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Now if <math>h is concave (i.e. diminishing returns to scale throughout) then <math>\frac{h(x)}{x} \ge h'{x}\;</math> and expected profits are always positive. They are only driven to zero in the limit of an infinite number of firms.
With an initial range of increasing returns to scale then returns can go to zero with a finite number of firms.
The basic setup is as follows:
*There are <math>n\;</math> identical firms, indexed by <math>i\;</math>
*Each firm invests <math>x_i\;</math> to buy a random variable <math>\tau(x_i)\;</math> which gives a completion date
*The firm with the earliest realised completion date wins <math>V\;</math>
*<math>\tau \sim F_{\tau}(h(x_i))\;</math> where <math>F_{\tau}\;</math> is the CDF for the exponential distribution: <math>F_{\tau}(h(x_i)) = 1 - e^{-h(x_i)t}\;</math>
*<math>h(x_i)\;</math> is the rate parameter, or the instantaneous probability of the innovation occuring.
Let :<math>\hatfrac{d \tau_iPi}{d n}= \frac{\partial \pi }{\partial a}\cdot (h(x^*) + (n-1)h'(x^*)) + \frac{\partial \Pi}{\partial x} \frac{\partial x}{\partial n} < 0\;</math> be an random variable giving the date of the earliest other firm:
<math>\hat{\tau_i} = \min_{j \ne i} \{ \tau(x_j) \}\;</math>
Assuming iid tau's (no externalities in innovation!), then we can use a [http://en.wikipedia.org/wiki/Exponential_distribution#Distribution_of_the_minimum_of_exponential_random_variables nice feature of the exponential distribution] which is that if <math>X_1,\ldots,X_N\;</math> are iid exponential with rates <math>\lambda_1,\ldots,\lambda_N\;</math>, then <math>\min(X_1,\ldots,X_N)\;</math> is distributed exponential with rate <math>\sum_1^N \lambda_i\;</math>.
Therefore We know, from the envelope theorem, that <math>\hatfrac{\tau_ipartial \Pi} {\frac \partial x} = 0\;</math>, and from the original profit function that <math>\sim F_frac{\hatpartial \Pi}{\tau}partial a}< 0\;</math>, where . By rearranging the other terms we can see that equilibrium profits decrease with more competition.
:There is a proof in the paper that shows that with initial increasing returns to scale the finite number of competitors in a zero profit equilibrium will be below <math>F_{\hat{\tau}} = 1 - e^{-\lefttilde( \sum_{j\ne i} h(x_j) \rightx) t}\;</math>, which is the point where firms are using their capacity.
For convenience we denote ===Welfare Considerations===
:<math>a_i= \sum_{j\ne i} h(x_j)\; </math>Ignoring the problem that social benefits may not equal private benefits, there are two other inefficiencies. The first arises from duplication of effort.
The firm discounts the future reciepts at Given a rate <math>r\;</math> (note that using continuous compoundingfixed market structure, social welfare is maximized with a choice <math>PV = FV \cdot ex^{-rt**})\;</math>.characterized by:
The firm wins the prize at time :<math>t\frac{\partial \pi}{\partial x}((n-1)h(x),x) + (n-1)h'(x) \cdot \frac{\partial \pi}{\partial a}((n-1)h(x),x) = 0\;</math> with probability:
:Whereas the individual firms choose an <math>pr(\tau(x_i) \le \min(\hat{\tau_i},t) = e^{-a_i t}(1-e^{-h)x_i)t}) + a_i \int_0^t (1-ex^{-h(x_i)s})e^{-a_i s})ds*\;</math>characterized by:
This is directly comparable to Since <math>\frac{\partial \pi}{\partial a contest success function:} < 0\;</math> it follows that <math>x^*(n) > x^{**}(n)\;</math>.
:<math>pr(\tau(x_i) \le \min(\hat{\tau_i},t)) = \underbrace{\left( \frac{h(x_i)}{\sum_{i=1}^{n} h(x_i)} \right) }_{\mbox{Firm i relative effort}} \cdot \underbrace{ \left ( 1-e^{-\left(\sum_{i=1}^{n} h(x_i)\right)t}\right ) }_{\mbox{Prob of innov at t}}\;</math> ===Solution concept=== The model second inefficiency is not actually solved, but comparative statics can be performed on an implicit solutionthat there are too many firms. The implicit solution is arrived at by noting that:#If a firms expectations are rational then the beliefs about the fastest competing firm are indeed formed using <math>\hatoverline{\tau_ix}\;</math>#<math>a_i\;</math> can be taken as constant by firm <math>i\;</math> (i.e. in equilibrium <math>a_i\;</math> will be correct)#<math>V\;</math> and <math>r\;</math> are exogenously given#As the firms are identical we can look for a symmetric solution! Each firm maximizes profit: :<math>\max_x \Pi (a_i,x,V,rpoint where increasing returns to scale stop) = \max_x \left (\frac{V h(x_i)}{r(a_i + r +h (x_i))} - x \right)\;</math> This is presumably constructed by taking: :<math>\Pi = \int_0^{\infty} \left( \underbrace{pr(\tau_i \le \min(\hat{\tau_i},t)}_{\mbox{Prob of winning at t}} \cdot \underbrace{PV_t (V)}_{\mbox{PV of V at t}} \right ) dt - \underbrace{x}_{\mbox{cost}}\;</math> The FOC for zero then infinite firms enter the profit maximization implicitly defines the equilibrium solutioncompetitive race. :If <math>\frac{h'(\hatoverline{x})(a+r)}{(a+r+h(\hat{x}))^2} - \frac{r}{V} = > 0\;</math> The SOC must also hold (the paper has the first term missing) :<math>\frac{a+r}{(a+r+h(\hat{x}))^3} \cdot \left ( h''(\hat{x}) (a+r+h(\hat{x})) - 2h'(\hat{x})^2 \right) \le 0\;</math> Howeverfinite firms enter, this only defines the partial equilibrium. To complete the equilibrium we need but continue to use the symmetry (which is also why the subscripts enter until all profits are dropped above): :<math>a = \sum_{j \ne i} h(x_j) = (n-1)h(x^*)\;</math> This equilibrium exists providing R&D is profitable absent rivalry (otherwise their may be a corner, not an internal solution)dissipated.
Loury (1979) - Market Structure And Innovation (view source)
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|Has article title=Market Structure And Innovation
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*This page is referenced in [[PHDBA602 (Innovation Models)]]
==Reference(s)==
*Loury G.C.(1979), "Market structure and innovation", Quarterly Journal of Economics, 93, pp. 395-410. [http://www.edegan.com/pdfs/Loury%20(1979)%20-%20Market%20structure%20and%20innovation.pdf (pdf)]
@article{loury1979market, title==Abstract==In the application of conventional economic theory to the regulation of industry, there often arises a conflict between two great traditions. Adam Smith's "invisible hand" doctrine formalized in the First Fundamental Theorem of Welfare Economics supports the prescription that monopoly should be restrained and competitive market structures should be promoted. On the other hand, Schumpeter, in his classic Capitalism, Socialism and Democracy, takes a dynamic view of the economy in which momentary monopoly power is functional and is naturally eroded over time through entry, imitation, {Market structure and innovation. Indeed the possibility of acquiring monopoly power and associated quasi rents is necessary to provide entrepreneurs an incentive to pursue innovative activity. As Schumpeter put it}, progress occurs through a process of "creative destruction." An antitrust policy that actively promotes static competition is not obviously superior to laissez faire in such a world. This leads one to ponder what degree of competition within an industry leads to performance that is in some sense optimal. This question has been extensively studied in the literature concerning the relationship between industrial concentration and firm investment in research and development.' Both theoretical and empirical studies have suggested the existence of a degree of concentration intermediate between pure monopoly and atomistic (perfect) competition that is best in terms of R & D performance... ==The Model== author===Basic Setup and Assumptions=====Reference(s)==*{Loury , G.C.(1979), "Market structure and innovation"}, journal={The Quarterly Journal of Economics}, 93, pp. pages={395--410. [http://www.edegan.com/pdfs/Loury%20(}, year={1979)%20-%20Market%20structure%20and%20innovation.pdf (pdf)]}, publisher={JSTOR} }
==Abstract==
*<math>h(0) = 0 = \lim_{x \to \infty} h'(x)\;</math>
*For some <math>\overline{x} \ge 0\;</math>, <math>h''(x) \ge 0\;</math> for <math>x \le \overline{x}\;</math>, and <math>h''(x) \le 0\;</math> for <math>x \ge \overline{x}\;</math>. This says that h is weakly convex prior to some point (possibly zero, so never convex) and concave after that point. If the point is away from zero then there is an initial range of increasing returns to scale, but after the point there is always diminishing returns to scale.
*<math>\tilde{x}\;</math> is defined as the point where <math>\frac{h(x)}{x}\;</math> is greatest- this is the point where a firm is using its full capactity.
:<math>a_i= \sum_{j\ne i} h(x_j)\; </math>
The firm discounts the future reciepts at a rate <math>r\;</math> (note that using continuous compounding, <math>PV = FV \cdot e^{-rt})\;</math>, but apparently this paper values the future price at <math>\frac{V}{r}\;</math>).
The firm wins the prize at time <math>t\;</math> with probability:
:<math>\Pi(a,x) = \frac{h(x^*)}{h'(x^*)} \left ( \frac{a+r+h(x^*)}{(a+r)} \right ) - x^* \quad \mbox{where}\; a = (n-1)h(x^*)\;</math>
Now if <math>h(x_i)\;</math> is assumed concave (i.e. diminishing returns to have the following properties:*scale throughout) then <math>h(0) = 0 = \lim_frac{x \to \infty} h'(x)\;</math>*For some <math>\overline}{x} \ge 0\;</math>, <math>h''(x) \ge 0\;</math> for <math>x \le \overline{x}\;</math>, and <math>h''(x) \le 0\;</math> for <math>x \ge \overline{x}\;</math>*<math>\tilde{x}\;</math> is defined as expected profits are always positive. They are only driven to zero in the point where <math>\frac{h(x)}{x}\;</math> is greatestlimit of an infinite number of firms.
With an initial range of increasing returns to scale then returns can go to zero with a finite number of firms. To see this we examine the change in profit with respect the number of firms, remembering that the expenditure each firm will make will depend upon the total number of competitors.
:<math>\therefore pr(frac{\tau(x_i) partial \le \min(\hatpi}{\tau_ipartial x},t) = \frac{h(x_i)}{a_i + h(x_in-1)} (1-e^{-(a_i+h(x_ix),x)t})= 0\;</math>
