Industry dynamics, technological regimes and the role of demand

Abstract

In this paper, we propose an industrial dynamics model to analyze the interactions between the price-performance sensitivity of demand, the sources of innovation in a sector, and certain features of the corresponding pattern of industrial transformation. More precisely, we study market concentration in different technological regimes and demand conditions. The computational analysis of our model shows that market demand plays a key role in industrial dynamics. Thus, although for intermediate values of the price-performance sensitivity, our results show the well-known relationships in the literature between technological regimes and industry transformation, we find surprising outcomes when demand is strongly biased either towards price or performance. Hence, for different technological regimes, a high performance sensitivity of demand tends to concentrate the market. On the other hand, under conditions of high price sensitivity, the industry generally tends to atomize. That is to say, for extreme values of the price-performance sensitivity of demand, we find concentrated or atomized market structures no matter the technological regime we are in. These results highlight the importance of considering the role of demand in the analysis of industrial dynamics.

Appendices

Appendix 1

Statement  In the deterministic version of the model (i.e. $u_{\textrm Max}$ $=$ $\sigma$ $=$ 0 and $\lambda$ $=$ 1), stationarity (i.e. $s_{it}=s_{it+1}$ and $r_{it}=r_{it+1}$, for all i and for all $t)$ implies that all firms present in the market end up becoming indistinguishable from each other. Specifically, using $n >$ 1 to denote the number of firms present in the market, in the long run (i.e. as time goes to infinity):

• (a) $s_{it}=s_{i}$ $=$ 1/n, $\forall i$.

• (b) $r_{it}=r_{i}=r$, $\forall i$.

• (c) $c_{it}=c_{i}=n$ /($n-r)$, $\forall i$.

• (d) $p_{it}=p_{i}$ $=$ (n +1)/($n-r)$, $\forall i$.

• (e) $R_{it}=R_{i}=r$/($n(n-r))$, $\forall i$.

• (f) $x_{it}=x_{i}=x^{\textrm Max}$, $\forall i$.

(The case where there is only one firm in the market is insignificant).

Proof

The proof is conducted in several steps:

• 1. $r_{it+1}=r_{it}$, $\forall i$, $\forall t$ $\Rightarrow$ {Eq. 8}$\Rightarrow$ $r_{it}=r$, $\forall i$, $\forall t$.

• 2. $s_{it+1}=s_{it}$, $\forall i$, $\forall t$ $\Rightarrow$ {Eq. 5}$\Rightarrow$ $\gamma _{it} =\overline \gamma _t$, $\forall i$, $\forall t$.

• 3. $c_{it} = 1 + \frac {R_{it}}{s_{it}} = 1 + {r_{it-1}} {\pi _{it-1}} \frac {s_{it-1}}{s_{it}} = 1 + {r_{it-1}} {s_{it-1}} {c_{it-1}} \frac {s_{it-1}}{s_{it}} = \{{{s_{it}} = s_{it-1} = {s_i} ;\;{r_{it}} =r} \}=1+r\cdot {s_i} \cdot {c_{it-1}}$

Noting that $r\cdot s_{i}<$ 1, we can solve the recursive equation to obtain:

$$c_{it} ={\left({1-\left({r\cdot s_i}\right)^t}\right)}/{\left({1-r\cdot s_i}\right)}+c_{i0} ({r\cdot s_i})^t $$

Thus, taking limits when t goes to infinity:

$$\lim\limits_{t\to \infty} c_{it} =1 / {( {1-r\cdot s_i} )}=c_i $$

• 4. $R_{it+1} =r\cdot s_i^2 \cdot c_{it}$ $\Rightarrow$ $\lim \limits _{t\to \infty } R_{it} ={r\cdot s_i^2} / {( {1-r\cdot s_i} )}=R_i$.

• 5. $p_{it} =( {1+s_i} )c_{it}$ $\Rightarrow$ $\mathop {\lim }\limits _{t\to \infty } p_{it} ={( {1+s_i} )} / {( {1-r\cdot s_i} )}=p_i$.

Also, note that, for all i, the convergence of prices $p_{it}$ to the corresponding limit $p_{i}$ is monotonous, and the series $\vert p_{it}-p_{i}\vert$ is geometric and monotonously decreasing (remember that $r\cdot s_{i} <$1):

$$| {p_{it} -p_i} |=\left|({1+s_i})\;\left[c_{i0} -1 / {({1-r\cdot s_i})}\right] \right|({r\cdot s_i})^t $$

In simple words, prices stabilize, i.e. for any time h:

$$| {p_{ih+1} -p_{ih}} |>| {p_{it+1} -p_{it}} |\quad \quad \quad \forall t>h. $$

• 6. $\gamma _{it} =\overline \gamma _t , \quad \forall i$, $\forall t$ $\Rightarrow$ $\gamma _{it} =\gamma _{jt}$, $\forall i$, $\forall j$, $\forall t$

Therefore:

$$ \begin{array}{rll} \gamma_{it} &=&( {1-\alpha} )\frac{x_{it} -\overline x_t} {\overline x_t} -\alpha \frac{p_{it} -\overline p_t} {\overline p_t} =( {1-\alpha} )\frac{x_{jt} -\overline x_t} {\overline x_t} -\alpha \frac{p_{jt} -\overline p_t} {\overline p_t} =\gamma_{jt} \\ &&( {1-\alpha} )\frac{x_{it} -x_{jt}} {\overline x_t} =\alpha\frac{p_{it} -p_{jt}} {\overline p_t}\\ &&x_{it} -x_{jt} =\overline x_t \frac{\alpha} {1-\alpha} \frac{p_{it}-p_{jt}} {\overline p_t} \end{array} $$

Since $x_{it} >$ 0, $\forall i$, $\forall t$, the equation above implies that beyond a certain time T (i.e. when prices stabilize and there are no changes in the ranking of prices anymore—see point 5 of the proof) there cannot be changes in the ranking of performances either, i.e. $\exists T$ such that $x_{iT} > x_{jT}$ $\Rightarrow$ $x_{it} > x_{jt}$, $\forall t > T$. Thus:

$\exists m$, T, such that $x_{mt} =x_t^{\text {Max}}$, $\forall t > T$. Using Eq. 10, this implies that $x_t^{\text {Max}} =x^{\text {Max}}$, $\forall t > T$.

Using Eq. 10 again, and noting that $\mathop {\lim }\limits _{t\to \infty } R_{it} ={r\cdot s_i^2} / {( {1-r\cdot s_i} )}=R_i$ and that ${\phi \cdot r\cdot s_i^2} / {( {1-r\cdot s_i} )}<1$ (since there cannot be changes in the ranking of performances $\forall t > T)$, it can be proved that:

$$\mathop{\lim}\limits_{t\to \infty} x_{it} =x^{\text{Max}} $$

Thus, in the long run: $x_{it}=x_{i}=x^{\textrm Max}$, $\forall i$. Bearing in mind also that in the long run $\gamma _{it} =\overline \gamma _t$, we then obtain $p_i =\overline p$, $\forall i$. Thus:

$$ \begin{array}{rll} p_i &=& {( {1+s_i} )} / {( {1-r\cdot s_i} )}={( {1+s_j} )} / {( {1-r\cdot s_j} )}=p_j \\ &\Rightarrow & \quad {\{}0 < s_{i} < 1{\}} \quad \Rightarrow \quad s_i =s_j =s=1 / n, \forall i, j. \end{array} $$

Substituting this last result in the equations derived above, we obtain the 6 propositions (a–f) included in the statement. □

Appendix 2

Given the difficulties involved in an exhaustive computational analysis for all the parameters of the model,¹⁸ we have taken high cumulativeness in Eq. 10 as an almost permanent assumption. However, it seems reasonable to check—even if only provisionally—what would happen to the patterns which emerge from entrepreneurial regimes, if we suppose low cumulativeness.

In Fig. 10 we present the results obtained for the Herfindhal index and the number of firms at t $=$ 30,000, for the case of entrepreneurial regimes with $\eta$ $=$ 0.1, considering the level of technological opportunities is medium, and the demand has an intermediate profile. This parametric configuration allows us to directly compare Fig. 10 with Fig. 5, and thus see how the results change when considering low cumulativeness in entrepreneurial regimes.

Fig. 10

Box-plots of the Herfindhal index (above) and the number of firms (below) at time step 30,000, for $\alpha$ $=$ 0.5, $\lambda$ $=$ 0.1, $\phi$ $=$ 0.9, $u_{\textrm Max}$ $=$ 0.5, $\eta$ $=$ 0.1 and different values of $\sigma$ and $\beta$. Each box-plot represents data from 1,000 simulation runs

As we can see in both figures, the expected results are maintained for the case of low cumulativeness. In this case, we also obtain low levels of industry concentration and a high number of firms in the limit states. This reinforces the results seen in Section 4.1.2 regarding the appearance of creative destruction patterns in entrepreneurial regimes.