informational externalities, strategic delay, and optimal investment subsidies

Informational externalities, strategic delay,and optimal investment subsidies

Matthew Doyle Department of Economics, Universityof Waterloo

Abstract. This paper examines optimal government policy when private investment gen-erates information, but investors cannot internalize the informational value their actionshave to others. Equilibrium exhibits inefficient delay, as investors adopt a wait-and-see ap-proach. The government can alter incentives via an investment subsidy, but complicationsarise, since future subsidies may induce investors to disregard current policy initiatives.The paper shows that the government achieves its desired outcome only when the theinvestment subsidy is financed by a non-distortionary, lump-sum tax. When taxationis distortionary, the government faces a time inconsistency problem that may preventeffective policy. JEL classification: D83, H2

Externalites informationnelles, delai strategique et subventions optimales a l’investissement.Ce texte examine la politique optimale du gouvernement quand l’investissement priveengendre de l’information, mais que les investisseurs ne peuvent pas internaliser la valeurinformationnelle que leurs actions ont sur les autres. L’equilibre est marque par des delaisinefficaces a proportion que les investisseurs adoptent une attitude attendre-pour-voir. Legouvernement peut modifier les incitations en creant une subvention a l’investissement,mais des complications sont possibles puisque des subventions possibles dans l’avenirpeuvent amener les investisseurs a ne pas se prevaloir des initiatives mises de l’avantpar la politique courante. Le texte montre que le gouvernement atteint le resultat desireseulement quand la subvention a l’investissement est finance par une taxe forfaitairenon distorsionnante. Quand la taxe cree des distorsions, le gouvernement fait face a unprobleme d’inconsistance dans le temps qui empeche que la politique soit effective.

I wish to thank Paul Beaudry, Francisco Gonzalez, Parikshit Ghosh, Mick Devereux, GuofuTan, John Leahy, and seminar participants at the University of British Columbia. Financialsupport from the SSHRCC is gratefully acknowledged. Email: [email protected]

Canadian Journal of Economics / Revue canadienne d’Economique, Vol. 43, No. 3August / aout 2010. Printed in Canada / Imprime au Canada

0008-4085 / 10 / 941–966 / C© Canadian Economics Association

942 M. Doyle

1. Introduction

Economic decision makers generally face a great deal of uncertainty when mak-ing choices. One way of reducing this uncertainty is to try to learn by observingthe actions of others. When decision makers can learn by observing the ac-tions of others, however, they may disregard their own information in favourof imitating others. In such cases informational cascades, or imitative herds inwhich much private information is lost, may arise.1 Recent research suggests thatinformational cascades may play a role in explaining a variety of economic phe-nomena, including financial panics (Chari and Kehoe 2003; Chari and Kehoe2004; Chamley 2003) technology adoption, and business cycles (Schivardi 2003).

The outcomes of models of informational cascades are generally inefficient.The ability to learn by observing others creates an informational externalitywhereby a private decision maker cannot capture the value to others of the infor-mation revealed by his or her actions. As a result, inefficiently little information isrevealed in equilibrium. This inefficiency-implies a role for efficiency-enhancinggovernment intervention. Given the apparently widespread presence of informa-tional cascades in the economy, understanding the problems facing policy makersoperating in such environments is an important research question.

This paper studies optimal government policy in an economy where infor-mation externalities create inefficient delay. Relative to the existing literature onoptimal policy and informational externalities, the contribution of the paper isto allow decision makers to choose the timing of their actions. Endogenous tim-ing of actions complicates the policy maker’s problem, since, when agents canchoose when to act, today’s policy intervention has to compete with future policyofferings.2 The extent to which this effect constrains the policy maker’s ability toinduce efficient outcomes is unclear a priori.

The environment is a version of the Chamley and Gale (1994) model ofinformation revelation and inefficient strategic delay, extended to the case whereinvestors possess idiosyncratic costs of starting a project. Potential investorspossess an option to invest in a project of uncertain profitability. The profitabilityof investment is correlated with the number of such potential investors. Thiscreates an incentive for all potential investors to adopt a ‘wait-and-see’ strategy, asobservations of others’ decisions reveal information about the likely profitabilityof one’s own investment opportunity. The government can alter incentives usinga simple investment subsidy.

To build intuition about the information structure, consider an economy inwhich the profitability of investment is generally high. In such an economy, the

1 Important early theoretical contributions include Banerjee (1992), Bikhchandani, Hirshleifer,and Welch (1992), and Chamley and Gale (1994). Examples of experimental work documentingthe existing of informationally driven herd behaviour include Anderson and Holt (1997), Celenand Kariv (2004), and Alevy, Haigh, and List (2007).

2 Caplin and Leahy (1996) argue that this mechanism explains the gradual nature of interest ratechanges undertaken by central banks.

Informational externalities, strategic delay 943

number of firms that think they see a profitable investment project will likely belarge. In contrast, in an economy in which the profitability of investment is likelyto be low, fewer firms are likely to find profitable-looking investment projects.An observer would be able to learn about the likely profitability of investmentby observing how many firms have investment projects lined up, where a largenumber of such firms would represent ‘good news.’ Since it is difficult to observepotential investors until they actually begin to invest, observing the number ofinvestment projects undertaken in the economy would reveal information aboutthe likely profitability of investment.

For simplicity, investors only interact informationally in this model, meaningthat there are no network or congestion effects associated with the investment ofothers. The model also has the attractive property that the equilibrium outcomein the absence of government intervention displays inefficient delay in everyperiod. As a result, the comparison between the equilibrium and the efficientoutcomes is relatively simple, as the socially optimal outcome always involves areduction of delay.3

The first result of the paper is that, in equilibrium, a policy maker withaccess to lump-sum, non-distortionary taxes is able to implement the sociallyoptimal investment profile. This is because the equilibrium subsidy is set so as tocompensate the investors not only for the informational value of early investment,but also for any future subsidies that they forgo by investing sooner. Because thestrategic behaviour of investors’ forces the policy maker to compensate investorsfor these forgone future subsidies, the optimal subsidy exceeds the subsidy thatwould be required were investors to behave myopically. The strategic behaviourof investors affects only the path of the policy instrument and does not preventthe policy maker from implementing the desired allocation of investment.

When subsidies are financed by distortionary taxation, these results change.Unsurprisingly, when the use of the policy instrument is costly, the policy makerno longer achieves the socially optimal investment profile as an equilibriumoutcome. More interestingly, the combination of distortionary taxation and thestrategic behaviour of investors creates a time inconsistency problem. Whensubsidies are costly, the policy maker would like to announce that subsidies inthe future will be low. This would enable the policy maker to increase investmenttoday at a fairly low cost, as there would be no incentive for investors to delayin hopes of enjoying future subsidies. However, an announcement of low orzero subsidies in the future will not be credible. When the future period arrives,the policy maker will optimally want to renege on the announced subsidies inorder to reduce delay in that period. Rational investors know this and ignore anyannouncement of low future subsidies made by the policy maker. This suggeststhat there will be gains to the policy maker from having access to a commitment

3 The equilibrium of the model is also unique, which avoids additional complications that wouldarise when studying policy making in environments exhibiting multiple equilibria, such as thoseof Chamley (2004), and Gossner and Melissas (2006).

944 M. Doyle

technology. This problem does not arise when the subsidy is costless to employ,because in that case the policy maker can reduce delay today in the presence ofhigh future subsidies by offering even higher subsidies today.

This time inconsistency problem means that policy intervention may actu-ally produce outcomes that are inferior to the no-intervention equilibrium. Thereason is that the inability to commit to low future subsidies means that thepolicy maker must pay an excessively high subsidy in the present, as part of to-day’s subsidy goes towards compensating investors for forgone future subsidies.Policy intervention can then be inferior to the no-intervention outcome, even ifdelay is reduced, because the policy maker overpays, in terms of excessive use ofdistortionary taxation, for the reduction in delay that is achieved.

Do firms delay investment so as to gather information by observing the ac-tions of others? Recent work finds evidence of this mechanism, at least in regardsto foreign direct investment. Using survey data on Japanese firms, Kinoshitaand Mody (2001) conclude that new investment in emerging economies is depen-dent in part on learning from earlier investors. Barry, Gorg, and Strobl (2003)find empirical support for the idea that information gleaned from previous in-vestors is an important determinant of the foreign direct investment decisionsof U.S. firms in Ireland. Country case studies also suggest that strategic delaycan be an important problem for developing countries. For example, Hausmann(2003), Hausmann and Rodrik (2005), and Hausmann, Rodrik, and Velasco(2005) present evidence suggesting that informational externalities and strategicdelay were important elements in the development of El Salvador.

There is ample evidence that governments use financial incentives to attractinvestment (examples include Noland and Pack 2003; Melo 2001; and Hanson2001). It is less clear to what extent these subsidies are motivated by concernsabout strategic delay of the type examined in this paper, but policy makers areaware of and concerned about the possibility that uncertainty causes investors todelay irreversible investments. The argument for the sort of intervention studiedin this paper goes back at least as far as Baldwin (1969), and the literature hasseen a recent renewal of interest in the problem (Rodrik 2005; Edwards and Alves2005; Nabli et al. 2005).

The time inconsistency problem highlighted by this paper could explain thecourse of the investment subsidy in East Germany following reunification. Con-sider this passage, from Thimann and Thum (1998):4

Initially [the] investment grant was planned to last for only 12 months until June1991 at 12 percent and then to decrease to 8 percent for another 12 months. How-ever, by the end of 1992 it had already been prolonged twice. Later, the Germangovernment even agreed upon raising the measure to a level of 20 percent, bowingto pressures arising from sluggish investment in the East . . . Investors who expected

4 Thimann and Thum report that another financial incentive to investors, accelerateddepreciation, suffered from the same problem. Intended as a temporary measure to apply onlyto goods bought in 1990, it was extended twice, until 1998.


the government to react to small investment were well advised to wait in order tocollect both the information from earlier investors and the (increased) governmentsupport.5

This suggests that time inconsistency may have been a problem. First, the pathof subsidies was increasing over time, which is likely to deter rather than encour-age early investment. The initial policy offering was intended to be temporary,precisely to entice investors to act earlier, but the government could not main-tain this policy in the face of a slow response by investors. Second, the subsidyappears to have been excessively large. Sinn (1995) estimates that the investmentsubsidies were so generous as to make the cost of capital negative for many typesof investment.

This paper is most closely related to a number of papers that study optimalpolicy models of herd behaviour, where the timing of agent’s decisions is ex-ogenously given. Smith and Sørensen (2008) show that a transfer scheme canimplement the socially efficient outcome in the herding model of Bikhchandani,Hirshleifer, and Welch (1992). Sgroi (2002) shows that forcing a subset of agents toact early can also improve informational flows in such an environment. Gossnerand Melissas (2006) study policy in a setting with both informational externali-ties and endogenous timing of decisions. However, theirs is a two-period modelwith no externality in the second period. Consequently, optimal policy involvesa one-time-only intervention, which rules out the time inconsistency issues. Rel-ative to the present paper, the possibility of time inconsistency of optimal policycannot arise in any of the settings examined by the previous literature, becauseof either exogenous sequencing of decisions or the lack of inefficiency in futureperiods.

While the present paper focuses solely on the case in which agents interactonly informationally, in many applications investor’s payoffs are likely to dependdirectly on the actions of other investors. For example, Lambrecht and Perraudin(2003) examine a case in which firms with an uncertain investment opportunityface a pre-emption threat, in that an early entrant is able to capture a largershare of the market.6 The policy implications in these kinds of setting are lessclear-cut, in part because it is not clear a priori whether the informational benefitof delay will cause equilibrium investment to be too slow or whether the pre-emption effect will cause equilibrium investment to be too fast. Furthermore,in cases in which investor payoffs are interdependent, individual investors maywish to actively conceal their information from other players. In these settings, the

5 This episode is reminiscent of the course of U.S. monetary policy in the early 1990s, as discussedin Caplin and Leahy (1996). During that episode, the Federal Reserve appeared to lower interestrates, watch to see if the cut induced the desired investment response, and then cut interest ratesfurther if the initial cut was unsuccessful.

6 Payoff interdependence might also take the form of complementarities, or agglomeration effects,where an investor’s payoffs depend positively on the number of other entrants in the market.Mason and Weeds (2008) examine the case where both pre-emption and agglomeration effectsare present and find that multiple equilibria exist.

946 M. Doyle

observation that future policy constrains a policy maker who is attempting to alterthe timing of investors’ decisions is likely to be only one of many complicationsof conducting policy.

The paper proceeds as follows. Section 2 presents the benchmark model anddiscusses the problem of an investor in both the presence and the absence of aninvestment subsidy. Section 3 presents the problem of a social planner in thisenvironment, shows that the social optimum coincides with the equilibrium ofthe policy game, and derives properties of the equilibrium outcome. Section 4provides some results for the case where the policy maker has access only todistortionary taxation and provides discussion. Section 5 offers concluding com-ments.

2. The model

There are N investors and a policy maker, where N is given and known to all.Of the N investors, n receive the option to invest in an investment project, wheren is a random variable drawn from a known distribution Go(n) with associateddensity go(n). The n projects are randomly assigned among the N investors, andthe realization of n is not observed.

An investor’s payoff is given by r(n) + s − c, where r(n) is the revenue generatedfrom investing, c is an idiosyncratic investor-specific cost, and s is a subsidy thatcan be conditioned on the history of the game but not on either n or c. Costs aredrawn independently from a known distribution F(c), with associated densityf (c) and support [c, c]. An investor’s cost is private information and is realizedafter the investor receives a project but before any decisions are made.

Revenue is strictly increasing in n, the unobserved state of the economy. Noticethat revenue depends on the number of available projects, not the number ofprojects that are actually undertaken. This assumption means that the revenueof investing does not depend directly upon the investment decisions of others,and that investors interact only informationally.

An investor who receives a project chooses whether or not to invest at anytime t = 1, 2, . . . , ∞, subject to the constraints that investment is irreversibleand a given investor can invest at most once. Investors discount the future by afactor δ < 1. Investors are risk neutral and choose the timing of any investmentto maximize expected profits. An investor who never invests receives a payoff ofzero. Investors who do not receive projects do nothing, trivially.

The number of investors with projects, n, is not observable, but the actions ofinvestors are public information. Investors may not communicate their privateinformation to one another. Given the lack of payoff interdependence, investorshave no strong incentive to conceal their private information from others (thoughthey do not have a strict incentive to report their information to others either).If communication is costly, as is likely to be the case in practice, investors wouldstrictly prefer to keep their information private.


The policy maker observes neither the number of investors with an option,n, nor the realized cost of any individual investor, c. The policy maker has ac-cess to an instrument, s, that affects the profitability of investment and that canbe conditioned on the observed history of outcomes. The policy maker’s prob-lem is to choose a path of history-dependent investment subsidies to maximizeeconomy-wide expected profits.

A perfect Bayesian equilibrium consists of a strategy and a probability assess-ment such that each investor’s strategy is a best response to the actions of thepolicy maker and the other investors at every information set and the probabilityassessments are consistent with Bayes’ rule at every information set reached withpositive probability. It is also assumed that beliefs satisfy a ‘no-signalling-what-you-don’t-know’ condition (Fudenberg and Tirole 1991). This paper restrictsattention to symmetric equilibria, in which an investor’s actions depend only onhis or her type and the publicly observed history of the game.

The policy maker chooses an investment subsidy at every history. It is impor-tant to note that the policy maker cannot commit to a path of future subsidies.This means that the policy maker effectively acts like a sequence of players,because the policy maker at a given history is not able to bind the actions offuture policy makers. Thus, today’s policy must be a best response both to thestrategies of private investors as well as to the actions of future policy makers.Since different versions of the policy maker cannot coordinate their actions, co-ordination failure can occur between different versions of the policy maker. Thiscoordination failure may prevent effective policy intervention.

2.1. Cutoff rulesIt is convenient to start by establishing a monotonicity result, namely, that thevalue of delay is increasing in an investor’s cost of investment.

LEMMA 1. An investor’s payoff from investing relative to waiting decreases as cincreases.

Proof. See the appendix. �

Since all investors possess the same beliefs and face the same subsidies, theirstrategies differ only to the extent that they have different costs. Since investorsdiscount the future, investors who invest should do so as soon as possible, in orderto minimize the losses due to discounting. The value of not investing immediatelylies in the fact that additional information acquired while waiting may allow theinvestor to avoid investing in cases in which the return is low. This option hasmore value to investors with high costs, as there are more states of the world forwhich a high-cost investor would like not to invest.

Lemma 1 implies that investors’ strategies take the form of a cutoff rule. Thatis, at any history ht all investors with c cost below some cutoff, denoted c(ht),invest immediately, and all investors with costs above c(ht) delay.

948 M. Doyle

2.2. InformationExpected revenue from investing at history ht is given by

R(ht) =∑

n

p(n | ht) · r(n), (1)

where p(n | ht) denotes the investor’s beliefs over n at ht.7 The undiscountedexpected profits of an investor with cost c who invests at ht therefore equal R(ht)+ s(ht) − c.

Conditional on a given cutoff, it is more likely that there will be a lot ofinvestment at date t − 1 when the number of projects, n, is high. The observationthat kt−1 is high therefore leads investors to conclude that higher values of n aremore likely. Since r(n) is increasing in n, the expected revenue from investing ishigher after the observation of a high value of kt−1.8

Observe that the amount of information learned by delaying depends directlyon the cutoff cost, c. Investors delay in hopes of learning about n. Intuitively, asc increases, an investor who delays observes more of the cost distribution andtherefore gets a more accurate signal about n. If c were equal to c, for example, adelaying investor would observe n with 100% accuracy.9

The investor’s problem is to decide when to invest so as to maximize expectedprofits. Note that, while the decision of an investor of type c to invest providesinformation to those who choose to delay, a private investor cannot internalizethe value of this information.

2.3. The policy makerThe policy maker influences investors’ behaviour through the investment subsidy.The policy maker wishes to maximize discounted, aggregate expected profitsin the economy. The policy maker chooses today’s subsidy, s(ht), prior to theinvestors’ decision on whether or not to invest, but cannot commit to future valuesof the subsidy. The key difference between the policy maker and private investorsis that the policy maker takes into account the effects of current investment onfuture information when deciding on how much investment to induce.

The policy maker’s problem is given by

V (ht) = maxs

N∑n=0

p(n | ht){∫ c

ct−1

[r(n) − c]f (c) dc

+ δ

N∑kt=0

p(kt | ht)V (ht, kt)}, (2)

7 For beliefs consistent with Bayes’ rule, only the probability of investment (which depends on thecutoff cost) and the number of investment decisions in each period matters.

8 See the appendix for a formal proof.9 See the appendix for a formal proof.


subject to the constraint that c is chosen by investors maximizing their expectedprofits. Note that both p(kt | ht) and V (ht, kt) depend on the equilibrium cutoff.

2.4. EquilibriumThis section characterizes the equilibrium behaviour of private investors in theeconomy with government subsidies. The problem is greatly simplified by thefact that, at any history, the marginal investor (i.e., the investor who is indifferentbetween investing and waiting at ht) finds it optimal to make a once-and-for-alldecision between investing and not investing at time t + 1. This means that thedecision of when to invest can be reduced from an infinite period to a two-periodproblem.

PROPOSITION 1. At any symmetric PBE, an investor indifferent between investingand waiting at ht finds it optimal to make a once-and-for-all decision at t + 1 both(i) when there are no subsidies, and (ii) when the subsidies are chosen optimally bythe government.


The intuition behind the proposition is as follows. In the absence of subsidies,the only cause of delay is the possibility of obtaining new information. Themarginal investor is the most eager to invest of all investors who have yet to investat t + 1. At t + 1, the marginal investor either invests or doesn’t. A decision toinvest is irreversible (i.e., once and for all). If this investor is unwilling to invest,given the information available at t + 1, then no other investor will be willingto invest either. As a result, no additional information will be forthcoming andtherefore the marginal investor will face the identical decision in the subsequentperiod, with the same results. Hence, a decision by the marginal investor not toinvest at t + 1 is a decision never to invest.

The presence of an investment subsidy introduces an additional incentive fordelay: the subsidy may rise in some future period. Under the optimal subsidies,however, it remains the case that the marginal investor at t finds it optimal tomake a once-and-for all decision at t + 1. Consider the problem facing the gov-ernment: if the expected value (including the informational value) of additionalinvestment at t + 1 is positive, then additional delay attenuates this value, owingto discounting. As long as the marginal investor delays, no additional informa-tion is produced, so this delay represents a cost with no offsetting benefits. Inthis case, the government will not offer subsidies that encourage the marginalinvestor to delay further. On the other hand, if the expected value of additionalinvestment is negative at ht, the government will not want to subsidize additionalinvestment, and the marginal investor will never invest.

An important implication of proposition 1 is that the game is finite. Thisis because an observation of no investment in a given period (i.e., kt = 0) isthe worst possible news. Since no one invests after an observation of the worst

950 M. Doyle

possible news,10 investment ceases after any period in which no one invests. Witha finite number of potential investors, this implies that the game is finite.

Proposition 1 also implies that the equilibrium cutoff cost can be found bycomparing the payoff from investing today to the discounted payoff to an investorwith cost c who waits exactly one period at ht before making an irrevocabledecision in period t + 1. The latter payoff is given by

δW (c, c, ht, {s(ht+1)}) = δ∑

kt

p(kt | ht) max{R(ht+1) + s(ht+1) − c, 0}, (3)

where c is the investor’s own cost, c is the cutoff cost played by other agents, ht

is the history, and {s(ht+1)} is the set of subsidies in the period following ht.11

It is important to observe that W (c, c, ht, {s(ht+1)}) is increasing in its secondargument. This is because a delaying investor receives more precise informationwhen others choose a higher cutoff cost. By Blackwell’s theorem (Blackwell 1951,1953), the investor prefers more over less precise information.

The equilibrium cutoff, c∗(ht), can be calculated in each of the three possiblecases that can occur along the equilibrium path:

Case (i) The expected value of additional investment is negative for the lowestremaining cost type: R(ht) + s∗(ht) + c∗(ht−1) < 0. In this case noadditional investment occurs, so that c∗(ht) = c∗(ht−1).

Case (ii) The expected value of additional investment exceeds the value of delayfor all costs: R(ht) + s∗(ht) − c > δW (c, c, ht, {s∗(ht+1)}). In this case,all remaining cost types choose to invest immediately, so that c∗(ht) =c.

Case (iii) The expected value of investment is at some intermediate level, so thatfor some c∗(ht) such that c > c∗(ht) > c∗(ht−1):

R(ht) + s∗(ht) − c∗ = δW (c∗, c∗, ht, {s∗(ht+1)}). (4)

In this case, some but not all of the remaining cost types choose toinvest at ht.12

10 Any investor who would be willing to invest, even following the worst possible news, would haveinvested in a previous period, as discounting attenuates a positive expected return.

11 There is one future subsidy, s(ht+1), associated with each possible realization of kt, where ht+1 ={ht, kt}.

12 Observe that for case (iii), there is a marginal cost type such that an investor with this cost isprecisely indifferent between investing and delaying. Consequently, there are many possibleequilibria depending on whether the marginal investor chooses to invest, to delay, to mixbetween the two, or to condition the decision on some other variable (e.g., the identities ofprevious investors). Note that all such equilibria are payoff equivalent, and that, with acontinuous cost distribution, the probability that any investor’s realized cost type exactly equalsone of the cutoff costs in any play of the game is zero.


It is worth considering, first, the case with no subsidies. In this case, equa-tion (4) becomes

R(ht) − co = δW (co, co, ht, 0), (5)

where co denotes the equilibrium cutoff in the absence of subsidies. From inspec-tion of equation (3), it is clear that the right-hand side of (5) is greater than orequal to zero and strictly positive if the return to investment is negative for somepossible states of the world. This implies that some investors choose not to investat ht even if the expected value of doing so is positive. This delay is inefficient.Individual investors trade off the returns to investing immediately against thepossibility of learning if they wait, but they do not internalize the informationalvalue of their actions to investors who have yet to act. This externality motivatespolicy intervention, since at any given history a policy maker will wish to reducedelay by encouraging a larger set of investors to invest.

Note that, from the policy maker’s perspective, the problem boils down tochoosing the optimal cutoffs and then figuring out which subsidies will implementthese cutoffs. In other words, the policy maker’s problem becomes

V (ht) = maxs

N∑n=0

p(n | ht){∫ c

ct−1

[r(n) − c]f (c) dc

+ δ

N∑kt=0

p(kt | ht)V (ht, kt)}, (6)

where the subsidies are determined by solving the investor’s problem at theoptimal cutoff, using equation (4), at an interior, and the analogous conditions(from cases (i) and (ii), above) if the optimal cutoff is a ‘corner.’

3. Efficiency

This section asks to what extent a policy maker is able to induce efficiencygains in the economy. The first-best outcome would be for the policy makerto collect all of the information ex ante and release it to all investors at thestart of play. However, this trivializes the coordination problem at the heart ofthe decentralized economy. This section presents a constrained efficient welfarebenchmark by studying the problem of a social planner who can direct theinvestment decisions of investors but must respect the information constraintsof the economy. By comparing the solution to the planner’s problem with theequilibrium investment profile induced by the policy maker, it is shown that thethe policy maker can attain the constrained efficient outcome.

952 M. Doyle

The welfare benchmark is constructed by considering a situation in which allinvestors gather at the start of the game, prior to the realization of n, and committo their future actions by means of a binding contract. These contracts specifythe action of an investor contingent on his or her cost type for every possibleobserved history of the game. This setup effectively internalizes the externaleffects of investment because, prior to the realization of cost types, investors donot know whether they will likely be early or late investors. Therefore, they takeinto account the informational value of early investment to investors who willmake their decisions later. Since at this ex ante stage all investors are identical andrisk neutral, they agree on the contract that maximizes the ex ante representativeinvestor’s expected payoff. The outcome chosen by this ex ante representativeagent constitutes the constrained first-best outcome with which the equilibriumoutcome is compared.

The planner knows the probability distribution over n, Go(n), and the distri-bution F(c) from which costs are drawn. The planner forms expectations aboutthe state of the economy by observing the history of the game and maximizesthe sum of discounted expected profits. Respecting the informational constraintsin the economy, the planner observes neither the number of investors with anoption, n, nor the realized cost type of any individual investor, c.

If we assume that the planner discounts the future at δ, the discount rate usedby investors, the planner’s objective function is

V =N∑

n=0

g(n)∫C(h1)

(r(n) − c)f (c) dc

+ δ

N−1∑k1=0

p(k1 | h1)

{N∑

n=0

p(n | h2)∫C(h2)

(r(n) − c)f (c)dc + · · ·}

. (7)

The planner’s problem is to maximize V by specifying a set of cost types for eachpossible history, C(ht), such that an investor invests at ht if his or her cost is inC(ht) and waits at ht if his or her cost is not in C(ht).

The solution to the planner’s problem involves cutoff rules. The intuition isthat the planner has more to lose by requiring that investors with lower costs delaylonger than agents with higher costs. The planner wants to influence informationgeneration through the choice of cutoff, and the least costly way of doing thiscorresponds to requiring that the lowest-cost investors are the first to invest.13

13 The planner never finds it optimal to have the investors play mixed strategies. For any desiredprobability of investment (which determines how much information is revealed) the plannerdoes best by having the lowest-cost agents invest earliest, rather than having a higher-costinvestor invest early with positive probability.


The planner’s problem can be written in recursive form:

V (ht) = maxc

N∑n=0

p(n | ht){∫ c

ct−1

[r(n) − c]f (c) dc

+ δ

N∑kt=0

p(kt | ht)V (ht, kt)}. (8)

PROPOSITION 2. Let {cp(ht)}ht∈H denote the cutoffs that solve the social planner’sproblem.

(i) The equilibrium cutoffs induced by the policy maker in the absence of a com-mitment technology are identical to the cutoffs that solve the social planner’sproblem.

(ii) The policy maker reduces delay (chooses a higher cutoff) at any history ht

relative to the equilibrium cutoff that would obtain at history ht in the absenceof subsidies.

Proof

(i) By inspection, problems (6) and (8) are identical. Hence, the cutoffs thatsolve the planner’s problem will also maximize the policy maker’s objectivefunction.

(ii) See the appendix. �

The intuition for part (ii) is as follows. Information is of positive value andwill be underprovided in the absence of policy intervention, owing to the exter-nality discussed in section 2.1. The optimal policy requires that the policy makerinduce investors to generate additional information. Since information is createdwhen potential investors actually invest in a project, the policy maker generatesinformation by inducing investors to invest sooner than they would choose to doin the absence of subsidies. In other words, the policy maker reduces delay.

Observe that there is no guarantee that the true value of n will be revealed evenunder the socially optimum investment strategy. To see this, simply note that it ispossible for the true value of n to be high but for all agents with projects to havehigh costs. In this case k1 can be zero and the game will end with no investment,despite the fact that investing may well be profitable for some or all investors.

3.1. Efficient subsidiesPROPOSITION 3. The equilibrium subsidy at ht, s∗(ht), equals the value of informa-tion plus the expected value of the subsidy E(s∗(ht+1)). Since the value of informationis positive, s∗(ht) ≥ E(s∗(ht+1)).

954 M. Doyle


In order to reduce delay in equilibrium the policy maker sets the optimalsubsidy to be declining in expected value. The optimal subsidy does not merelycompensate investors for the value of information created by their early invest-ment. Since investors understand that there will be future subsidies, there is anadditional incentive to delay, which the policy maker must account for whenchoosing the optimal subsidy. The policy maker offsets this effect by increasingthe earlier subsidies to compensate investors for future subsidies forgone by in-vesting early. Hence, the optimal subsidy equals the informational value of extrainvestment plus the expected value of next period’s subsidy. Since the subsidydoes not enter the planner’s objective function, the planner is willing to offerhigh levels of the subsidy early on. As a result, the willingness of the planner tomove subsidies ahead does not compromise his or her ability to provide incentivesearlier in the game.

In short, with access to a costless policy instrument, the strategic responseof investors to policy initiatives forces the policy maker to act more aggressivelytoday to offset the willingness of strategic investors to wait for future policychanges.14 However, this does not prevent the policy maker from implementingeffective policy. Even in the absence of any commitment device, the policy makeris able to overcome complications caused by the strategic behaviour of investorsand to induce the efficient outcome.

4. Discussion

4.1. Distortionary taxationThis section relaxes the assumption of the previous sections that the investmentsubsidy can be financed costlessly, via lump-sum taxation. This assumption isreplaced with the assumption that, whenever the policy maker subsidizes invest-ment, distortionary taxes are raised elsewhere in the economy, imposing someadditional cost on the policy maker. The simplest way to model this is to assumethat the policy maker faces a constant marginal cost of using the subsidy, denotedby θ , where θ > 0.15

The main results are summarized in the following proposition.

PROPOSITION 4. When subsidies are costly, the policy maker never attains the first-best outcome. Furthermore, if the policy maker cannot commit to future subsidies,

14 This result closely resembles Caplin and Leahy’s (1996) result. Their model, however, does notalow them to address the issue of efficiency of the equilibrium investment profile.

15 The linearity assumption abstracts away from issues of tax smoothing and any desire on the partof the policy maker to borrow or lend to finance the investment subsidy. Furthermore, the costof intervention is assumed to depend on the absolute value of the subsidy. This allows the policymaker to employ taxes on future investment as a disincentive to delay, without setting up asituation in which the policy maker taxes investment as a source of revenue for unspecified andunmodelled reasons.


TABLE 1Cutoffs and subsidies, θ = 0.1

No-intervention Optimum No-commitment

History Cutoffs Cutoffs Subsidies Cutoffs Subsidies

h1 = {∅} 0.48 0.55 0.06 0.55 0.05h2 = (0) 0.48 0.55 0 0.55 0h2 = (1) 0.68 0.71 0.03 0.69 0.02h2 = (2) 0.95 0.95 0 0.95 0h3 = (0, 0) 0.48 0.55 0 0.55 0h3 = (1, 0) 0.68 0.71 0 0.69 0h3 = (1, 1) 0.95 0.95 0 0.95 0h3 = (2, 0) 0.95 0.95 0 0.95 0

Welfare 0.3091 0.3137 0.3083

the equilibrium outcome may be worse, in terms of welfare, than the equilibriumoutcome that would obtain with no policy maker. If the policy maker can commit,then welfare is greater than in the no-intervention case.

The first result – when θ > 0, the policy maker is never able to attain the firstbest outcome (as characterized in section 3.1) – arises simply because the policymaker wants to economize on the costly subsidy, at the expense of leaving someinefficient delay in the investment process.

That the policy maker can do worse than the no-intervention outcome isshown through the use of numerical examples. Table 1 presents the solution to agame where N = 3.16 The calculations are based on the assumptions that F(c) isuniform in the interval [0, 1], δ equals 0.9, go(n) is uniform (taking on the value0.25 for each value of n), and r(n) takes on the values {0.0, 0.1, 0.4, 0.95}.

The table reports the cutoffs at each ht with t ≤ 3 that is reached in equilibriumwith positive probability for three scenarios. The first column of the table reportsthe cutoffs for the no-intervention equilibrium (i.e., when all subsidies are zero).The second column reports the socially optimal cutoffs, and the third columnreports the subsidies that would be required to implement the optimal cutoffs.These are the subsidies that a policy maker would set in the case in which θ wasequal to zero. The final two columns report the cutoffs and subsidies that occurwhen θ = 0.1 and the policy maker cannot commit. Note that in the case of nocommitment there is less delay (i.e., higher cutoffs) than in the no-interventionequilibrium but greater delay than the socially optimal level.

The key result from table 1 is that the policy maker does worse than the no-intervention outcome, despite reducing delay. This can be observed in the final

16 The game is still finite when subsidies are costly: if no one invests, then the policy maker facesthe same set of future histories and finds it optimal to offer a lower subsidy than was the case inthe initial period. Hence, a three-investor game will have a maximum of three periods.

956 M. Doyle

row of the table, which lists expected profits in each case net of any distortionsdue to the use of subsidies. The intuition is that the inability to commit tolow future subsidies makes it expensive to subsidize investment today. Recall(from proposition 3) that today’s subsidy equals the informational value of anyreduction in delay plus an additional payment that compensates investors forforgoing future subsidies. Since the subsidy exceeds the informational value ofadditional information, the cost to the policy maker may exceed the value ofany reduction in delay. Therefore, the policy maker can do worse than the no-intervention outcome, depending on the value of θ (the severity of the distortionsassociated with using the subsidy).

Note that the inclusion of costs associated with distortionary taxation intro-duces a problem of time inconsistency. Since investors’ decisions to invest ordelay depend on both the current and the future subsidy, the policy maker couldreduce the cost of inducing current investment by announcing that future subsi-dies will be low. Announcements of low future subsidies, however, are not timeconsistent. This is because the policy maker will renege on the announced lowsubsidy in favour of a higher subsidy, in order to reduce future delay.

Finally, if the policy maker has access to a commitment technology, he orshe can always do at least as well as the no-intervention equilibrium simply byannouncing that all subsidies will be zero. The policy maker can strictly improveon the no-intervention equilibrium outcome whenever the marginal value ofinformation at the no-intervention equilibrium cutoffs exceeds θ at some ht

by offering a positive subsidy at ht and zero subsidies at all other histories.This is strictly preferred to the no-intervention equilibrium outcome becausethe expected current profit and the option value of delay are equal at the no-intervention cutoffs, which means that the benefit of the reduction in delay is themarginal value of information, which exceeds the marginal cost of the subsidy.

4.2. Period lengthChanges to the discount factor can be interpreted as changing the length of aperiod, as a longer wait between periods implies more losses due to discounting(i.e., smaller δ). In Chamley and Gale (1994), the equilibrium outcome exhibitsinefficient delay for low period lengths, but when the period lengthens the equi-librium outcome can switch to one in which all investors invest in the first period.Basically, if it is more costly to wait, then in order to generate any delay the in-formation revealed in the first period must be more accurate (i.e., the probabilityof investment must rise). If it gets costly enough to delay, then investment todayis preferable to delay even if delay would reveal the true value of n. In this case,everyone invests immediately and there is no delay.

In the model studied here, where investors have heterogeneous costs, thismechanism is still present. However, with heterogeneous costs there may be someagents who have projects with negative expected value, based on the informationavailable at period 1. In this case, there is no value of δ that induces all cost typesto invest immediately, and the equilibrium therefore always exhibits delay.


In cases where the equilibrium does exhibit delay, changes in the period lengthhave the intuitively expected effects. When the period length is longer, informationfrom given investment is the same, but the costs associated with discounting rise.Consequently, there is less delay (represented by higher cutoffs) in the economywith a longer period length relative to the case where the period length is short.This is the case both for the the model with no policy intervention as well as forthe investment profile induced by a policy maker.

4.3. Policy with a large number of playersIt is worth considering the implications for policy when N, the number of players,becomes very large. As in Chamley and Gale (1994), an increase in N does notfundamentally alter the behaviour of private investors. The amount of investmentis determined by the condition that, for the marginal agent, the cost of delay dueto discounting is exactly balanced by the option value.

It would appear that when N becomes very large, the information revealed byobserving a fixed portion of the cost distribution would increase. This is indeedthe case.17 In the absence of policy intervention, however, this does not meanthat full information is revealed immediately. The investor who is indifferentbetween investing and delay when N is small would prefer to delay when N islarge, because the information obtained is more accurate. Consequently, as Nbecomes large, the cutoff costs have to fall to rebalance the value of investingimmediately versus delay. These cutoff costs fall sufficiently that information isrevealed only slowly in equilibrium, so that the no-intervention equilibrium ofthe large game retains the flavour of the game with finite N.

The implications for the model with optimal policy intervention are quitedifferent. In the limit as N → ∞, the optimal policy can approach the first-bestoutcome rather than the constrained efficient outcome discussed in section 3. Thefirst-best outcome would be for n to be revealed immediately and for all agentsto make the correct investment decision in the first period. In the case in whichN is large, observing a small fraction of the cost distribution reveals a great dealof information. The policy maker can therefore set a first-period subsidy suchthat the payoff to investing is greater than delay for some cost types, even underthe assumption that the true value of n is revealed in the first period. Since theflow of investment in the first period would be small relative the total numberof players, the total cost of the subsidy would be small relative to the size of theeconomy, and future subsidies could be set to zero.

4.4. Asymmetric equilibriaThe paper restricts attention to symmetric equilibria, in which an investor’sactions depend only on his or her type and the publicly observed history ofthe game. Given the environment of the paper, which presumes that investors are

17 By the law of large numbers, when N is very large, the frequency of investors with a cost belowsome cutoff approximately equals the probability that an investor has a cost below some cutoff.Since the number of people investing equals n times the probability that an investor has a costbelow the cutoff, n can be inferred with great accuracy.

958 M. Doyle

unable to coordinate sufficiently to share their private information, the restrictionof attention to symmetric equilibria is reasonable. This focus may not be entirelyinnocuous, however, as there definitely exist asymmetric equilibria of the model.18

Most interestingly, there may exist asymmetric equilibria that exhibit markedlydifferent properties from symmetric equilibria. In particular, there may existasymmetric equilibria with no delay under some parameterizations of the model.

For example, Gale (1996) shows that in a two-player, continuous time model ofinformational externalities with endogenous timing of actions there always existasymmetric equilibria which do not exhibit delay. Essentially, if player 1 alwaysexpects player 2 to act in the immediate future, then player 1 will always delay,as observing player 2’s action is always worth an infinitesimal wait. Player 2, onthe other hand will have no incentive to wait, given than player 1 always delays.Thus, player 2’s optimal strategy is to invest immediately. Note, however, thatthis argument depends on the continuous time assumption, as it is always worthan infinitesimally short wait in order to obtain information. In a discrete timesetting, delay may not always be worthwhile, which breaks the equilibrium.19

5. Conclusion

This paper has examined optimality in an economy with strategic delay driven byan informational externality. In this setting, policy makers have to be aware thatsubsidizing future investment provides private agents with additional incentivesto delay. How does this affect optimal policy making? When the investmentsubsidy can be financed through non-distortionary taxation, policy is effectiveand the strategic behaviour of investors influences only the profile of the optimalinvestment subsidy.

If the subsidy must be financed through distortionary taxation, however, thetime inconsistency problem created by policy maker’s inability to commit tolow future subsidies may create policy failure. Interestingly, this failure is notnecessarily associated with an increase in delay, as welfare losses can arise whenthe policy maker overpays for the reduction in delay, since current subsidies (andtherefore taxes) must be set at an inefficiently high level in order to compensatetoday’s investors for future subsidies forgone.

Appendix

A.1. Proof of lemma 1Consider an investor at some history ht. An investor investing today receives anexpected payoff of π (ht) = R(ht) + s(ht) − c, where R(ht) is the expected revenue

18 Versions of the asymmetric equilibrium presented in section 7.2 of Chamley and Gale (1994)exist in the current model.

19 On the other hand, Rogers (2005) shows that in a two-player, discrete time setting withheterogeneous investors no asymmetric equilibria exist. His argument, however, relies on asufficient degree of heterogeneity that at least some types of agents never delay.


of the investment project given the information available at ht and s(ht) is thesubsidy. Note that the expected profit decreases one to one with an increase in c.

Consider the payoff of an investor who delays investment at ht, but may investat some future time. Let H+ denote the set of continuation games starting at ht.Let HI

t+j denote the set of continuation games in which an agent invests in periodt + j. Similarly, HO

t+1 denote the set of continuation games in which an agentdoes not invest in period t. Let P(ht+j) denote the probability of arriving at somefuture history, given ht. Then the value of not investing at ht can then be written

D(ht) = δ∑HI

t+1

P(ht+1)π (ht+1) + δ∑HO

t+1

P(ht+1)D(ht+1). (A1)

Note that the payoff from investing today can be written

π (ht) =∑HI

t+1

P(ht+1)π (ht+1) +∑HO

t+1

P(ht+1)D(ht+1). (A2)

Then, the relative value of investing today can be written

π (ht) − D(ht) = (1 − δ)∑HI

t+1

P(ht+1)π (ht+1)

+∑HO

t+1

P(ht+1(ht))π (ht+1) − δ∑HO

t+1

P(ht+1(ht))D(ht+1(ht)).

(A3)

Iterative substitution using (A1) and (A2) gives

π (ht) − D(ht) = (1 − δ)∑HI

t+1

P(ht+1(ht))π (ht+1)

+∑HO

t+1

P(ht+1)

{{(1 − δ2)

∑HI

t+2

P(ht+2(ht+1))π (ht+2(ht+1))

+ · · ·

+∑

HOT−1

P(hT )

{(1 − δT−t)

∑HI

T

P(hT )π (hT )

+∑HO

T

P(hT )π (hT )

}}, (A4)

where T is the ending period, which may be infinity.

960 M. Doyle

Note that delaying at T corresponds to never investing, which has a payoff ofzero (and therefore doesn’t appear in the above expression). The benefit of notinvesting immediately, however, relies on the existence of continuation games inwhich the investor never invests. If there were no such games, then the investorwould always invest eventually. In this case, however, the fact that future payoffsare discounted implies that the investor would be better off investing immediately.The benefit of delay is that the investor may acquire new information that allowshim to avoid investing in states where investment is not profitable.

From the definition of π (ht), we can see that the right-hand side of (A4)is decreasing in c. Hence, for fixed decisions the relative benefit of investingimmediately versus delay is decreasing in c. This is because a low-cost investorloses more because of discounting than a high-cost investor.

However, the decision of when to invest is not fixed. Delay has value preciselybecause it allows investors to not invest in some states of the world. The factthat expected profits are decreasing in c implies that high cost investors are morelikely to exercise the option to never invest. That is, HO

T is increasing in c. Thisfurther implies that the relative benefit of investing immediately versus delay isdecreasing in c, as weight is being taken from positive expected value situationsand being applied to zero expected value events.

Hence, the right-hand side of (A4) is decreasing in c, which implies that agentswith a lower cost find investing immediately more attractive relative to delay thando high-cost investors.

A.2. Proof that R(ht) is increasing in kt

Conditional on the realization of n, each investor has an equal chance, n/N, ofreceiving a project. An investor who does receive a project uses Bayes’ Theoremto derive posterior beliefs on n as follows:

g(n) = go(n) · n

[N∑

n′=0

go(n′) · n′]−1

. (A5)

Since investors play cutoff rules in equilibrium, at a given history ht all remain-ing investors will have costs above the cutoff of the previous period, denoted byc(ht−1). Suppose that all remaining investors with costs below some cutoff c thenchoose to invest at ht. Then the probability that an investor with an option whohas not yet invested invests at ht is given by

F(c) =∫ c

c(ht−1)f (c | c(ht−1))dc = F(c) − F(c(ht−1))

1 − F(c(ht−1)), (A6)

Let K(ht) ≡ ∑t−1τ=0 kτ be the total number of investments that have been made

at ht. Then the probability of observing kt investment decisions at ht, given c(ht),


follows the following binomial distribution:

p(kt | n, ht) = (n−K(ht)−1kt

)[Ft(c)]kt · [1 − Ft(c)]n−K(ht)−kt−1

= b(kt | n − 1, ht; c). (A7)

Agents use Bayes’ rule to update beliefs, which results in the following poste-rior assessment:

p(n | ht+1) = b(kt−1 | n − 1, ht−1; c)) · p(n | ht)∑Nn′=0 b(kt−1 | n′ − 1, ht−1; c) · p(n′ | ht)

. (A8)

If 0 < F ≤ 1 and p( · ) is non-degenerate, then p( · |ht) is increasing in kt in thesense of first-order stochastic dominance. To see this, let n and m be two numbersin the support of p( · ). Then, for any n > kt + K(ht),

p(n | {ht, kt})p(n + m | {ht, kt}) = (n + m − kt − Kt)! · n! · p(n | ht)

(n − kt − K(ht))!(n + m)! · Ft(c)mp(n + m | ht). (A9)

This is decreasing in kt.Compare p( · |{ht, kt}) and p( · |{ht, kt + 1}). For n = 1, . . . , kt + K(ht) − 1,

p(n|{ht, kt}) = p(n|{ht, kt + 1}) = 0. For n = kt + K(ht), p( · |{ht, kt}) > 0 andp( · |{ht, kt + 1}) = 0. For any n > kt + K(ht), if p(n|{ht, kt}) ≤ p(n|{ht, kt + 1})then,

p(n + m | {ht, kt})p(n + m | {ht, kt + 1}) <

p(n | {ht, kt})p(n | {ht, kt + 1}) ≤ 1. (A10)

Since we know, for some n, that p(n|{ht, kt}) > p(n|{ht, kt + 1}), the twodensity functions have the single crossing property. That is, there exists some n0

such that

p(n | {ht, kt}) ≤ p(n | {ht, kt + 1}) if n ≤ n0 (A11)

and

p(n | {ht, kt}) ≥ p(n | {ht, kt + 1}) if n ≥ n0 (A12)

This implies first-order stochastic dominance, which further implies that R(ht) =∑np(n|ht) · r(n) is increasing in kt.

A.3. Proof that information is increasing in the cutoffRecall from (equation (14)) that, for arbitrary cutoff c, F(c) measures the prob-ability that an investor with an unknown cost invests in this period. In a precisesense, F measures the amount of information revealed by observing the numberof investors who invest.

962 M. Doyle

Let X and X′

denote the random number of investors who invest when theprobabilities are F ′ and F , respectively. When F ′ < F , X ′ is equal to X plusnoise, so that observing X is more informative than observing X

′in the sense of

Blackwell (1953) (see also Chamley and Gale 1994, 1070). This implies that as Fincreases from zero to one, the amount of information revealed increases. SinceF is (weakly) increasing in c, this means that the amount of information revealedincreases as the cutoff, c, increases from c(ht−1) to c.

A.4. Proof of proposition 1Part (i) If c∗(ht+1) > c∗(ht), then the investor with type c∗(ht) prefers investmentover no investment at ht+1, because investors with lower costs have more to lose bywaiting than investors with higher costs, and therefore they have more incentiveto invest early.

It remains to consider the case where c∗(ht+1) ≤ c∗(ht), which can be dividedinto two possible cases:

(a) If R(ht+1) − c(ht) > 0 and c∗(ht+1) = c∗(ht), then an investor withc∗(ht) must be planning to invest at some future information set. Lethτ = (ht+1, 0, . . . , 0) be the first information set after ht+1 at whichc∗(hτ ) > c∗(ht). The probability of investment is zero between ht+1 andhτ , because if the investor with the lowest remaining cost, c∗(ht), doesn’tfind it optimal to invest, then no one does. Therefore, the informa-tion is the same at both information sets, so that R(ht) − c∗(ht) =R(ht+1) − c∗(ht) > 0. Discounting makes the investor better off if he investsat ht+1, meaning that c∗(ht+1) > c∗(ht).

(b) If R(ht+1) − c∗(ht) ≤ 0 and c∗(ht+1) = c∗(ht), then at hτ no new informationhas been revealed and R(hτ ) − c∗(ht) < 0. Therefore, it is not optimal to investat any such information set hτ , so that it is never optimal to invest followinght+1.

Part (ii) Suppose at ht+1 future subsidies {s(ht+1+j)}∞j=0 are such that investorswhose costs are such that they are indifferent between investing and waiting at ht

do not invest at ht+1 but wish to invest at some future information set hτ = (ht+1,0, . . . , 0).

That is, at hτ , the indifferent investor facing a set of future subsidies {sτ+j}∞j=0chooses to invest. Since the probability of investment is zero between ht+1 andhτ , no information is revealed, to either the investor or the policy maker, betweenht+1 and hτ .

This implies that the policy maker’s expected payoff from inducing investmentat hτ is the same as the policy maker’s expected payoff at t + 1 but is discounted.Since the policy maker induces the indifferent investor to invest at hτ , the expectedpayoff to the policy maker from such an investment must be positive. Therefore,the expected payoff at ht+1 must also be positive.

Since information is the same at ht+1 and hτ , the policy maker is able inducethe investor to invest at ht+1 rather than at hτ by setting st+j = sτ+j for all j =


{0, 1, . . . }. Since the policy maker and the investors face exactly the same situationat ht+1 as at hτ . If sτ+j induces the indifferent investor to invest at hτ , then thosesame continuation subsidies at ht+1 would induce the indifferent investor to investat ht+1. Since the policy maker loses from discounting, the policy maker is betteroff by inducing the investor to invest at ht+1.

A.5. Proof of proposition 2, part (ii)Denote by co(ht) the equilibrium cutoff when the value of the subsidy is equalto zero in every possible state. There are three possibilities for a solution toequation (8) at ht: (i) a corner solution in which cp(ht) = c, so that all remaininginvestors invest regardless of their costs; (ii) an interior solution in which cp(ht) ∈[cp(ht−1), c], where any remaining investors will invest if their costs are below thecutoff cp(ht); or (iii) a corner solution in which cp

t = cp(ht−1), where no investorwith a cost in the set of remaining possible costs invests.

Case (i) It is clear that cp(ht) ≥ co(ht), since cp(ht) = c which is the maximumpossible that co(ht) can take.

Case (ii) This occurs when there is an interior solution to equation (8). In thiscase, cp(ht) is that value of c which satisfies the following first-order condition:

0 =∑

n

p(n | ht)(r(n) − c)f (c) − δ∑

k

p(kt | ht)∑

n

p(n | ht+1)(r(n) − c)f (c)

+ δ∑ ∂p(kt | ht)

∂c· V (ht, kt), (A13)

where the summation is over those values of kt for which cp(ht) ≥ cp(ht−1).The first term in equation (A13) is the additional expected profit (or loss)

from current investment that will accrue to the planner if the cutoff is raised.The second term is the negative of the option value of delay and represents acost of increasing the current cutoff, c. The third term is the marginal benefitof increased information to the planner. This term is greater than or equal tozero, as information is valuable.

The first two terms equal the two terms of equation (4) and therefore sumto zero when cp(ht) = co(ht). The sum of these terms is negative when cp(ht) isgreater than co(ht). Since the third term is positive, it follows that any interiorsolution to the planner’s problem involves cp(ht) > co(ht).

Case (iii) This can be proven by contradiction. Assume that cp(ht) = cp(ht−1),and co(ht) > cp(ht−1). If co(ht) > cp(ht−1), then

0 <∑

n

p(n | ht)(r(n) − cp(ht−1))f (cp(ht−1))

− δ∑

kt

p(kt | ht)∑

n

p(n | ht+1)(r(n) − cp(ht−1))f (cp(ht−1))0. (A14)

Since δ∑

k[∂p(kt | ht)/∂c] · V (ht, kt) ≥ 0, the right-hand side of equation (A13)must be greater than zero. Therefore, the planner’s utility can be increased by

964 M. Doyle

increasing cp(ht), which contradicts the assumption that cpt = cp(ht−1) is the

solution to the planner’s problem. Therefore, if cp(ht) = cp(ht−1), it must be thecase that co(ht) = cp(ht−1).

A.6. Proof of proposition 3From equation (4):

s∗(ht) = δW (c∗(ht), c∗(ht), ht, {s∗(ht+1)}) − (R(ht) − c∗(ht))

=∑

p(kt | ht)s∗(ht+1) + [δW (c∗(ht), c∗(ht), ht, {0}) − (R(ht) − c∗(ht))],

where the summation is over those values of kt following which c∗({ht+1}) ≥c∗(ht).

The term δW (c∗(ht), c∗(ht), ht, {0}) − (R(ht) − c∗(ht)) equals the value of delayin the case where the subsidy is equal to zero in every period, evaluated at c∗(ht).This can be signed. First, note that, by proposition 2, the equilibrium cutoff withoptimal subsidies (c∗) is greater than or equal to the equilibrium cutoff that wouldobtain in the absence of subsidies (co) at any history. Hence, by discounting:

δW (c∗(ht), c∗(ht), ht, {0}) − (R(ht) − c∗(ht))

≥ δW (co(ht), c∗(ht), ht, {0}) − (R(ht) − co(ht)),

and by Blackwell’s theorem:

δW (co(ht), c∗(ht), ht, {0}) − (R(ht) − co(ht))

≥ δW (co(ht), co(ht), ht, {0}) − (R(ht) − co(ht)).

By inspection of equation (3), δW (co(ht), co(ht), ht, {0}) − (R(ht) − co(ht)) ≥0 and is strictly positive if the return to investment is negative for some possiblestates of the world. This implies that δW (c∗(ht), c∗(ht), ht, {0}) − (R(ht) − c∗(ht))is greater than or equal to zero at c∗(ht). Therefore, the equilibrium subsidy in ht

is greater than or equal to the expected value of the period t + 1 subsidy to themarginal agent.

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