7 To increase the number of goods it produces a firm invests in innovative

7 to increase the number of goods it produces a firm

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To increase the number of goods it produces, a firm invests in innovative effort, i.e. spends resources on R&D. A firm’s R&D choice is two-dimensional. Each instant, it chooses both the frequency of arrival of new innovations λ t [0 , ¯ λ ] and the quality of new innovations θ t [0 , 1]. The arrival intensity of a new innovation λ t determines the Poisson rate at which innovations arrive. Conditional on an innovation, the number of new product lines that this innovation generates is given by X t = min ( Y t , ¯ p - P t - ) with Y t Bin ( n, θ ) , where n is an exogenous upper bound on the number of new product lines that can be developed following an innovation, θ measures the expected quality of the innovation, and Bin ( n, θ ) is the binomial distribution. The expected number of new product lines is approx- imately given by . Therefore, a higher quality θ leads to a higher expected number of new product lines. Bounding the number of new product lines X t from above by ¯ p - P t - ensures that P t never exceeds ¯ p . These assumptions imply that the total number of product lines the firm has developed up to time t , denoted by I t , evolves as dI t = X t dN I t , where dN I t is a Poisson process with intensity λ t . A firm’s existing product lines can become obsolete because some other firm innovates on a good it is currently producing. In this case, the incumbent producer loses the good from its portfolio due to creative destruction. Since any firm is infinitesimal, we can ignore the possi- bility that it innovates on a good it is currently producing. Because of creative destruction, each product becomes obsolete at an exponentially distributed time with intensity f . We call f the rate of creative destruction, that each firm takes as given. Subsection C embeds the single-firm model into an industry equilibrium and endogenizes the rate f of creative destruction. The total number O t of product lines lost by the firm up to time t 0 because of creative destruction evolves as dO t = dN O t , 8
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where dN O t is a Poisson process with intensity fP t - . The total number product lines in a firm’s portfolio P t is therefore given by P t = I t - O t . A firm with zero product lines exits the economy at time τ 0 inf { t > 0 : P t = 0 } . A firm performing R&D with intensity and quality ( λ t , θ t ) incurs flow costs q ( P t , λ t , θ t ). To ensure that shareholders are better off with more product lines we impose that the R&D cost function does not increase too fast in the number of product lines, in that q ( p + 1 , λ, θ ) - q ( p, λ, θ ) < 1 . (1) An incumbent firm’s operating profit is the profit that comes from the operation of the product lines minus the costs of performing R&D: P t - q ( P t , λ t , θ t ) . Profits are taxed at the constant rate π > 0. As a result, firms have an incentive to issue debt to reduce corporate taxes.
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