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Spence (1984) - Cost Reduction Competition And Industry Performance (view source)
Revision as of 18:35, 3 December 2010
, 18:35, 3 December 2010no edit summary
*This page is referenced in [[PHDBA602 (Innovation Models)]]
==Reference(s)==
*Spence, Michael (1984), "Cost Reduction, Competition, and Industry Performance', Econometrica, Vol. 52, No. 1 (Jan.), pp. 101-122 [http://www.edegan.com/pdfs/Spence%20(1984)%20-%20Cost%20Reduction%20Competition%20and%20Industry%20Performance.pdf (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 <math>n\;</math> firms, indexed by <math>i\;</math>, with costs depending on time <math>t\;</math> of <math>c_i(t)\;</math>.
'''Costs''' depend on accumulated investment the firm and possibly other firms:
:<math>c_i(t) = F(z_i(t))\;</math>
where <math>F(\cdot)\;</math> is a decreasing function of <math>z_i\;</math>, which is accumulated knowledge obtained by firm <math>i\;</math>.
Suppressing the <math>t\;</math>, we denote <math>c = (c_1,\ldots,c_n) = (F(z_1),\ldots,F(z_n))\;</math>
:<math>m(t)\;</math> is the '''current expenditure on R&D''' at time <math>t\;</math>, and it is assumed that:
:<math>\dot{z}_i(t) = m_i(t) + \theta \sum_{j \ne i} m_j(t)\;</math>
where <math>\theta \in [0,1]\;</math> measures the degree of spillovers, and <math>\dot{z}_i(t)\;</math> is <math>\frac{\partial z_i(t)}{\partial t}\;</math>.
Letting <math>M_i = \int_0^t m_i(\tau) d\tau\;</math> be the '''accumulated investment in R&D''' by time <math>t\;</math>, we have:
:<math>z_i(t) = M_i(t) + \theta \sum_{j \ne i} M_j(t)\;</math>
Suppressing the <math>t\;</math>, we denote <math>z = (z_1,\ldots,z_n)\;</math>. Also, we note that <math>z_i(0) = 0\;</math>.
The '''benefits''' from selling <math>x units are <math>B(x)\;</math>, and the '''inverse demand''' is <math>B'(x)\;</math>.
The '''output''' by firm <math>i\;</math> is <math>x_i\;</math>, such that <math>x = \sum_i x_i\;</math>
The '''profits''' for firm <math>i\;</math> are:
:<math>E^i = x_i B'(x) - c_i x_i\;</math>
It is assumed that an equilibrium exists and is unique given <math>c\;</math> or <math>z\;</math>.
Let <math>x_i(z)\;</math> and <math>x(z)=sum_i x_i(z)\;</math> be the '''equilibrium'''.
The '''consumer surplus''' is:
:<math>H(z) = B(x(z)) - x(x)\cdot B'(x(z))\;</math>
The '''earnings gross of R&D expenditure''' for a firm are:
:<math>E_i(z) = x_i(z)B'(x(z)) - c_i(z_i)x_i(z)\;</math>
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 <math>(1-s)\;</math>, the '''earnings net of R&D''' is:
:<math>V^i = E^i(z) - (1-s)M_i\;</math>
As the firm takes the <math>M_j\;</math> of its rivals as given, it maximizes <math>V^i\;</math> wrt <math>M_i\;</math> by setting:
:<math>E_i^i + \theta \sum{j \ne i} E_j^i = (1-s)\;</math>
Where <math>E_j^i\;</math> denotes the derivitive of <math>E^i\;</math> wrt <math>a_j\;</math>.
Given the equilibrium values <math>M = (M_1,\ldots,M_n)\;</math>, the '''total surplus''' is:
:<math>T(M) = H(z(M)) + \sum_i V^i (M) - s \sum_i M_i\;</math>
where <math>s \sum_i M_i\;</math> is the cost (to the public sector) of the subsidies (note that the paper says <math>x\;</math> instead of <math>s\;</math> 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 <math>\dot{z}\;</math>
*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.
