Spence (1984) - Cost Reduction Competition And Industry Performance

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Reference(s)

• Spence, Michael (1984), "Cost Reduction, Competition, and Industry Performance', Econometrica, Vol. 52, No. 1 (Jan.), pp. 101-122 (pdf)

Abstract

In many markets, firms compete over time by expending resources with the purpose of reducing their costs. Sometimes the cost reducing investments operate directly on costs. In many instances, they take the form of developing new products that deliver what customers need more cheaply. Therefore product development can have the same ultimate effect as direct cost reduction. In fact if one thinks of the product as the services it delivers to the customer (in the way that Lancaster pioneered), then product development often is just cost reduction. There are at least three sorts of problems associated with industry performance. They occur simultaneously, making the problem of overall assessment of performance quite complicated. The problems are these. Cost reducing expenditures are largely fixed costs. In a market system, the criterion for determining the value of cost reducing R & D is profitability, or revenues. Since revenues may understate the social benefits both in the aggregate, and at the margin, there is no a priori reason to expect a market to result in optimal results. Second and related, because R & D represents a fixed cost, and depending upon the technological environment, sometimes a large one, market structures are likely to be concentrated and imperfectly competitive, with consequences for prices, margins, and allocative efficiency.

The Model

There are $n\;$ firms, indexed by $i\;$, with costs depending on time $t\;$ of $c_i(t)\;$.

Costs depend on accumulated investment the firm and possibly other firms:

$c_i(t) = F(z_i(t))\;$

where $F(\cdot)\;$ is a decreasing function of $z_i\;$, which is accumulated knowledge obtained by firm $i\;$.

Suppressing the $t\;$, we denote $c = (c_1,\ldots,c_n) = (F(z_1),\ldots,F(z_n))\;$

$m(t)\;$ is the current expenditure on R&D at time $t\;$, and it is assumed that:
$\dot{z}_i(t) = m_i(t) + \theta \sum_{j \ne i} m_j(t)\;$

where $\theta \in [0,1]\;$ measures the degree of spillovers, and $\dot{z}_i(t)\;$ is $\frac{\partial z_i(t)}{\partial t}\;$.

Letting $M_i = \int_0^t m_i(\tau) d\tau\;$ be the accumulated investment in R&D by time $t\;$, we have:

$z_i(t) = M_i(t) + \theta \sum_{j \ne i} M_j(t)\;$

Suppressing the $t\;$, we denote $z = (z_1,\ldots,z_n)\;$. Also, we note that $z_i(0) = 0\;$.

The benefits from selling $x\;$ units are $B(x)\;$, and the inverse demand is $B'(x)\;$.

The output by firm $i\;$ is $x_i\;$, such that $x = \sum_i x_i\;$

The profits for firm $i\;$ are:

$E^i = x_i B'(x) - c_i x_i\;$

It is assumed that an equilibrium exists and is unique given $c\;$ or $z\;$.

Let $x_i(z)\;$ and $x(z)=\sum_i x_i(z)\;$ be the equilibrium.

The consumer surplus is:

$H(z) = B(x(z)) - x(x)\cdot B'(x(z))\;$

The earnings gross of R&D expenditure for a firm are:

$E_i(z) = x_i(z)B'(x(z)) - c_i(z_i)x_i(z)\;$

Given that R&D occurs at the outset only, we don't need to worry about intertemporal considerations. We can treat the gross earnings as a present value.

Supposing that the subsidy for R&D is s, so that 1 unit of R&D costs $(1-s)\;$, the earnings net of R&D is:

$V^i = E^i(z) - (1-s)M_i\;$

As the firm takes the $M_j\;$ of its rivals as given, it maximizes $V^i\;$ wrt $M_i\;$ by setting:

$E_i^i + \theta \sum_{j \ne i} E_j^i = (1-s)\;$

Where $E_j^i\;$ denotes the derivitive of $E^i\;$ wrt $a_j\;$.

Given the equilibrium values $M = (M_1,\ldots,M_n)\;$, the total surplus is:

$T(M) = H(z(M)) + \sum_i V^i (M) - s \sum_i M_i\;$

where $s \sum_i M_i\;$ is the cost (to the public sector) of the subsidies (note that the paper says $x\;$ instead of $s\;$ here, but that is surely a typo).

Maintained Assumptions

There are the following maintained assumptions, which merit some discussion:

• There are no diminishing returns to current R&D expenditure, implicit in $\dot{z}\;$
• The goods produced are the same (i.e. the product is homogenous)
• R&D is done at the outset in a lump (this comes from the model) - which allows us to treat everything as present values.

The Symmetric Case

Taking:

$E_i^i + \theta \sum_{j \ne i} E_j^i = (1-s)\;$

And making $E\;$ a function of both a scalar $v\;$ and a vector of ones $e\;$, we have:

$E_i^i(ve) + \theta \sum_{j \ne i} E_j^i(ve) = (1-s)\;$

Then we define a new function:

$R(v) = \int_0^v \left (E_i^i(ve) + \theta \sum_{j \ne i} E_j^i(ve) \right )\;$

Which is the total benefits from R&D as a function of a single parameter $v\;$, such that the marginal benefits from R&D are:

$R'(v) = 1-s\;$

And therefore the market acts as if it were maximizing (wrt to $v\;$):

$R(v) = (1-s)v\;$

Using symmetry so that $M_i = M\;$, and $z_i = z\;$, we then have:

$v = (1+\theta (n-1))\cdot M = K(\theta,n)\cdot M \quad K = (1+\theta (n-1))\;$

Expenditures per firm are:

$M = \frac{z}{K}\;$

And total surplus is:

$T(M) = H(z,n) + nE(z,n) - n M\;$

Properties of the Market Equilibria

The function $R\;$ (which I labelled the total benefits from R&D), captures the market incentives with respect to R&D. Assuming $E_j^i \lt 0 , \; i \ne j\;$, then:

• $R_{\theta} \lt 0\;$: Benefits are decreasing in spillovers
• $R_{z \theta} \lt 0\;$: ...