Product Innovation and Macroeconomic Dynamics

Abstract

We develop an agent-based macroeconomic model in which product innovation is the fundamental driver of growth and business cycle fluctuations. The model builds on a hedonic approach to the product space and product innovation developed in Georges (A hedonic approach to product innovation for agent-based macroeconomic modeling, 2011).

Appendix: Representative Agent Benchmark

Consider the case in which all firms are identical and each starts with a 1/n share of the total market. Suppose further that all firms engage in R&D in every period and experience identical innovations over time.

In this case, the equal individual market shares will persist, since there is no reason for consumers to switch between firms with identical product qualities and identical prices. Further, there is a unique equilibrium for the real production and sales of consumer goods Y at which demand and supply are in balance. At this equilibrium, aggregate real activity depends on the markup η, the per firm overhead labor costs H and R, the wage rate W (for production workers), labor productivity A (for production workers), and the number of firms n. Specifically, at this equilibrium $Y = ({1\over \eta -1}) \cdot {A\over W}\cdot (H+R) \cdot n$. See below for details. Further, this equilibrium is a steady state of the agent dynamics in the model and is locally stable under those dynamics. If, for example, firms all start with production less than steady state production, then since the markup η > 1, demand will be greater than production for each firm, and production will converge over time to the steady state equilibrium.

We can see this as follows. Since all firms are identical, they produce identical quantities q of their goods. Then total labor income is:

$$\displaystyle \begin{aligned} E=n\cdot \left[{W\over A}\cdot q + H + R\right] {} \end{aligned} $$

(4)

Each firm also charges an identical price p for its good, which is a markup η on marginal cost

$$\displaystyle \begin{aligned} \begin{array}{rcl} p = \eta\cdot \text{MC} \\ \text{MC} = {W\over A}{} \end{array} \end{aligned} $$

(5)

and so produces and sells

$$\displaystyle \begin{aligned} q = {E\over n\cdot p} {} \end{aligned} $$

(6)

units of its good.

These relationships yield the following steady state equilibrium value for (per firm) production.

$$\displaystyle \begin{aligned} q^* = {(H+R)\cdot {A\over W} \over \eta-1} {} \end{aligned} $$

(7)

We can see that ∂q ^∗∕∂(H + R) > 0, ∂q ^∗∕∂A > 0, ∂q ^∗∕∂W < 0, and ∂q ^∗∕∂η < 0. These are all demand driven. An increase in the cost of overhead labor (H or R) raises the incomes of overhead workers, raising AD and thus equilibrium output. An increase in labor productivity A will cause firms to lower their prices, raising aggregate demand and equilibrium output. Similarly an increase in the wage rate W of production workers or in the mark-up η will cause firms to raise their prices, lowering AD and equilibrium output.

Further, if in this representative agent case firms follow simple one period adaptive expectations for demand, then firm level output dynamics are given by:

$$\displaystyle \begin{aligned} q_t = {1\over \eta}\cdot q_{t-1} + {1\over \eta}\cdot (H+R)\cdot {A\over W} {} \end{aligned} $$

(8)

Thus, given η > 1, the steady state equilibrium q ^∗ is asymptotically stable.

Total market output in the steady state equilibrium is just

$$\displaystyle \begin{aligned} \begin{array}{rcl} Y^* = n\cdot q^* \\ = {n\cdot (H+R)\cdot {A\over W} \over \eta-1} {} \end{array} \end{aligned} $$

(9)

Clearly, this will be constant as long as there is no change in the parameters H, R, A, W, η and n. If we were to allow them to change, growth in the number of firms n or the productivity of production labor A would cause equilibrium total production Y ^∗ to grow over time, while balanced growth in production wages W and labor productivity A would have no impact on equilibrium production. Note that, in the full heterogeneous agent model, while all of the parameters above are fixed, the fraction of firms adopting R&D investment varies endogenously, contributing to the disequilibrium dynamics of the model.

Importantly for the present paper, note that the representative agent steady state equilibrium production Y ^∗ above is entirely independent of product quality . Improvements in product quality will, however, increase consumer utility, or equivalently, the quality adjusted value of total production at this equilibrium. Specifically, given the nested CES formulation of utility, if the magnitudes of all product characteristics grow at rate g due to innovation, then the growth rate of consumer utility will converge in the long run to g. If the magnitudes of different characteristics grow at different rates, the growth rate of utility will converge to the rate of growth of the fastest growing characteristic. All else equal, the long run growth path of utility will be lower in the latter case than in the former case.

It is also worth noting that, at the representative agent steady state equilibrium above, firms make zero profit. Revenues net of the cost of production labor are just great enough to cover all overhead labor costs. Once we move to the heterogeneous firm case, in the comparable steady state equilibrium, profits are distributed around zero across firms. Firms will vary as to R&D investment status, with firms who are not engaging in R&D investment saving overhead cost R per period. Firms will also vary with respect to demand shares (driven by product qualities which are themselves related to past investments in R&D ), and firms with relatively high demand shares are able to spread overhead cost over greater sales. Thus, for two firms with the same overhead cost (i.e., the same current R&D investment status), the firm with greater demand for its product will have higher profit, while for two firms with the same product demand shares, the one with the lower overhead cost (lower current R&D investment) will have higher profit. Firms that face chronic losses will eventually fail and be replaced.