Recursively applying this argument ensures that the entrant value E e ( f ) goes to zero as f → ∞ . 2. If ∃ f such that E e ( f ) > H then Lemma 2 , the previous step, and the intermediate value theorem imply there exists an f * such that E e ( f * ) = H, which is an industry equilibrium. 3. If @ f such that E e ( f ) > H then f * = 0 is an industry equilibrium. B General Equilibrium Setup In this appendix, we embed our model into a general equilibrium setup. This endogenizes the growth rate of the economy, the labor supply, and the interest rate. The general equilibrium setup is similar to Klette and Kortum ( 2004 ) and leads to a stationary equilibrium with a balanced growth path. Production There is a unit mass of differentiated goods in the economy, which are indexed by i ∈ [0 , 1]. A measure L P of labor is used for production, a measure L R & D of labor performs R&D, and a meausre L E of labor is used to generate entrants. Labor supply L S is perfectly elastic, and it receives a wage w per unit supplied in each of these activities. Incumbent firms use labor and installed product lines to produce goods. An improvement in the production technology increases the amount of the consumption good that one unit of labor produces. For each type of product there is a leading producer, as in the industry equilibrium model. The production technology of good i ’s leading producer is q i t and determines the number of products that one unit of labor produces. A firm that innovates on product i improves the production technology and becomes the leading producer. Each innovation is a quality improvement applying to a good drawn at random. The innovation increases the production technology proportionally. That is, when an innovation arrives at time t , the production technology increases from q i t - to q i t = (1+ δ ) q i t - with δ > 0. 52
A firm that is the leading producer for product i is a monopolist for that good and can choose to supply or not supply that good. If the firm supplies the good then it uses one unit of labor to generate q i t units of the product. If the firm does not supply the good, its output and profits are zero. 13 Let y i t be the amount of good i produced at time t . As in Klette and Kortum ( 2004 ) or Aghion, Bloom, Blundell, Griffith, and Howitt ( 2005 ), the aggregate consumption good is produced using a logarithmic aggregator ln( Y t ) = Z 1 0 ln ( y i t ) di, with Y t the aggregate production of the consumption good. 14 Innovation Firms can invest in R&D. Investment in R&D leads to product innovations, which improve the amount of a product that one unit of labor produces. R&D investment costs come in the form of labor costs. Innovation costs are a function of the wage rate multiplied by the number of hours spend on R&D: q ( p, λ, θ ) = w * ˜ q ( p, λ, θ ) . (8) Therefore, a firm with p products that has an R&D policy ( λ, θ ) requires ˜ q ( p, λ, θ ) units of labor.
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