I'm working on a Schumpeterian R&D growth model, and I assume that the quality gap between a new machine and the replaced one, i.e. $$\lambda$$ is sufficiently large such that I have drastic innovation (the monopolist can charge its unconstrained price). In particular, it satisfies:

$$\lambda \ge [\frac{1}{1-\beta}]^{\frac{1-\beta}{\beta}}$$

Q1:How can I derive such a condition?

In this economy, we have a perfectly competitive final output sector producing a homogeneous good. The aggregate production function is given by:

$$Y(t) = \frac{1}{1-\beta}\left(\int_{0}^{1} q(j,t)x(j,t |q)^{1-\beta} \; dj\right) L^{\beta}$$

where $$x(j,t|q)$$ is the quantity of machine variety $$j$$ of quality $$q$$ used in the production process, and $$q(j,t)$$ is the quality of machine $$j$$ at time $$t$$. Here, we are implicitly assuming that only one quality of any machine is used, i.e., the leading-edge machine (the last vintage one).

The quality ladders determine the quality of each machine variety as follows:

$$q(j,t) = \lambda^{n(j,t)}q(j,0)$$ , with $$\lambda > 1$$, and $$n(j,t)$$ denotes the number of innovations occurred on machine $$j$$ from time zero to $$t$$. This notation implies that there is a quality ladder for each machine variety, and each innovation pushes the machine quality up by one rung on this ladder. These rungs are proportionally equidistant so that each innovation leads to a proportional increase in quality by an amount $$\lambda >1$$. Q2:Why the increase is proportional? it seems exponential.

Also, we assume that the number of innovations in machine line $$j$$ by time $$t$$, i.e., $$n(j,t)$$ is a random variable. This implies that also $$q(j,t)$$ is a random variable.Then, we set $$Q(t) = \left(\int_{0}^{1} q(j,t) dj\right)$$ and we assume that $$Q(t)$$ is deterministic. Q3:Why?. It is something about the LLN(law of large numbers)?

The R&D sector, which develops blueprints for new machines, is perfectly competitive. Then machines, once invented, can be manufactured, under monopolistic competition, at a marginal cost $$\psi q(j,t)$$, with $$\psi >0$$ units of the final output (final good is the numeraire good).

It's straightforward to see that the optimal demand from final good producers for machines is given by:

$$x(j,t|q) = {\frac{q(j,t)}{p^{x}(j,t|q)}}^{1/ \beta} L$$ , with $$p^{x}(j,t|q)$$ denoting the price of a machine of variety j of quality $$q(j,t)$$ at time $$t$$

Hint from the book to answer Q1:

First, suppose that the innovator sets the monopoly price $$\psi q/(1− \beta)$$ for a product of quality $$q$$. Then consider the firm with the next highest quality, $$\lambda^{-1}q$$, and suppose that this firm sells at marginal cost, $$\psi \lambda^{-1} q$$. Find the value of $$\lambda$$ such that final good producers are indifferent between buying a machine of quality $$q$$ at the price $$\psi q /(1 - \beta)$$ versus a machine of quality $$\lambda^{-1}q$$ at the price $$\psi \lambda^{-1}q$$