Innovation, competition and firm size distribution on fragmented markets

Abstract

This paper presents a simple model of firm and consumer behavior. We formulate a sub-market entry game, where boundedly rational firms decide on investing in R&D for inventing new products that will appeal to targeted groups of consumers. The success depends on the amount of resources available for the project as well as on the firm’s familiarity with market characteristics. Successful innovation feeds back into the firm size and (potentially into) market knowledge and increases the future R&D productivity. A new product decreases the market-shares of incumbents. However, this business stealing effect is asymmetric across incumbent population. We identify the section of parameter space where firms have an incentive to diversify horizontally. In this section, the model results in rich industrial dynamics. Firm size heterogeneity emerges endogenously in the model. Equilibrium firm size distributions are heavy tailed and skewed to the right. The heaviness of the tail depends on submarket specificity of firm’s market knowledge. This relationship is non-monotonic, emphasizing two different effects of innovation on industrial dynamics (positive feedback and asymmetric business stealing).

Appendix

1.1 Solution to the firm’s problem at t = 0 for linear functional forms

Here we use linear functional forms for the decay of consumer preferences and firm knowledge productivity in terms of distance. Namely, we assume that

$$ {k_{i}^{s}} = 1 - \frac{2(1-\tilde c)}{S}{d_{i}^{s}}. $$

(13)

And that

$$ {\kappa_{i}^{n}} = 1 - \frac{2(1-\tilde a)}{S}{d_{i}^{n}}. $$

(14)

In contrast to the setup discussed in the paper, the higher the parameter controlling the preference decay $\tilde c$, the lower the consumer taste specificity. The same applies to $\tilde a$.

For the analytical tractability of the results, we introduce another simplification of a functional form. We assume that

$$ r^{d} = \frac{\kappa^{d} \beta Y}{1+\kappa^{d} \beta Y}. $$

(15)

Without loss of generality, to simplify calculations we also assume that N÷4=0.

Then, we know that

$$ r^{d} = \frac{S-2(1-\tilde a)d}{S-2(1-\tilde a)d+\frac{S}{\beta Y}}, $$

(16)

which for d = 0 collapses to

$$ r^{0} = \frac{\beta Y}{1+ \beta Y}. $$

(17)

Producers can take three types of decisions: (i) do not try to innovate / no R&D, so in this case they expect to earn π ^ø; (ii) innovate on the submarket where you operate, so they expect profits π ^o; (iii) enter other submarkets / diversify, so they expect π ^d.

Given the initial conditions (the same as used in the paper), we know that

$$ \pi^{\o}=Y/p. $$

(18)

The new product will split the demand equally with the incumbent product on that submarket where it is located.Initially all submarkets are symmetric and therefore the demand for the new product (D _{n e w} ) will not depend in the submarket in which it is located, and thus on d. Therefore,

$$ \pi^{o} = 2r^{0}D_{new}+(1-r^{0}-\beta)Y. $$

(19)

We can also express the expected profits from diversification by investing in the R&D project at the distance d from the currently operational submarket. This is

$$ \pi^{d} = r^{d}(D_{old}^{d} + D_{new}) + (1 - r^{d} - \beta)Y. $$

(20)

The demand for the new product coming from consumer s is

$$ D_{new}^{s}=\frac{1-2\frac{1-\tilde c}{S}d^{s}_{new}}{\frac{S}{2}(1+\tilde c)+1-2\frac{1-\tilde c}{S}d^{s}_{new}} \frac{Y}{p}, $$

(21)

where $d^{s}_{new}$ is the distance from consumer s’s preferred submarket to the submarket where the new product is located.

Using this expression we can calculate

$$ D_{new}= \frac{Y}{p} \left (2 \sum\limits_{d_{i}}^{S/2} \frac{1-2\frac{1-\tilde c}{S}d_{i}}{\frac{S}{2}(1+\tilde c)+1-2\frac{1-\tilde c}{S}d_{i}} - \left( \frac{1}{\frac{S}{2}(1+\tilde c)+1} + \frac{\tilde c}{\frac{S}{2}(1+\tilde c)+\tilde c} \right) \right). $$

(22)

Note that the demands from consumers on the submarket where the new product is placed and on the one the most distant from it are counted twice in the first summand. Hence, the second summand.

Using the same principles we can write down the demand from consumer s for the old product of the innovator after the new product has been placed

$$ D^{s}_{old} = \frac{1-2\frac{1-\tilde c}{S}d^{s}_{old}}{\frac{S}{2}(1+\tilde c)+1-2\frac{1-\tilde c}{S}d^{s}_{new}} \frac{Y}{p}, $$

(23)

where $d^{s}_{old}$ is the distance from consumer s’s preferred submarket to the submarket where the innovator’s old product is traded.

Now we can calculate the total demand for the old product that is traded at distance d from the new product.

$$\begin{array}{@{}rcl@{}} D^{d}_{old} = & & \frac{Y}{p} \sum\limits_{d_{i}=0}^{S/2-1} \frac{1-2\frac{1-\tilde c}{S}d_{i}}{\frac{S}{2}(1+\tilde c)+1-2\frac{1-\tilde c}{S}|d-d_{i}|} \end{array} $$

(24)

$$\begin{array}{@{}rcl@{}} & + & \frac{Y}{p} \sum\limits_{S/2}^{S-1} \frac{1-2\frac{1-\tilde c}{S}(S-d_{i})}{\frac{S}{2}(1+\tilde c)+1-2\frac{1-\tilde c}{S}\min(d_{i}-d;S-(d_{i}-d))} \end{array} $$

(25)

Substituting definitions of D _{n e w} and $D^{d}_{old}$ back in Eqs. 19 and 20, results into the Eqs. 18–20 fully characterizing the producers problem in only one choice variable - d.

We can numerically analyze the consumer’s problem. For that we fix Y = 200, p = 1 and β = 0.15. It turns out that, in this setup, for any values of $\tilde a$ and $\tilde c$ there are always innovation incentives (π ^ø is always dominated). However, if R&D becomes really expensive, β→1 and research incentives disappear gradually. For example, if β = 0.888 we have all three possible regimes in the space $\tilde a\in [0;1] \times \tilde c\in [0;1]$. These regimes are plotted in Figs. 6 and 7.

Fig. 6

Three regimes. If the value of the function is 1 - no innovation (R1), if it is 2 - innovation on own submarket (R2), if it is 3 - entry into new submarkets (R3)

Fig. 7

Expected profits from all three types of actions

1.2 Pseudo-code of a single simulation run

• 1. Initialization

  • (a) Choose parameter a

  • (b) Choose parameter c

  • (c) Create submarkets

  • (d) Create consumers

  • (e) Match consumers with submarkets (e.g. assign each consumer a unique submarket as her ideal variety)

    Create firms

  • (f) Match firms with submarkets (e.g. assign each firm a unique submarket where it sells its first product)

  • (g) For each firm, set last period’s sales to Y

• 2. For each t

  • (a) For each submarket

    • i. Calculate demand on a new product on the submarket ($D_{\iota }^{i}$)

  • (b) For each firm

    • i. For each submarket

      • A. Calculate the total demand on the products in case the new product is placed in given submarket ($\sum {{D^{i}_{j}}}$)

      • B. Calculate profits from potential innovation by adding $D_{\iota }^{i}$ to $\sum {{D^{i}_{j}}}$

      • C. Calculate expected profits from innovation by taking into account r

      • D. Factor in the costs of innovation R

    • ii. Choose the most profitable submarket to innovate

    • iii. Compare the expected profits from innovating there to profits in t − 1

    • iv. If former is greater, start R&D project in that submarket

    • v. Else – do not invest in R&D

    • vi. Draw a random number from the uniform distribution to determine whether R&D project was successful.

    • vii. Depending from the outcome, calculate new total demand and revenues

  • (c) If none of the firms conducted R&D this period – end simulation

  • (d) Else go to t + 1