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Setting the Budget for Targeted Research Projects

Alessandro De Chiara ORCID logo and Elisabetta Iossa

Abstract

We consider a funding competition for targeted projects. Potential participants have stochastic opportunity costs, and do not know the number of competitors. The funding agency sets a budget cap indicating the maximum funding that participants may request. We show that raising the budget cap helps to attract more participants but causes an increase in the requested funds. A higher budget cap is optimal when the preferences of researchers and the funding agency are more congruent, competition is lower, targeted projects have larger social value, the cost of public funds is smaller, or bidding preparation costs are lower.

JEL Classification: D8; O25; O30; O31; O38; L2

Corresponding author: Alessandro De Chiara, Universitat de Barcelona and Barcelona Economic Analysis Team (BEAT), Avinguda Diagonal 696, 08034, Barcelona, Spain; and Central European University, 9 Nádor utca, Budapest1051, Hungary, E-mail:

Funding source: Italian Ministry of Education

Award Identifier / Grant number: PRIN 2017 Y5PJ43-001

Acknowledgements

For useful comments, we wish to thank Gianni De Fraja, Ester Manna, Markus Reisinger, and Emanuele Tarantino. An earlier version of this paper was circulated under the title: “How to set budget caps for competitive grants”. We gratefully acknowledge financial support from the Italian Ministry of Higher Education, and Research, PRIN 2017, Grant n. 2017Y5PJ43_001.

Appendix

Proof of proposition 1

First, the agency must set RπH if it wants to induce participation of agents in state (AB); in fact, a budget cap greater than πH is suboptimal as it would generate an (extra) rent for agents in state (A) and (AB) without effects on their participation. When R = πH, agents in state (AB) participate and submit a price bid t = πH. In the rest of the proof, we focus on the price-bidding behavior of agents in state (A) and we restrict attention to symmetric equilibria where each agent in state (A) follows the same bidding strategy.

Agents in (A) will never submit bids below k, for if they win with such a bid, they will not be willing to implement the project. In addition, they will not submit bids above πH or their bids will be discarded. Furthermore, there is no equilibrium in which all agents submit t = k as they would get no surplus whereas an agent could secure a payoff (1PA)n1πHϵk>0 by submitting a bid arbitrarily close to πH. They will not submit bids equal to πH, either, as they would also face the competition of agents in (AB), which they could avoid by undercutting them even slightly by bidding t = πHϵ, with ϵ > 0.

We now argue that there is no symmetric equilibrium in which all agents in (A) submit the same bid t̃(k,πH). With a bid equal to t̃ agent i in state (A) expects to get:

Πi(t̃,t̃|A)=j=0n11j+1n1jPAj(1PA)n1jt̃k.

An agent in (A) could profitably deviate by submitting t̃ϵ, with t̃k>ϵ>0, as he would obtain t̃ϵk>0 with probability one. We now claim that there cannot be point masses in the equilibrium bid strategy. Suppose that an agent bids t ∈ (k, πH) with some positive probability. Akin to the argument put forward above, another agent could then gain by placing zero weight on t and positive weight on tϵ. Moreover, the agents randomize over a connected support. Let f(t) be the probability that an agent in (A) submits a bid t. It cannot be that f(t̂)=0 for t̂(t1,t2), with t2 > t1 and f(t1) > 0, f(t2) > 0. In each instance in which a bid t1 wins, also t̂ wins but yields a strictly higher payoff. Therefore, t1 would not be part of the equilibrium strategy and f(t1) = 0. Hence, the equilibrium in mixed strategy is described by the cumulative distribution function of price bids F(t). To pin down this c.d.f. we need the expected payoff of an agent i who is in state (A) when other agents adhere to the same strategy. The probability that exactly n − 1 − j agents are in state (A) is:

n1jPAn1j(1PA)j.

In this instance, i’s expected payoff if he bids t ∈ (k, πH) is given by tk times the probability that each of the n − 1 − j competitors submit more than t, i.e. [1F(t)]n1j(tk). Agent i’s expected payoff from submitting t is

Πi(t,F(t)|A)=j=0n1n1j(1PA)j[PA(1F(t))]n1j(tk)=[1PAF(t)]n1tk.

Agent i must obtain the same expected payoff from each bid in which f(t) > 0, or else it would be better off submitting those bids associated with a higher expected payoff. To find the equilibrium bids, we take the derivative of the above expected payoff with respect to t and set it equal to zero:

[1PAF(t)]=(n1)PAf(t)tk.

This is a differential equation whose solution yields the expression reported in the statement of the proposition. To see this, rewrite the above expression as:

PAf(t)1PAF(t)=1(n1)tk.

Adopting this change of variables, y = F(t) and dy = f(t)dt, and integrating:

y0yPAdy1PAy=1(n1)t0t1tkdtln1PAyln1PAy0=1(n1)ln(t0k)ln(tk).

Because at t0 = πH, it holds that y0 = 1,

ln[1PAy]=ln(1PA)πHktk1n1,

from which it is immediate to recover the expression reported in the proposition once variable y is transformed back into F(t) and the solution is unique. QED

Proof of corollary 1

Take the derivative of the right hand side of condition (6):

nPA(1PA)n1(1PA)nPBnn=PA1PAn1[1PAnPBn]2HPB,PAB0,

where

HPB,PABPB+PABnPBnPBnnlnPB+PABlnPB.

The above expression is zero for PAB = 0 and it is increasing in PAB, as:

HPB,PABPAB=nPAB+PBnPBnPAB+PB0.

Therefore the derivative of the right hand side of Condition (6) is nonnegative, which implies that the condition in Proposition 2 is more difficult to satisfy the greater is n. QED

Proof of proposition 3

We begin by recovering Fc(t). The agency can set Rk + c such that an agent in (A) wins if and only he is the only one in state (A) and gets no rent:

(1PA)n1(Rk)c=0.

From this we recover the budget cap and the endpoint of the equilibrium bids. The cumulative distribution function of the bids by agents in (A) is obtained following the same procedure as in Proposition 1. It is easy to see that no agent has incentive to deviate. To obtain the expected winning bid, that we call tA′, consider that the sum of the expected payoff of all agents in (A) is:

ΠR=k+c(1PA)n1,c=[1(1PA)n](tAk)nPAc=0,

where 1(1PA)n is the probability that at least one agent is in (A). Therefore,

tA=k+nPAc1(1PA)n.

The agency’s utility is:

(A1)UR=k+c(1PA)n1,c=[1(1PA)n][uHtA(1+λ)]=[1(1PA)n][uHk(1+λ)]nPAc(1+λ).

Suppose that the agency induces agents in state (AB) to participate. We have already shown that R=πH+cPBN1. The other endpoint of the interval over which agents in state (AB) bid is obtained from this equation:

(1PA)n1(xπH)c=0.

That is, by bidding x=πH+c(1PA)n1, an agent in state (AB) would undercut all the other agents in (AB) but none in (A). To determine FABc(t) we follow the same approach as in the Proof of Proposition 1. See that an agent in (AB) who submits a bid tπH+c(1PA)n1,πH+cPBn1 gets tπH if he wins, which occurs if his competitors are either in state (B) or in (AB) but submit a bid higher than t. Thus, agent i’s expected payoff from submitting such t is:

Πi(t,FABc(t)|AB)=j=0n1n1jPBj[PAB(1FABc(t))]n1j(tπH)c=[1PAPABFABc(t)]n1tπHc.

To find the equilibrium bids, we take the derivative of the above expected payoff with respect to t and set it equal to zero:

[1PAPABFABc(t)]=(n1)PABfABc(t)tπH.

The above equation can be rewritten as:

PABfABc(t)1PAPABFABc(t)=1(n1)tπH.

Adopting this change of variables: y=FABc(t) and dy = fAB(t)dt, and integrating

y0yPABdy1PAPABy=1(n1)t0t1tπHdtln1PAPAByln1PAPABy0=1(n1)ln(t0πH)ln(tπH).

Because y0 = 1 at t0=πH+cPBn1, and PB = 1 − PAPAB, we have:

ln[1PAPABy]=lnPBcPBn1(tπH)1n1.

We then transform y back into FABc(t).

Consider now the bidding behavior of agents in state (A). By bidding t=πH+c(1PA)n1 an agent in (A) expects to get:

(1PA)n1πH+c(1PA)n1kc=(1PA)n1(πHk).

The left endpoint of the interval, y, is a bid which wins with probability 1 and gives an expected payoff equal to (1PA)n1(πHk); it therefore solves:

(yk)c=(1PA)n1(πHk)y=k+c+(1PA)n1(πHk).

The cumulative distribution function of the bids over the interval is obtained following the same steps as in Proposition 1.

To check whether these are equilibrium bidding strategies, let us see whether an agent would have an incentive to deviate. If an agent in state (AB) bids below πH+c(1PA)n1, that is, say, he submits some bid

tk+c+(1PA)n1(πHk),πH+c(1PA)n1,

then he wins with probability p(t′) and he obtains an expected payoff equal to:

p(t)(tπH)c=p(t)(tk)cp(t)(πHk)=(1PA)n1(πHk)p(t)(πHk)<0.

To see why this is negative, notice that p(t)(tk)c=(1PA)n1(πHk) is the expected payoff of an agent in state (A) who submits bid t′ and that p(t)>(1PA)n1, where the latter is the probability of winning if an agent bids πH+c(1PA)n1. Similarly, let us show that an agent in (A) does not want to bid above πH+c(1PA)n1. Specifically, suppose that such an agent submits a bid tπH+c(1PA)n1,πH+cPBn1, and let p(t″) denote the associated probability of winning. The agent in state (A) would get:

p(t)(tk)c=p(t)(tπH)c+p(t)(πHk)=p(t)(πHk)<(1PA)n1(πHk).

To understand why, consider that p(t″)(t″ − πH) − c = 0 because agents in state (AB) get no rent and p(t)<(1PA)n1 because the latter is the probability of winning with a bid πH+c(1PA)n1<t. It is also straightforward to show that an agent would not bid above the budget cap or otherwise his bid would be discarded, nor below k+c+(1PA)n1(πHk) as it would decrease the payoff conditional on winning without affecting the probability of submitting the lowest bid.

The expected utility of the agency is:

(A2)UR=πH+cPBn1,c=1PBnuH1PAnPBnπH+11PAn×tA(1+λ)n(1PB)c(1+λ).

Notice that the agency’s new utility functions, (A1) and (A2), differ from the ones in the baseline model (1), and (5), only by the expected participation costs. As n(1 − PB)c(1 + λ) > nPAc(1 + λ), the scope for setting a high budget cap decreases with c. QED

References

Aghion, P., M. Dewatripont, and J. C. Stein. 2008. “Academic Freedom, Private-Sector Focus, and the Process of Innovation.” Rand Journal of Economics 39: 617–35. https://doi.org/10.1111/j.1756-2171.2008.00031.x. Search in Google Scholar

Armstrong, M., and J. Vickers. 2010. “A Model of Delegated Project Choice.” Econometrica 78 (1): 213–44. Search in Google Scholar

Azoulay, P., J. S. G. Zivin, and G. Manso. 2011. “Incentives and Creativity: Evidence from the Academic Life Sciences.” The RAND Journal of Economics 42 (3): 527–54. https://doi.org/10.1111/j.1756-2171.2011.00140.x. Search in Google Scholar

Azoulay, P., J. S. G. Zivin, D. Li, and B. N. Sampat. 2019. “Public R&D Investments and Private Sector Patenting: Evidence from NIH Funding Rules.” The Review of Economic Studies 86 (1): 117–52. https://doi.org/10.1093/restud/rdy034. Search in Google Scholar

Banal-Estañol, A., I. Macho-Stadler, and D.-P. Castrillo. 2019. “Evaluation in Research Funding Agencies: Are Structurally Diverse Teams Biased against?” Research Policy 48 (7): 1823–40. Search in Google Scholar

Baye, M. R., and J. Morgan. 2001. “Information Gatekeepers on the Internet and the Competitiveness of Homogeneous Product Markets.” The American Economic Review 91 (3): 454–74. https://doi.org/10.1257/aer.91.3.454. Search in Google Scholar

Baye, M. R., D. Kovenock, and C. G. De Vries. 1992. “It Takes Two to Tango: Equilibria in a Model of Sales.” Games and Economic Behavior 4 (4): 493–510. https://doi.org/10.1016/0899-8256(92)90033-o. Search in Google Scholar

Beaudry, C., and S. Alloui. 2012. “Impact of Public and Private Research Funding on Scientific Production: The Case of Nanotechnology.” Research Policy 41 (9): 1589–606. https://doi.org/10.1016/j.respol.2012.03.022. Search in Google Scholar

Berkovitch, E., and R. Israel. 2004. “Why the NPV Criterion Does Not Maximize NPV.” Review of Financial Studies 17 (1): 239–55. https://doi.org/10.1093/rfs/hhg023. Search in Google Scholar

Bhattacharya, V. 2018. An Empirical Model of R&D Procurement Contests: An Analysis of the DOD SBIR Program. Mimeo, Department of Economics, Northwestern University. Search in Google Scholar

Bloch, C., and M. Sørensen. 2015. “The Size of Research Funding: Trends and Implications.” Science and Public Policy 42: 30–43. https://doi.org/10.1093/scipol/scu019. Search in Google Scholar

Bucciol, A., R. Camboni, and P. Valbonesi. 2020. “Purchasing Medical Devices: The Role of Buyers Competence and Discretion.” Journal of Health Economics 74: 102370. https://doi.org/10.1016/j.jhealeco.2020.102370. Search in Google Scholar

Che, Y.-K., and I. Gale. 2003. “Optimal Design of Research Contests.” The American Economic Review 93 (3): 646–71. https://doi.org/10.1257/000282803322157025. Search in Google Scholar

Che, Y.-K., E. Iossa, and P. Rey. 2020. “Prizes versus Contracts as Incentives for Innovation.” The Review of Economic Studies. https://doi.org/10.1093/restud/rdaa092. Search in Google Scholar

De Chiara, A., and E. Iossa. 2019. “Innovation Policy: Procurement vs Grants.” In C.E.P.R. Discussion Paper 13664. Search in Google Scholar

De Fraja, G. 2016. “Optimal Public Funding for Research: a Theoretical Analysis.” The RAND Journal of Economics 47 (3): 498–528. https://doi.org/10.1111/1756-2171.12135. Search in Google Scholar

Decarolis, F., L. M. Giuffrida, E. Iossa, V. Mollisi, and G. Spagnolo. 2020. “Bureaucratic Competence and Procurement Outcomes.” Journal of Law, Economics, and Organization 36 (3): 537–97. https://doi.org/10.1093/jleo/ewaa004. Search in Google Scholar

Decarolis, F., G. de Rassenfosse, L. M. Giuffrida, E. Iossa, V. Mollisi, E. Raitieri, and G. Spagnolo. 2021. “Buyers’ Role in Innovation Procurement: Evidence from U.S. Military R&D Contracts.” Journal of Economics and Management Strategy. forthcoming. Search in Google Scholar

European Commission (EC). 2017. HORIZON 2020 in Full Swing: Three Years on. Key Facts and Figures 2014-2016. https://ec.europa.eu/. Search in Google Scholar

Fedderke, J. W., and M. Goldschmidt. 2015. “Does Massive Funding Support of Researchers Work?: Evaluating the Impact of the South African Research Chair Funding Initiative.” Research Policy 44 (2): 467–82. https://doi.org/10.1016/j.respol.2014.09.009. Search in Google Scholar

Fu, Q., and J. Lu. 2010. “Contest Design and Optimal Endogenous Entry.” Economic Inquiry 48 (1): 80–8. https://doi.org/10.1111/j.1465-7295.2008.00135.x. Search in Google Scholar

Gavious, A., B. Moldovanu, and A. Sela. 2002. “Bid Costs and Endogenous Bid Caps.” The RAND Journal of Economics 33 (4): 709–22. https://doi.org/10.2307/3087482. Search in Google Scholar

Hicks, D. 2012. “Performance-Based University Research Funding Systems.” Research Policy 41 (2): 251–61. https://doi.org/10.1016/j.respol.2011.09.007. Search in Google Scholar

Janssen, M., and E. Rasmusen. 2002. “Bertrand Competition under Uncertainty.” The Journal of Industrial Economics 50: 11–21. Search in Google Scholar

Levin, D., and J. L. Smith. 1994. “Equilibrium in Auctions with Entry.” The American Economic Review 84 (3): 585–99. Search in Google Scholar

Lu, J. 2009. “Auction Design with Opportunity Cost.” Economic Theory 38: 73–103. https://doi.org/10.1007/s00199-008-0331-2. Search in Google Scholar

Menezes, F. M., and P. K. Monteiro. 2000. “Auctions with Endogenous Participation.” Review of Economic Design 5 (1): 71–89. https://doi.org/10.1007/s100580050048. Search in Google Scholar

Milgrom, P., and R. Weber. 1982. “A Theory of Auctions and Competitive Bidding.” Econometrica 50: 1089–11. https://doi.org/10.2307/1911865. Search in Google Scholar

Moreno, D., and J. Wooders. 2011. “Auctions with Heterogeneous Entry Costs.” The RAND Journal of Economics 42 (2): 313–36. https://doi.org/10.1111/j.1756-2171.2011.00135.x. Search in Google Scholar

Myers, K. 2020. “The Elasticity of Science.” American Economic Journal: Applied Economics 12 (4): 103–34. Search in Google Scholar

Myerson, R. B. 1981. “Optimal Auction Design.” Mathematical Operation Research 6: 58–73. https://doi.org/10.1287/moor.6.1.58. Search in Google Scholar

National Science Foundation (NSF). 2019. Science and Engineering Indicators 2018. National Science Foundation. Search in Google Scholar

OECD. 2011. Demand Side Innovation Policy. Paris: Organization for Economic Cooperation and Development. Search in Google Scholar

Taylor, C. R. 1995. “Digging for Golden Carrots: An Analysis of Research Tournaments.” The American Economic Review 85 (4): 872–90. Search in Google Scholar

Varian, H. R. 1980. “A Model of Sales.” The American Economic Review 70 (4): 651–9. Search in Google Scholar

Viner, N., P. Powell, and R. Green. 2014. “Institutionalized Biases in the Award of Research Grants: A Preliminary Analysis Revisiting the Principle of Accumulative Advantage.” Research Policy 33 (3): 443–54. Search in Google Scholar

Received: 2020-10-10
Revised: 2021-02-22
Accepted: 2021-05-20
Published Online: 2021-06-07

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