Costly R&D in a model of competitive equilibrium in which technology is completely non-excludable

Recently the academic discussion of whether R&D can occur if competition is perfect has spilled over into the blogosphere. The standard view is that no one will pay the cost of R&D if they don't obtain some market power from trade secrets or patents or something as a result. This is not a new view -- the only regulatory power specifically and explicitly granted to Congress in the US constitution is the ability to award patents and copyrights to encourage innovation.

The aim of this silly note is to describe an exception to this general rule. The point, if any, is to respond to the idea that we know something based on economic theory with the claim that the conclusion follows from assumptions beyond the standard core assumptions of rational utility maximization. I don't think the alleged example presented here is of any relevance to the real world -- the exercize is purely mathematical.

The idea is that investing in R&D may be rational even if the resulting technology is completely non-excludable if the technology complements a rival factor of production owned by the agent who invests in R&D. The math is simple -- the challenge is to decide if I have done violence to the meanings of "R&D", "technology" and "non-excludable".

This post is a sin. For my sins I am trying to post a note about less than, greater than etc as html. Blogger was very unhappy. Anyway, the reader is warned it is pointless (and will be warned again when it gets pointlesser).

Is it possible for rational agents to engage in costly research and development even though the resulting technology is non excludable so there can be no revenue generating ownership of it ? The answer is obviouisly yes -- rational altruistic agents may do so in order to help humanity as a whole, rational highly curious agents may do so for their own amusement. It is not obvious that rational agents who care only about consumption (or consumption and leisure) might conceivably do so.

I aim to present a model in which agents care only about consumption, in which technology is developed through labor which could otherwise be used to earn wages, in which the technology is freely available to everyone so the inventor gains nothing from the fact that she rather than someone else is the inventor, in which agents are rational utility maximizers, and in which technology advances and output grows without bound. The incentive for the production of non rival technology is that is is a complement for the inventors endowement of rival laboring capacity.

In the model there is one consumption good which is produced using labor and technology -- there is no capital. There is a continuum of workers each of which has a unique type of ability and there is a continuum of technologies. There is one unit of workers indexed by i which is a real number in the interval [0,1]. Each worker is endowed with L units of labor which she supplies inelastically so labor supplied L_i=L. Output of the homogenous non durable consumption good Y is given by equation 1

Y = integral from i=0 to 1 of L_iA_i di = L integral from i =0 to 1 of A_i di

This means that technology i can be used only by worker i. A_i is labor augmenting technology which augments only labor of worker i. It is not necessary that worker i produces exactly zero using technology j !=i . So far it is enough that worker i produces more with technology i than with technology j . Later it will be necessary that this difference is large enough and in any case I will assume that worker i can produce the good only using technology i.

The workers may be self employed or employed by competitive firms. It doesn't matter. In either case they get income w_i = A_i L. I am going to assume that L = 1 so w_i = A_i. I introduced L then set it to one for some reason which I forget. The workers can do three things with this income. They can consume it. They can, in theory, lend it to other workers, although in equilibrium all lend nothing and borrow nothing. Finally they can take it home and feed it into an R&D machine which causes some technology A_j to increase. I tell this crazy story to simplify notation. The model works the same if workers employ part of their time producing the good and part of their time researching and developing so long as the productivity of worker i as a researcher and developer is proportional to A_it like her productivity as a goods producer.

I assume worker i dedicates di to R&D. In theory worker i could devote resources to improving some technology other than technology i. Obviously they don't do this -- it would have no effect on their income ever and they care about nothing other than consumption. So by optimal choice d_it is devoted to improving technology i. The effect is

dA_it/dt = dit/eta

so eta is the cost in terms of the good of one unit of technology.

I guess that workers neither borrow nor lend so C_it = A_it - d_it , nonetheless I define a market real interest rate r_t such that they all choose this optimally.

R_t = integral from t=0 to 1 of r_t

Worker i chooses d_it and,in principal C_it to maximize

integral from t=0 to infinity of e^{-rho t} ln(C_it) dt

subject to

integral t = 0 to infinity e_^{-R_t} (C_it + d_it - A_it) dt less than or equal to 0

and

dA_t/dt = d_it/eta for all t.

It is fairly clear (as I will guess and check) that r_t = 1/eta -- in exchange for eta units of consumption good now, worker i gets a flow of one unit of income forever.

so dln(C_it)/dt = 1/eta-rho

dln(A_it)/dt = d_it/(eta A_it)

if d_it/A_it = (1-rho eta)

then dLn(A_it/dt) = dln(C_it)/dt = 1/eta-rho.

If c_it = rho eta A_it then the budget constraint trivially holds with equality and c_it grows at rate 1/eta-rho so the Euler equation holds. Finally for any i and j the ratio C_it/C_jt is constant so agents i and j can't obtain any gains from lending one to the other.

Therefore a solution to the model is for all i and t

c_it = rho eta A_it

and d_it = (1-rho eta)A_it

and there is balanced growth at rate 1/eta-rho.

It is reasonably obvious that this solution is unique.

I think this is a case of costly R&D investment in completely non-excludable technology and a model in which growth results from costly R&D even though there is perfect competition.

One might argue that the technology A_i is excludable, because it can be used only by worker i. This is not true, a firm can be set up which uses technology A_i and hires worker i and also uses other technologies and hires other workers. This makes no difference at all, since the firm will pay exactly all of the good produced by worker i to worker i. With perfect competition and constant returns to scale it doesn't matter how production is divided between firms -- each worker can be self employed or each can work for a firm which employs many workers.

Worker i has no market power. the reason is that effective labor of type i = A_iL_i is a perfect substitute for effective labor of any other type A_jL_j . If worker i threatens to quit, she doesn't scare her employer which can costlessly switch to using another technology and hire another worker. Since technology is non rival and non excludable, workers are perfect substitutes. I think that one would be foolish to argue that technology A_i is excludable, because it can be used only by worker i. However, I will present an absolutely pointless model to illustrate this claim.

------------------------------- from here on the note is even morepointless than it was above------------------------------------

Here is a totally pointless model in which a given technology augments the labor of N=10 workers not just one. It works equally well for any finite N including a trillion. The analysis is similar to that of the model I just described above with a bit more irritating notation and a continuum of identical inequality constraints to check.

There is a continuum of agents of measure 10 indexed by i which goes from 0 to 10. Agents with iless than1 inelastically supply a total of 2 units of labor, while agens with i in [1,10] supply a total of 9 units of labor. The goods may be consumed or used to finance research and development -- of course a more sensible model would have some labor devoted to goods production and some to R&D -- the odd assumption is made just to simplify notation.

there is a continuum of measure of different technologies A_j where j goes from 0 to 1. Each agent has the ability to use one and only one technology. This is a strange assumption which is key to the results. Obviously it is not realistic. Each technology can be used by 10 different agents. If i'-i is an integer, then agent i and agent i' use the same technology. define j(i)= i-floor(i).

Goods are produced by perfectly competitive firms and total output at time t Y_t is given by Yt = (integral from i = 0 to 1 A_it di)11 =

integral from 0 to 1 (A_it di) + integral from i = 0 to 10 A_{j(i) t} di

Total spending on R&D at time t = D_t = intergral from j=0 to 1 (integral from i=0 to 10 D_ijt di dj where D_ijt is the rate of investment by agent i in technology j at time t

Total consumption C_t = Y_t-D_t

the consumption of agent i at time t is C_it

Note that firms do not invest in R&D. They pay all of their revenues to their employees no matter what and so have profits zero whether they have advanced technology or not. Workers do invest in R&D. In the model they feed consumption good into the R&D sausage grinder and technology comes out. In an only slightly more realistic interpretation (which is identical except for notation) they spend some of their time working for a wage and some conducting household production of technology through household based research and development.

Clearly agent i invests in further development of technology j only if j= i - floor(i). define d_it = D_{i (i-floor(i) t} which is obviously equal to all the investment in R&D by agent i. It should also be fairly obvious that in equilibrium only agents with iless than1 invest in research and development. the reason for the horrible assumption about the double endowement of labor of these agents is to make this equilibrium unique. This is, in principle, a result not an assumption so formally

d A_j/ d_t = (sum k = 0 to 9 d_{j+k t})/eta

where eta is the usual cost of R&D parameter.

r_t is the instantaneous real interest rate at time t. R_t = integral s = 0 to t r_t ds

Agent i chooses C_it and d_it to maximize

integral t=0 to infinity e^-(rho t)ln(C_it)dt

subject to integral t= 0 to infinity e^-(R_t)(((C_it+d_it) - 2A_it)dt less than or equal to 0 if i less than 1

integral t= 0 to infinity e^-(R_t)(((C_it+d_it) - A_it)dt less than or equal to 0 if i greater than or equal to 1

and d A_j/ d_t = (sum k = 0 to 9 d_{j+k t})/eta for every t.

In equilibrium agents with i less than1 are indifferent between investing and R&D and consuming. Clearly agents with igreater than or equal to1 strictly prefer to consume so d_i=0 if i greater than or equal to 1

The two problems become the very boring and standard

if i greater than or equal to 1

agent i choose C_it to maximize

integral t=0 to infinity e^-(rho t)ln(C_it)dt

subject to

integral t= 0 to infinity e^-(R_t)(C_it - A_it)dt less than or equal to 0

and if i less than 1

Agent i chooses C_it and d_it to maximize

integral t=0 to infinity e^-(rho t)ln(C_it)dt

subject to

integral t= 0 to infinity e^-(R_t)(((C_it+d_it) - 2A_it)dt less than or equal to 0 and

d A_j/ d_t = d_{i t})/eta

A solution ( I think the solution) is a balanced growth path in which

D_t/C_t is a constant

lets see what happens if it is 0.1 . Can I find an eta so that D_t/C_t = 0.1 if agents make optimal consumption/saving/R&D investment choices ?

d_it = A_it if i less than 1

D A_it/dt = A_it/eta

if r_t = rho+1/eta then growth is balanced.

OK first order conditions. Agents with iless than1 must be indifferent between consumption and R&D while agents with igreater than1 must be indifferent or strictly prefer consumption to R&D (so dit=0 is a corner solution for igreater than or equal to1) . I will check this last.

Also, there are consumption saving choices for both agents with iless than1 and igreater than or equal to1 so both must have consumption grow at the same rate (all will have saving equal to zero in equilibrium, but I must check that all choose neither a borrower nor a lender to be). This is satisfied if d_it/C_it is constant for every i. For any constant r_t = r that constant can be chosen so A_it grows at the same rate as C_it which means that d_it/C_it is constant and must be the same constant for every iless than1 (it is zero for every igreater than1). So things look OK so far so long as r_t is constant and d_it/C_it is the same for every i and for every t. This means that Ajt grows at the same constant rate for every j and t.

Now I need to check the choice between consumption and R&D. I know agents are indifferent between consumption and saving (and all save exactly zero) so I need that the return on an investment in R&D = r.

For agents with iless than1 the return is 2/eta for one unit of investment in R&D in technology i, the agent has a flow labor income higher by 2/eta per unit of time. So r = 2/eta. Oh it is constant. For agents with igreater than or equal to0 the return is 1/etaless thanr so they don't invest in R&D.

In the guess that dit/Cit = 1 for iless than1, I need

r-rho = 1/eta and r= 2/eta so 1/eta = rho.

more generally I get the rate of growth of consumption C_it is 2/eta-rho (for all i and t) so the rate of growth of each A_j must be 2/eta-rho so,

for i less than1, d_it/A_it = eta(2/eta-rho) = 2-eta*rho

they neither borrow nor save so C_it/A_it = (w_it-d_it)/A_it = 2 - (2-(eta*rho)) = eta*rho. I get balanced growth at rate 2/eta-rho so long as C_it = w_it for i greater than or equal to 1 and C_it = eta*rho*A_it for i less than1, that is C_it = (eta*rho/2)w_it for i less than 1.

OK so I think the model solves out for a constant growth rate = 2-eta*rho for any rho and eta.