zero as well 7 1 1 Technology The technology for final goods is given by a neoclassical production function of labor L and a composite factor Z Yt F Xt Lt A Lt 1 Xt The composite factor is given by a CES aggregator of intermediate inputs Xt Z Nt Xt j dj 0 1 where Nt denotes the number of di erent intermediate goods available in period t and Xt j denotes the quantity of intermediate input j employed in period t In what follows we will assume which implies Z Nt 1 Xt j dj Yt A Lt 0 Note that means the marginal product of each intermediate input is independent of the quantity of other intermediate inputs 1 Yt Lt A Xt j Xt j 130 Lecture Notes More generally intermediate inputs could be either complements or substitutes in the sense that the marginal product of input j could depend either positively or negatively on Xt We will interpret intermediate inputs as capital goods and therefore let aggregate capital be given by the aggregate quantity of intermediate inputs Z Nt Kt Xt j dj 0 Finally note that if Xt j X for all j and t then Yt AL1 Nt X t and Kt Nt X implying Yt A Nt Lt 1 Kt or in intensive form yt ANt1 kt Therefore to the extent that all intermediate inputs are used in the same quantity the technology is linear in knowledge N and capital K Therefore if both N and K grow at a constant rate as we will show to be the case in equilibrium the economy will exhibit long run growth like in an Ak model 7 1 2 Final Good Sector The final good sector is perfectly competitive Firms are price takers Final good firms solve max Yt wt Lt 131 Z 0 Nt pt j Xt j dj George Marios Angeletos where wt is the wage rate and pt j is the price of intermediate good j Profits in the final good sector are zero due to CRS and the demands for each input are given by the FOCs wt and pt j Yt Yt 1 Lt Lt 1 Yt Lt A Xt j Xt j for all j 0 Nt 7 1 3 Intermediate Good Sector The intermediate good sector is monopolistic Firms understand that they face a downward sloping demand for their output The producer of intermediate good j solves max t j pt j Xt j Xt j subject to the demand curve Xt j Lt A pt j 1 1 where X represents the cost of producing X in terms of final good units We will let the cost function be linear X X The implicit assumption behind this linear specification is that technology of producing intermediate goods is identical to the technology of producing final goods Equivalently 132 Lecture Notes you can think of intermediate good producers buying final goods and transforming them to intermediate inputs What gives them the know how for this transformation is precisely the blueprint they hold The FOCs give pt j p 1 1 for the optimal price and Xt j xL for the optimal supply where 1 2 x A 1 1 Note that the price is higher than the marginal cost p 1 0 X 1 the gap representing the mark up that intermediate good firms charge to their customers the final good firms Because there are no distortions in the economy other than monopolistic competition in the intermediate good sector the price that final good firms are willing to pay represents the social product of that intermediate input and the cost that intermediate good firms face represents the social cost of that intermediate input Therefore the mark up 1 gives the gap between the social product and the social cost of intermediate inputs Hint The social planner would like to correct for this distortion How The resulting maximal profits are t j L where p 1 x 1 x 133 1 2 1 1 A 1 George Marios Angeletos 7 1 4 The Innovation Sector The present value of profits of intermediate good j from period t and on is given by Vt j X q t qt j or recursively Vt j t j Vt 1 j 1 Rt 1 We know that profits are stationary and identical across all intermediate goods t j L for all t j As long as the economy follows a balanced growth path we expect the interest rate to be stationary as well Rt R for all t It follows that the present value of profits is stationary and identical across all intermediate goods Vt j V L L R 1 R R Equivalently RV L which has a simple interpretation The opportunity cost of holding an asset which has value V and happens to be a blueprint instead of investing in bonds is RV the dividend that this asset pays in each period is L arbitrage then requires the dividend to equal the opportunity cost of the asset namely RV L New blueprints are also produced using the same technology as final goods In e ect innovators buy final goods and transform them to blueprints at a rate 1 Producing an amount N of new blueprints costs N where 0 measures the cost of R D in units of output On the other hand the value of these new blueprints is V N where V L R Net profits for a research firm are thus given by V N 134 Lecture Notes Free entry in the sector of producing blueprints then implies V 7 1 5 Households Households solve max X t u ct t 0 s t ct at 1 wt 1 Rt at As usual the Euler condition gives u0 ct 1 Rt 1 u0 ct 1 And assuming CEIS this …