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ASSET PRICE BUBBLES AND TECHNOLOGICAL INNOVATION.

I. INTRODUCTION

This paper introduces credit constraints into a two-sector Schumpeterian growth model and examines the interplay between R&D specific productivity shocks, bubbles, and technological innovation. Conventional wisdom has it that bubbles typically emerge in an economy during periods of high productivity growth (Gordon 2015; Lansing 2009), the argument essentially being that in such an environment, expectations of higher profits coupled with low interest rates facilitate their creation. On the other hand, we use a two-sector growth framework to demonstrate that adverse shocks to productivity in the R&D sector, the intermediate good sector in our model, can trigger bubbles in the economy. More importantly, we demonstrate that in such an environment bubbles play a constructive role by alleviating credit constraints and fostering innovation and growth. (1)

We introduce credit constraints into a two-period overlapping generation version of the Schumpeterian growth model. Agents are risk neutral and maximize their expected old-age consumption. There are two types of agents: workers and entrepreneurs. In the beginning of the first period, young workers provide labor service in the final goods sector. They save wages for old-age consumption. At the end of the first period, young entrepreneurs are given an opportunity to become innovators in the R&D sector, the intermediate good sector in our model. They finance R&D investment with borrowings from the savers. In addition, entrepreneurs face endogenous credit constraints because they cannot credibly pledge more than a fraction of the expected output due to an ex post moral hazard problem. Their borrowing limit is capped at a fraction of their expected profits. These credit constraints are introduced in the spirit of the "financial accelerator" models pioneered by Bernanke and Gertler (1989) and Kiyotaki and Moore (1997), where the presence of informational frictions in the economy results in investment by entrepreneurs being restricted to a fraction of pledgeable collateral.

Next, we proceed to demonstrate that rational bubbles can emerge in such a financially constrained economy. Essentially, bubbles represent a prorata share in a Ponzi scheme. Savers will wish to purchase bubbles as long as they offer a return at least equal to the interest rate in the economy. Due to the presence of credit constraints, any decrease in the value of their expected profits reduces demand for investment. The consequent fall in interest rates lowers the rate of return that bubbles have to offer so that they are attractive to savers. In this sense, binding credit constraints make bubbles viable in our framework.

We show that periods of permanently low productivity in the intermediate good R&D sector are conducive for the emergence of rational bubbles. Essentially, a decline in productivity in this intermediate good sector reduces the value of collateral, causing credit constraints to bind. The resultant excess supply of credit and lower interest rates opens the door for creation of bubbles. In such an environment, bubbles help circumvent the moral hazard problem and facilitate transfer of resources from savers to entrepreneurs. This reallocation then stimulates higher innovation and growth in the economy.

The longevity and path of bubbles are critically dependent upon market expectations and investor sentiments. Savers in our economy will hold bubbles only if they expect the next generation of savers to buy them from them. The fragility of these bubbles therefore creates a natural role for the government in helping facilitate the transfer of resources from savers to entrepreneurs in a credit constrained economy. We show that government credits to the R&D sector, financed through issue of public debts, can be a credible policy instrument in alleviating credit constraints and facilitating innovation.

This paper is closely related to Martin and Ventura (2016 [hereafter MV]), who examine a one-sector economy where firms face credit constraints. Negative productivity shocks cause investment demand and interest rates to fall thereby facilitating the emergence of bubbles. Furthermore, bubbles serve as collateral in their framework and can potentially "crowd in" and "crowd out" investments. Their paper focuses on business cycles and goes on to examine the optimal size of the bubbles in such an environment. Our framework on the other hand involves a two-sector growth model where the R&D sector which produces intermediate goods faces borrowing constraints. Our focus, unlike MV, is on the impact of bubbles on long-term growth and not business cycles. Crucially, we show that in such a framework bubbles arise only when there are adverse productivity shocks specific to this intermediate good sector. These shocks cause credit constraints in the sector to bind and reduce the demand for funds while leaving supply unchanged. The consequent drop in the interest rates facilitates bubbles. Importantly, in contrast to MV, adverse shocks to productivity in the final good sector do not give rise to bubbles in our model. Such a shock impacts both demand and supply of funds symmetrically and therefore leaves interest rates unchanged. MV would be a special case of our setup where there is only one sector and no growth. Our policy prescription for an economy with rational bubbles is also different from that obtained in MV. They show that in an economy where bubbles serve as collateral a "lean against the wind (LAW)" credit policy helps stabilize business cycles and hence is welfare enhancing. We, on the other hand, use our endogenous growth framework to show that a constant credit rule akin to the Friedman rule stimulates long-term growth and thereby trumps the LAW credit policy. Put differently, our analysis suggests that while a LAW policy might succeed in mitigating business cycles, it could have adverse consequences for long-term growth.

Our policy analysis here also contributes to the larger debate which has sought to analyze the causes of the Great Recession and the slow pace of recovery that has followed. Anzoategui et al. (2016) point out that conventional demand side-based explanations are unable to account for the extraordinarily sluggish movement of the economy back to the precrisis trend. Consistent with the explanation in our paper, they attribute the recessions and the sluggish recovery to reduce R&D expenditure. (2) Our policy prescription of government subsidized R&D expenditure therefore offers a new insight into the "secular stagnation" debate.

The rest of the paper is organized as follows. In Section II, we present our basic model without bubbles, Section III discusses equilibrium properties of our economy with bubbles, and Section IV examines the impact of productivity shocks on bubbles and innovation. In Section V, we examine the role of government policy, Section VI carries out stochastic simulations, and Section VII concludes the paper.

II. THE BASELINE MODEL-NO BUBBLES

This section extends the simple Schumpeterian two-sector growth model of Aghion and Howitt (2009) to include financial frictions. We begin by analyzing the model without bubbles. People consume one good, called the final good, which is produced by perfectly competitive firms using labor and a continuum of intermediate goods. There are overlapping generations of agents who live for two periods. Each generation consists of two types--workers and innovators. The mass of workers and innovators born in each period is one and e, respectively. They maximize the old-age consumption. Young workers are employed in the final good sector and earn wages at the beginning of the period. They save wages for old-age consumption.

Output growth results from technological innovations that enhance the productivity of intermediate products. Each young innovator-entrepreneur is given an opportunity to improve one type of intermediate goods at the end of the period. Successful innovation creates a new version of the intermediate product to be used in the next period, which is more productive than the previous versions. Importantly, we assume that entrepreneurs finance their R&D investment expenditure through risky borrowings from the market. Figure 1 summarizes the sequence of events.

A. Final Good Sector

The economy has one multipurpose final good. It can be consumed, used as an input to R&D, and used for the production of intermediate goods. The final good is produced by the technology

(1) [Z.sub.t] = [H.sup.1-[alpha].sub.t] [[integral].sup.[epsilon].sub.0] [A.sub.t] [(i).sup.1-[alpha]] [x.sub.t] [(i).sup.[alpha]] di

where [alpha] [member of] (0, 1), [H.sub.t] denotes labor input, [x.sub.t](i) is input of the latest generation of intermediate goods at t, and [A.sub.t](i) is productivity associated with it. The final goods market is perfectly competitive. Taking the final good as numeraire, the price of intermediate good equates its marginal product.

(2) [p.sub.t](i) = [alpha][[[A.sub.t](i)/[x.sub.t](i)].sup.1-[alpha]].

B. Intermediate Sectors

For each intermediate sector i, one innovator is born in each period. At the end of t, e mass of the young innovators engage in R&D to create a new version of the intermediate goods to be used in the next period. If the R&D activity is successful, the innovator is given the monopoly power in the sector at t + 1. If not, the monopoly passes to another old person at random. Let [[mu].sub.t](i) denote the probability of successful R&D for such an innovator working in the intermediate sector i. The productivity of intermediate good i is then, for [gamma] > 1,

(3) [mathematical expression not reproducible]

where [A.sub.t] = [[integral].sup.[epsilon].sub.0] [A.sub.t] (i)di is the average technology level at t. Given the homogeneity and the law of large numbers, the fraction of successful innovators is [[mu].sub.t] = [[mu].sub.t](i). The average technology evolves according to

(4) [A.sub.t+1] = [[mu].sub.t][gamma][A.sub.t] + (1 - [[mu].sub.t]) [A.sub.t] + [u.sub.t+1][A.sub.t]

where u denotes the direct shock to the productivity in the final good sector. The growth rate of the average productivity is

(5) [g.sub.t+1] = [A.sub.t+1] - [A.sub.t]/[A.sub.t] = [[mu].sub.t]([gamma] - 1) + [[mu].sub.t+1].

C. R&D Sector

The innovation technology for any sector i is given by

(6) [[mu].sub.t](i) = [square root of ([lambda][N.sub.t](i)/[A.sup.*.sub.t+1])] [less than or equal to] 1

where [[mu].sub.t](i) denotes the probability of successful innovation for young innovator i at period r, [lambda] is a R&D productivity parameter, [N.sub.t](i) is the R&D spending required for [[mu].sub.t](i), and [A.sup.*.sub.t+1] = [gamma][A.sub.t] is the target productivity level. The probability of innovation depends inversely on [A.sup.*] because as technology advances, it becomes more difficult to improve upon. Furthermore, the higher is the R&D efficiency, [lambda], the smaller is the R&D expenses to achieve a given [[mu].sub.t]. Rewriting Equation (6), we obtain the R&D spending as function of innovation probability [[mu].sub.t](i)

(7) [N.sub.t](i) = [[A.sup.*.sub.t+1]/[lambda]] [[mu].sub.t][(i).sup.2].

Since innovators have no income, they must borrow R&D expenditure [N.sub.t]. The prospective entrepreneurs finance their investment activity by selling credit contracts to the savers. To motivate the credit constraint, following Aghion, Banerjee, and Piketty (1999), we assume that there exists ex post moral hazard. In particular, the innovators can incur a cost and hide the proceeds from successful R&D activities. The hiding cost has to be paid when the borrowing is made. It is proportional to the expected profits and given by q[[mu].sub.t](i)[pi][A.sup.*.sub.t+1]. The parameter q [member of] (0, 1) reflects factors such as the lender's effectiveness in monitoring, legal costs of bankruptcy, policymakers' stance on protection of financial contracts, and other institutional elements. Borrowers would choose to be honest if the incentive compatibility condition below holds:

[[mu].sub.t](i) [R.sub.t+1](i)[N.sub.t] (i) [less than or equal to] [R.sub.t+1]q[[mu].sub.t](i)[mu] [A.sup.*.sub.t+1]

where [R.sub.t+1] (i) is the gross risky interest rate on a loan to entrepreneur i between t and t + 1 and [R.sub.t+1] denotes the risk-free interest factor. The left-hand side of the equation above represents the expected benefits from being dishonest while the right-hand side is the expected costs. Since the hiding cost has to be paid regardless of the success of R&D, the relevant interest rate is the risk-free interest factor, [R.sub.t+1].

Savers from the young generation are the only lenders in this overlapping generation's economy. Everyone has access to the storage technology. Thus, the expected return on lending must satisfy the following incentive compatibility

1 [less than or equal to] [[mu].sub.t](i)[R.sub.t+1](i) = [R.sub.t+1].

The last equality follows from the risk neutrality of lenders and competitive loan markets. Combining the incentive compatibility conditions of borrowers and lenders, we obtain the maximum amount which innovator-entrepreneurs can invest as follows:

(8) [N.sub.t](i) [less than or equal to] q[[mu].sub.t](i)[pi][A.sup.*.sub.t+1]

One could interpret the parameter q in the above equation as the loan-to-value (LTV) ratio. The entrepreneurs choose innovation probability [[mu].sub.t](i) to maximize their expected profit

(9) [mathematical expression not reproducible]

subject to the technology constraint (7) and the financial constraint (8).

Nonbinding Financial Constraint. We begin by considering the case where the financial constraint (8) is not binding. Solving (9) for this case yields the unconstrained innovation probability, which is given by

(10) [[mu].sup.*.sub.t] = [pi][lambda]/2[R.sub.t+1].

Combining (7) and (10), the unconstrained optimum R&D expense is

(11) [N.sup.*.sub.t](i) = [[[pi].sup.2][lambda][gamma]/4[R.sup.2.sub.t+1]][A.sub.t].

It follows from (11) that R&D expenditure in the unconstrained economy varies directly with R&D productivity [lambda] and inversely with the interest rate [R.sub.t+1].

The sole input to the intermediate sector is the final good. Using a linear technology, the monopoly producer can convert one unit of general good into one unit of the latest available version of intermediate good. Given the inverse demand (2), the monopolist i at period t chooses the quantity [x.sub.t](i) to maximize profit, [[PI].sub.t](i) = [alpha][A.sub.t][(i).sup.1-[alpha]][x.sub.t][(i).sup.[alpha]] - x(i). This implies the equilibrium quantity is

(12) [x.sub.t](i) = [[alpha].sup.1/1-[alpha]][A.sub.t](i).

Then, the equilibrium profit of the monopoly is

(13) [[pi].sub.t] (i) = [pi][A.sub.t] (i)

where [pi] = (1 - [alpha]) [[alpha].sup.1+[alpha].sub.1-[alpha]]. Combining (1) and (12), we find that gross output of the general good is given as

[Z.sub.t] = [PHI][A.sub.t]

where [PHI] = [[alpha].sup.[alpha]/1-[alpha]]. The gross domestic product (GDP) in our economy grows at the same rate as the technology growth rate [g.sub.t].

Binding Constraint. Under binding financial constraints, the R&D expenditure [[??].sub.t+1] is determined by (8) as follows:

(14) [[??].sub.t] = [[mu].sub.t](i)q[pi][A.sup.*.sub.t+1].

D. Market Equilibrium

We are now ready to solve the model. Young workers supply a unit of labor inelastically and thus [H.sub.t] = 1. In the competitive market for labor, old monopolists demand labor until the wage equals its marginal product, thus,

(15) wt = [omega][A.sub.t]

where [omega] = (1 - [alpha])[PHI].

Combining (7), (8), and [A.sup.*.sub.t+1] = [gamma][A.sub.t], the constrained innovation probability becomes

(16) [mathematical expression not reproducible].

Combining (14) and (16) constrained investment can be expressed as

[[??].sub.t+1] = v[A.sub.t],

where v = [??]q[pi][gamma] = [q.sup.2][[pi].sup.2][lambda][gamma]. We summarize our results in the lemma below.

LEMMA 1. The constrained investment increases with R&D efficiency [lambda], monopoly profits [pi] and the LTV ratio, q.

Essentially, higher values of [lambda], [pi], and q raise the pledgeable "collateral." This in turn relaxes the restraint on borrowing and drives up constrained investment.

Savers are willing to supply credit to the entrepreneur as long as the return is greater than the opportunity cost. Each saver can supply their wages earned as credit as long as their incentive compatibility is satisfied. Therefore, the supply of credit available at the end of t is described as follows:

(17) [mathematical expression not reproducible].

The demand for credit follows from (11) and (14). Formally,

(18) [mathematical expression not reproducible].

It follows from (18) that due to the borrowing constraint faced by the entrepreneurs, the credit demand curve is kinked at

[R.sub.t+1] = [bar.R] = [square root of ([[pi].sup.2][lambda][gamma]/4v)] = 1/q.

To streamline the discussion, we focus on the case with [bar.R] [greater than or equal to] 1 or q [less than or equal to] 1/2. If not, the lender's incentive compatibility is never satisfied when the borrowing is constrained and trivially there will be no lending in equilibrium.

The dynamics of the credit market can be analyzed with (17) and (18). By construction, the unconstrained credit market equilibrium occurs where the demand and supply curves intersect along the downward sloping section of [D.sub.t] as in Figure 2A.

Hence, the unconstrained equilibrium interest rate is

(19) [R.sup.*] = [square root of ([[pi].sup.2][lambda][gamma][epsilon]/4[omega])].

The constrained financial market emerges when the borrowing is restricted to the size of collateral. This is depicted in Figure 2B.

Since [S.sub.t] - [D.sub.t] > 0, savers compete for existing lending opportunities. This drives the interest rate down to the opportunity cost. It follows that in the constrained economy the interest rate R is pinned down by the lender's incentive compatibility condition and given by

(20) [??] = 1.

The credit market equilibrium can be summarized by the following excess credit supply curve:

(21) [mathematical expression not reproducible].

The key findings in this section can be summarized in the Lemma below.

LEMMA 2. The constrained equilibrium arises if and only if

[xi] = [omega] - [epsilon]v > 0.

Combined with Lemma 1, this result indicates that the constrained equilibrium is more likely to arise when the LTV ratio (q) is low or when the value of "collateral" is small due to low R&D productivity ([lambda]) or low monopoly profits ([pi]).

Next, we compare growth rates in the constrained and unconstrained economies. Combining (5), (10), and (19), the unconstrained innovation probability and the corresponding growth rate are

[[mu].sup.*] = [square root of ([omega][lambda]/[epsilon][gamma])], [g.sup.*] = [square root of ([omega][lambda]/[epsilon][gamma])]([gamma] - 1) + u.

Analogously, using (6) and (14), the innovation probability and the growth rate in a constrained economy are given by

(22) [mathematical expression not reproducible].

Given the condition in Lemma 1, it can be seen that [g.sup.*] > [??]. Since the output growth rate in the economy is a function of investment in the R&D sector, it is not surprising that the growth rate in a constrained economy is lower than the unconstrained case.

III. BUBBLES

This section introduces rational bubbles to the model. Essentially, bubbles represent a pro-rata share in a Ponzi scheme. In the market for bubbles, some of the bubbles are old since they have been initiated by previous generations of entrepreneurs and sustained by a chain of savers. Some are newly initiated by the current generation of young entrepreneurs. We use [B.sub.t] to denote the total size of the old bubbles carried over from t - 1 to t and [B.sup.N.sub.t] to denote the new bubbles issued at time t, respectively. Bubbles start randomly and are purchased by young savers in the expectation that the future generation will pay more to buy them.

Formally, rational bubbles are represented by a (stochastic) process [{[B.sub.t], [B.sup.N.sub.t]}.sup.[infinity].sub.t=0] which satisfies the following conditions.

(23) [E.sub.t][[R.sup.B.sub.t+1]] = [E.sub.t] [[B.sub.t+1]/[B.sub.t] + [B.sup.N.sub.t]] = [R.sub.t+1]

(24) [B.sub.t] + [B.sup.N.sub.t] [less than or equal to] [w.sub.t]

(25) [B.sub.t], [B.sup.N.sub.t] [greater than or equal to] 0

(26) [mathematical expression not reproducible].

The first condition (23) indicates the no-arbitrage condition. It requires that bubbles must deliver the same return as the alternative investment opportunity in equilibrium to attract investors. The next line (24) implies that the size of bubbles cannot exceed the resources available for bubble purchase. Equation (25) reflects the assumption that bubbles can be freely disposed and hence must be non-negative. Finally, (26) implies that the growth rate of bubbles cannot exceed the GDP growth rate in the long run. If this were not to be the case, then the aggregate bubble would eventually be too large for the next generation to purchase and therefore would not be sustainable.

A. Bubbles in the Capital Market

Traditionally, the literature on bubbles has focused on dynamically inefficient economies. The reasoning follows easily from our discussion in the previous section. As (23) makes it clear, savers wish to purchase bubbles if these grow at least as fast as the interest rate. Equation (26) implies that as long as the growth rate of bubbles does not exceed the output growth rate, savers will be able to mobilize enough resources to procure the bubble. Put together, it follows that bubbles are possible in equilibrium only if the interest rate does not exceed the output growth rate. Such a situation arises naturally when the frictionless economy is dynamically inefficient. Low interest rates in these economies arise due to excessive savings which in turn result in inefficiently high investment and low consumption.

The seminal work by Tirole (1985) among others has established that bubbles can be welfare enhancing in such a scenario by reducing inefficient investment and increasing consumption. However, Martin and Ventura (2012) point out that the conclusions of this literature are at odds with the data as bubbles are typically characterized by rising and not falling investment. They instead show that the interest rate in an otherwise dynamically efficient economy can fall below the growth rate when the agents are shut out of credit markets. Here, the interest rate is low because the demand for funds is low and, if anything, there might be underinvestment.

This is also the focus of our paper. Importantly, we examine how productivity shocks to the innovation sector alter borrowing constraints leading to the emergence or collapse of bubbles in dynamically efficient economies. To ensure dynamic efficiency in the unconstrained steady state, we make the following assumption:

ASSUMPTION 1. (Dynamic efficiency) [R.sup.*] [greater than or equal to] 1 + [g.sup.*].

This inequality ensures that the frictionless economy is dynamically efficient and investment is always productive. The left-hand side of the inequality represents the average return earned by the savers, which is also equal to the ex ante marginal productivity of the investment in the unconstrained regime. In the absence of borrowing limits, no excess credit supply exists and the returns on financial instruments such as savings and bubbles directly reflect the productivity of real investment. Due to the availability of excess credit, in the constrained case, the marginal product of investment is larger than the average return earned by savers. But the endogenous borrowing constraints arising from the moral hazard issue prevents the savers from offering further loans. Private sector bubbles can play a constructive role in this environment by transferring resources from savers to credit constrained entrepreneurs. As in other standard rational bubble models, savers purchase bubbles in the expectation that the future generation will buy it.

Formally, the maximum R&D investment that entrepreneurs can make in period tin the presence of bubbles is given by

(27) [N.sub.t](i) [less than or equal to] q[[mu].sub.t](i) [pi][A.sup.*.sub.t+1] + [b.sup.N.sub.t] (i)

where [b.sup.N.sub.t] (i) is the bubble initiated by entrepreneur i in time t. (3) This means that the borrowing in the presence of bubbles is [N.sub.t] (i) - [b.sup.N.sub.t (i). Thus, the capital demand curve in the economy with bubbles can be rewritten as

(28) [mathematical expression not reproducible]

where [B.sup.N.sub.t] = [[integral].sup.[epsilon].sub.i=0][b.sup.N.sub.t](i)di denotes the total amount of new bubbles made in period t. It can be seen from (28) that the introduction of new bubbles [B.sup.N.sub.t] reduces the borrowing required for investment, causing a downward parallel shift in the credit demand curve [D.sup.B.sub.t] at all ranges of the interest rate.

Savers allocate their wage income into lending and purchasing bubbles. The credit supply curve in the economy with bubbles is given by

(29) [mathematical expression not reproducible].

Since a part of the resources of the savers is diverted to purchase bubbles (both old and new), the nonzero part of the credit supply curve (29) shifts by the total size of the bubbles, [B.sub.t] + [B.sup.N.sub.t].

B. Equilibrium in an Economy with Bubbles

This section characterizes the equilibrium in an economy with bubbles and analyzes related dynamics. Combining (19), (28), and (29), we obtain an expression for the excess supply of credit:

(30) [mathematical expression not reproducible].

Here, the interest factor becomes a function of the existing bubbles and given by

(31) [R.sub.t+1] = [E.sub.t][R.sup.B.sub.t+1] = [epsilon][[pi].sup.2][lambda][gamma]/4 ([omega] - [B.sub.t]/[A.sub.t]) [greater than or equal to] [R.sup.*].

The last inequality follows from the non-negativity of bubbles and [partial derivative][E.sub.t][R.sup.B.sub.t+1]/[partial derivative][B.sub.t] > 0.

Note that the excess credit supply and interest rate are functions of [B.sub.t] but not of [B.sup.N.sub.t]. This happens because the introduction of new bubbles, [B.sup.N.sub.t], reduces the demand and supply of credit by exactly the same amount, leaving the excess credit supply equation in the economy unaltered. The old bubble [B.sub.t], on the other hand, impacts only the supply curve, thereby reducing the availability of credit in the economy.

To study the equilibrium dynamics of bubbles, it is useful to redefine our bubble variable in order to make the model recursive. Let [z.sub.t] = [B.sub.t]/[A.sub.t] and [z.sup.N.sub.t] = [B.sup.N.sub.t]/[A.sub.t] be the stock of old and new bubbles normalized by the level of aggregate productivity in the economy. Then, we can rewrite (23) and (24) as (4)

(32) [mathematical expression not reproducible]

(33) [mathematical expression not reproducible]

(34) [z.sub.t] + [z.sup.N.sub.t] [member of] [0, [omega]].

Now, we are ready to analyze the dynamics of bubbles in our economy. We begin by establishing that bubbles cannot arise in an unconstrained economy.

LEMMA 3. Bubbles in the economy are possible if and only if the economy is credit constrained ([xi] [greater than or equal to] 0).

We provide an intuitive graphical proof for the above lemma. Figure 3 uses (32) to plot [E.sub.t][z.sub.t+1] against [z.sub.t] assuming [z.sup.N.sub.t+s] = 0 for all s [greater than or equal to] 0. The top panel illustrates the case when the economy is financially unconstrained, implying there exists no excess credit supply, [xi] < 0. Note, in Figure 3A, that the dynamic efficiency condition (Assumption 1) and (31) ensure that the slope d[z.sub.t+1]/d[z.sub.t] = [E.sub.t][R.sup.B.sub.t+1]/1 + [g.sub.t+1] > 1. This implies that the expected return Sfrom holding the bubble in equilibrium exceeds the growth rate in the economy. It immediately follows that this is a violation of (33), which implies the size of the bubble [z.sub.t] will eventually overtake the availability of resources making it nonviable. Hence, z, = [z.sup.Nt.sub.t] = 0 for all t is the only feasible equilibrium for the unconstrained case.

On the other hand, the bottom panel shows the bubble dynamics for the constrained economy in which [xi] > 0. Due to the availability of positive excess credit, the interest rate is low and the slope is d[z.sub.t+1]/d[z.sub.t] = 1/1+[g.sub.t+1] < 1 for [z.sub.t] [less than or equal to] [xi], in Figure 3B. This implies the growth rate of the economy exceeds the return on investment. Thus (33) is satisfied. The economy stays in this constrained regime as long as [z.sub.t] [less than or equal to] [xi] = [omega] - [epsilon]v < [omega]. This satisfies the resource constraint (34). Therefore, the process of such bubbles is indeed sustainable.

To summarize. Lemma 3 shows that the existence of rational bubbles in the presence of dynamic efficiency is possible when financial constraints are binding, in which the return on savings are uncoupled from the marginal productivity of borrowers. The excess supply of funds together with low required returns on savings support environments for rational bubbles.

We complete the characterization of bubbles by establishing the upper bound of the bubble size.

COROLLARY 1. In the credit constrained economy in which [xi] [greater than or equal to] 0: (i) The old bubble [z.sub.t] cannot exceed max {0, [xi]}. (ii) Total size of bubbles z, + zf is bounded by max {0, min ([xi] (1 + [g.sup.B.sub.t+1]), [omega]}} where [g.sup.B.sub.t+1] is the growth rate of the economy after introduction of a new bubble of size [z.sup.N.sub.t].

Proof. The proof follows from examining Figure 3B. In the range [z.sub.t+1] > [xi], there is no excess credit and bubbles compete with real investments for the savings in the unconstrained economy. The dynamic efficiency condition and (31) imply that the required return in this range exceeds the GDP growth rate. This is reflected in the jump in the slope of the graph. The high rates of return in turn make bubbles nonviable in this region following the arguments in Lemma 3. The maximum size of the bubble in this region is therefore zero. In contrast, the range [z.sub.t+1] < [xi], is characterized by excess credit. The slope of graph is below the 45[degrees] line because the required return on the bubbles is lower than the economic growth rate. The low rates of return in turn make bubbles viable in this region. Since this argument holds for each time period t, (i) follows. To establish the maximum size of the total bubbles ([z.sub.t] + [z.sup.N.sub.t]), first observe that (32) implies [E.sub.t][z.sub.t+1] = ([z.sub.t] + [z.sup.N.sub.t]) / (1 + [g.sub.t+1]). Further, (i) implies [E.sub.t][z.sub.t+1] [less than or equal to] [xi] for bubbles to be sustainable. Put together, it follows that the size of the total bubble must be bounded by [z.sub.t] + [z.sup.N.sub.t] [less than or equal to] [xi] (1 + [g.sub.t+1]) for some [g.sub.t+1] [greater than or equal to] 0. Finally, [z.sub.t] + [z.sup.N.sub.t] [less than or equal to]< [omega] to due to the resource constraint (34).

This corollary establishes the upper bound on bubbles which an economy can support. The maximum size of bubbles is determined by the amount of excess credit supply, [xi], which would prevail in the constrained equilibrium without bubbles. The low interest rates in a credit constrained economy make bubbles viable. When the economy is unconstrained, the rate of return in the economy is too high for bubbles to be sustainable in the presence of dynamic efficiency assumption.

The next section characterizes economic growth under the bubble and derives the upper bound of [g.sub.t+1] in such settings.

C. Bubbles and Economic Growth

In the previous section, we have shown that the bubbles can exist only in a credit constrained economy. This section examines the link between bubbles and economic growth in such an economy.

COROLLARY 2. For a given new bubble [z.sup.N.sub.t] = [delta][epsilon]v where [delta] > 0, the maximum growth rate is [g.sup.B] = [??][(1 + [delta]).sup.0.5].

Proof. Note, under decreasing returns to scale technology in the R&D sector, an equal distribution among entrepreneurs is the most efficient use of new bubbles. An equal distribution of [B.sup.N.sub.t] = [delta][epsilon]v[A.sub.t] among e innovators yields the R&D spending per entrepreneur of v(1 + [delta])[A.sub.t]. Substituting this in Equation (6), the result follows. Since [delta] > 0, it follows that the growth rate of the economy under bubbles, [g.sup.B] is strictly larger than the constrained growth rate [??]. (5)

The above discussion implies that new bubbles by reallocating savings to entrepreneurs in a credit constrained economy are expansionary in nature. Having examined the effects of new bubbles on economic growth, we proceed to establish the maximum size of new bubbles in steady state.

COROLLARY 3. The maximum size of steady state new bubbles is [mathematical expression not reproducible], the maximum steady state growth rate in the economy with bubbles, is defined in the Appendix.

Proof. See Appendix.

Corollary 3 establishes that bubbles are not only feasible in constrained steady state but also, by facilitating growth, create room for further bubbles in the future. Intuitively, new bubbles "crowd in" investment by transferring idle resources from the savers to entrepreneurs in the credit constrained economy. The consequent increase in growth results in excess credit, paving the way for the emergence of new bubbles in the constrained economy.

To see this graphically, consider the economy initially located at X in Figure 4. The creation of a new bubble, [DELTA]z, increases productivity and economic growth causes the [z.sub.t+1] line to rotate downward, moving the economy to point W. Note, here, there are two effects in play. First, the rise in economic growth means that the size of the existing bubble shrinks relative to the size of the economy, in other words, the size of the normalized bubble, z, is lower at point W relative to Y. Second, the rise in wages due to increased productivity means there is an increase in the supply of savings. This excess credit supply in turn opens the door to creation of further bubbles in the economy provided the total size of bubbles is smaller than the upper limit in Corollary 1.

It, however, bears emphasis that in addition to "crowding in" productive investment, new bubbles can potentially also "crowd out" investment. This is easily seen by noticing that the creation of new bubbles increases the total size of bubbles in the economy. Consequently, future generations have to devote increasing amount of their resources in procuring existing bubbles. This has the effect of crowding out productive investment. On balance, as long as the economy is credit constrained and the size of the bubble does not exceed the maximum permissible limit, bubbles facilitate innovation and growth in our framework.

IV. BUBBLES, R&D PRODUCTIVITY, AND FINANCIAL CONSTRAINTS

This section discusses the link between bubbles and permanent shocks to the R&D sector and the final goods sector in our model. The literature has largely focussed on productivity shocks u to the final good sector, in explaining bubbles in the economy (see Martin and Ventura 2012, 2016 for example). Instead, our model has two sectors and we assume that the R&D sector is subject to a financial constraint due to intangibility of output and associated moral hazard issues whereas the final good sector is not. In this setting, bubbles may arise when the financially constrained R&D sector is subject to adverse productivity shocks. Importantly, adverse shocks to productivity in the unconstrained final good sector u, do not result in bubbles in our framework. The key results in this section are summarized in the corollary below:

COROLLARY 4. 1. Adverse shocks to X or the productivity in the intermediate good producing R&D sector may result in rational bubbles.

2. Adverse shocks to u, the final good productivity, do not result in rational bubbles.

To see this, consider an economy initially in the unconstrained steady state which faces an unanticipated permanent negative shock to R&D productivity, X. The consequent reduction in the value of collateral causes investment demand to fall. When the shock is large, the borrowing constraint becomes binding, which in turn opens the door to rational bubbles by Lemma 3. It therefore follows that a sufficiently large increase in X, by increasing the demand for investment and reducing excess credit can eliminate existing bubbles. Conversely, it follows from Lemmas 1-3 that the conditions for bubbles are not affected when there are productivity shocks specific to the final good sector. A decline in u induces a parallel shift in both supply and demand for funds leaving the (normalized) excess supply of funds and hence interest rates unchanged. In other words, unlike the shock to X, it does not have asymmetric effects on borrowers and lenders in our two-sector model. Our results here therefore differ from MV where productivity shocks to the final good sector trigger bubbles in the economy.

Interestingly, a mean-reverting shock in X does not shift the economy from a fundamental into a bubble regime. Consider an economy that is initially in unconstrained steady state. Now suppose that a large unexpected negative temporary shock in [lambda] hits the economy so that the innovators are financially constrained temporarily. Everyone in the economy knows that this shock will diminish eventually at some point t + s and revert to the initial unconstrained steady state. Given (26) and Assumption 1, the bubbles will have no value at t + s. By backward induction, no one will demand the bubble at period t.

V. GOVERNMENT INTERVENTION

We have shown that a negative permanent shock to the productivity in the R&D sector can result in credit constraints binding, causing a fall in the innovation probability and aggregate growth rate. In such a milieu, bubbles generated in the private sector help "crowd in" investments and raise the growth rate of the economy above the constrained growth rate.

However, since bubbles originating in the private sector are largely driven by investor sentiment, there are the obvious fragility concerns associated with them. Is there therefore anything governments can do to help alleviate credit constraints and facilitate R&D investment? In this section, we propose government fiscal policy in the form of R&D credit financed by perpetually rolled-over public debts as a way of mitigating investment credit constraints in dynamically efficient economies. Below, we establish that such a regime alleviates credit constraints much like a private sector bubble without running the risk of bubble crashes.

We now proceed to describe an economy without private bubbles but one where there is government fiscal policy in the form of credit contracts to the R&D sector financed by perpetually rolled-over public debts. At the end of t, the government issues new one-period discount bonds [d.sub.t+1]/[R.sup.d.sub.t+1] = [d.sub.t] + [d.sup.N.sub.t]. These funds are then used to redeem old bonds, [d.sub.t], held by the old savers as well as distribute research credits among young entrepreneurs. [R.sup.d.sub.t] is the gross return promised on the debt. Formally, the path of government debt satisfies the following conditions:

(35) [mathematical expression not reproducible]

(36) [d.sub.t] + [d.sup.N.sub.t] [less than or equal to] [w.sub.t]

(37) [mathematical expression not reproducible]

where (35)-(37) are analogous to (23)-(26).

One concern that may arise with the above proposal is that public debt in this scenario is a bubble in itself and hence raises the same fragility concerns surrounding a bubble. If the next generation refuses to roll over the debt, the bubble crashes. Caballero and Krishnamurthy (2006) emphasize that, unlike the bubble itself, the debt rollover is not subject to stochastic breakdown. The key is that the government can pledge to fill any shortfalls in the debt rollover with income taxes. This off-equilibrium commitment is sufficient to ensure that the debt is always repaid, even though no taxes are ever actually collected in equilibrium.

In the next section, we carry out a simulation exercise where we proceed to show that such a tax payer-funded initiative implemented by a credible government does a better job of facilitating growth in a credit constrained economy than private sector bubbles. Intuitively, the fragility of bubbles in the private sector means that the ex post return on surviving bubbles must be higher than that obtained under the government-funded credit. This leads to higher investment and growth under the government regime.

VI. NUMERICAL SIMULATION

In this section, we run a series of stochastic simulation exercises where we compare dynamics in the face of temporary shocks to R&D productivity ([lambda]) under the following three scenarios: (a) the baseline economy described in Section II, (b) the economy with private sector bubbles described in Section III, and (c) the economy with government-funded credits described in Section V.

For this exercise, we assume that the steady state of the economy is financially constrained, thus satisfying the condition in Lemma 3 so that the existence of bubbles is possible. The R&D-specific productivity shock is modeled as mean reverting 2-state Markov chain. In particular, assume that [[lambda].sub.t] = [bar.[lambda]] + [DELTA][[lambda].sub.t] > 0 and [DELTA][[lambda].sub.t] [member of] {-[sigma], [sigma]} where [sigma] > 0 is the size of shock. Bubbles emerging in the private sector are subject to a sentiment shock, represented by a dummy variable [D.sup.s.sub.t]. If [D.sup.s.sub.t] = 1 with the probability of [[delta].sup.s.sub.t] the sentiment is favorable and savers accept bubbles as long as (32) is met. If [D.sup.s.sub.t] = 0, with the probability of 1 - [[delta].sup.s.sub.t], existing bubbles burst ([B.sub.t] = 0) and new bubbles are not created. Due to the noarbitrage condition (32), the bubbles must pay the expected return on alternative investment, Rt. In the presence of sentiment shock, this means that the realization of the return on bubbles must satisfy

(38) [mathematical expression not reproducible].

The difference [R.sup.B.sub.t] - [R.sub.t] = (1 - [[delta].sup.s.sub.t](v)/[[delta].sup.s.sub.t][R.sub.t] represents the risk premium compensating for exposure to sentiment shock. The creation of new bubbles is subject to a similar shock. When [D.sup.n.sub.t] = 1, with probability of [[delta].sup.n.sub.t], entrepreneurs are able to sell new bubbles conditional on [D.sup.s.sub.t] = 1. With probability 1 - [[delta].sup.n.sub.t], [D.sup.n.sub.t] = 0 and [z.sup.N.sub.t] = 0.

The realization of the shocks is revealed to all agents at the beginning of each period. To simplify, we assume that transition probabilities for productivity and bubble shocks are independent. For simplicity, the normalized size of new bubbles is fixed at [z.sup.N.sub.t] = n. In the simulation where the government debt policy is introduced, we set [d.sup.N.sub.t]/[A.sub.t] = n for all t, to facilitate comparison. The proceeds from selling bubbles and the government debts are distributed equally among young entrepreneurs. The values of parameters used in the simulation exercise and short description are reported in Table 1.

A. Private Bubbles versus Government Debt Policy in Simulated Economy

Figure 5 plots the sample time series paths of key variables under the three regimes over 100 periods. The initial GDP is normalized to unity. To facilitate comparison across regimes, in this graph, we remove the log-linear time trend in B, d, N, and A using the sample GDP growth rate for the baseline economy. Thus, the plots represent deviation from the benchmark economy's log-linear GDP growth trend and could be interpreted as a counterfactual outcome under private sector bubbles and debt policy, respectively.

In the baseline economy (the blue solid line), both R&D investment and the GDP growth rate follow the path of the R&D productivity, [lambda]. Next, we consider an economy with bubbles (the red broken line). It is clear that both investment and economic growth are higher in this case relative to the baseline economy since bubbles channel idle funds into R&D investment. Note that the realized return on bubbles fluctuate between 0 and R/8s. Periods in which investors expect the bubble to collapse (low 8) result in high values of Rb, since investors expect to be compensated for the higher risk undertaken. Finally, the green dotted line represents the case where R&D investment is subsidized by government credits. Here, as expected, R&D investment and therefore output growth is higher than in the other two regimes.

To summarize, the key friction or inefficiency in our framework is the presence of borrowing constraints in the R&D sector. Private sector bubbles can play a constructive role in this environment by transferring resources to credit constrained entrepreneurs. However, the fragility of private sector bubbles calls into question their effectiveness. Instead, credible tax-funded government policy in the form of credits to the R&D sector does a much better job at addressing the fundamental inefficiency in the economy. Our results indicate after 100 periods, the output under the prescribed government policy is about thrice higher than the baseline case.

B. Credit Policy

Interestingly, our framework also allows us to weigh in on the debate on credit policy in the presence of rational bubbles. In an economy where bubbles serve as collateral, MV advocate a LAW credit policy to stabilize business cycles. Gall (2014) and Gall and Gambetti (2015), however, point out that such a policy in the presence of nominal rigidities will end up increasing volatility in the economy. We have a slightly different take on this debate as we focus on the implications of credit policy on long-term growth. We contrast long-term growth under LAW with that obtained under a constant credit policy. Specifically, under the LAW policy credit, d/A, rises (falls) when R&D productivity is low (high). Formally, the LAW credit rule is given by

[d.sup.N.sub.t]/[A.sub.t] =-[epsilon]v[DELTA][[lambda].sub.t]/[bar.[lambda]].

This policy would supply (withdraw) credit during [DELTA][[lambda].sub.t] < 0 (> 0). It can be shown that this policy offsets the effect of the R&D shock on the growth rate completely. On the other hand, the constant credit rule increases d/A at a constant rate and is given by

[d.sup.N.sub.t]/[A.sub.t] = [epsilon]v[sigma]/[bar.[lambda]]/2.

Figure 6 illustrates that the constant credit policy leads to higher long-term growth when compared with the LAW policy. (6) The intuition once again follows from our discussion in Section C. As long as the economy is borrowing constrained and the credit in the economy does not exceed the maximum permissible limit, a constant credit rule by alleviating borrowing constraints will succeed in delivering a higher long-term growth rate in the economy.

VII. CONCLUSION

In this paper, we introduce borrowing constraints into a standard two-sector Schumpeterian growth model and examine the impact of bubbles on R&D expenditure and innovation. Bubbles in our model emerge when there are permanent adverse shocks to productivity in the credit constrained intermediate good sector. Importantly, in contrast to much of the existing literature, bubbles do not appear when there are adverse shocks to productivity in the final good sector. In such a framework, we establish that bubbles drive innovation during periods of low productivity by alleviating credit constraints. We also contribute to the secular stagnation debate by advocating a supply side debt-financed credit policy. Our bubble-mimicking credit policy that subsidizes R&D expenditure alleviates credit constraints, thereby "crowding in" investment. Finally, our results suggest that while a LAW credit policy might succeed in mitigating business cycles, it could potentially lower long-term growth.

Our analysis on the impact of bubbles is largely normative. It would be interesting to empirically investigate the role bubbles have played in fostering innovation, particularly during periods of low productivity. We leave this extension for future research.

APPENDIX: PROOF OF COROLLARY 3

The proof consists of three parts. We first establish that the steady state path of growth maximizing bubbles is characterized by bubbles of size [mathematical expression not reproducible]. Suppose otherwise. Then, a chain of new bubbles can be additionally issued by Corollary 1. Given Corollary 2, adding [DELTA][z.sup.N] is growth enhancing and sustainable until the total size of bubble reaches [mathematical expression not reproducible]. This is a contradiction.

To see that maximum size of the new bubble is given by [mathematical expression not reproducible], observe that growth maximization requires [mathematical expression not reproducible] and z = [xi] by Corollary 1. Thus, [mathematical expression not reproducible].

To solve for [mathematical expression not reproducible], once again, consider [z.sup.N.sub.t] = [delta][epsilon]v, where new bubble per innovator is a fraction [delta] of the bubbleless constrained investment. Thus, the maximum bubbly growth rate can be determined by noticing [mathematical expression not reproducible]. This system of equations is straightforward to solve. ?

ABBREVIATIONS

GDP: Gross Domestic Product

LAW: Lean against the Wind

LTV: Loan-to-Value

MV: Martin and Ventura (2016)

doi: 10.1111/ecin.12695

REFERENCES

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Aghion, P, P. Howitt, and D. Mayer-Foulkes. "The Effect of Financial Development on Convergence: Theory and Evidence." Quarterly Journal of Economics, 120(1), 2005, 173-222.

Anzoategui, D., D. Comin. M. Gertler, and J. Martinez. "Endogenous Technology Adoption and R&D as Sources of Business Cycle Persistence." National Bureau of Economic Research Working Paper No. 22005, 2016.

Barlevy, G. "On the Cyclicality of Research and Development." American Economic Review, 97(4), 2007, 1131-64.

Bernanke, B. S., and M. Gertler. "Agency Costs, Net Worth, and Business Fluctuations." American Economic Review, 79(1), 1989, 14-31.

Caballero, R. J., and A. Krishnamurthy. "Bubbles and Capital Flow Volatility: Causes and Risk Management." Journal of Monetary Economics, 53(1), 2006, 35-53.

Farhi, E., and J. Tirole. "Bubbly Liquidity." Review of Economic Studies, 79, 2011, 678-706.

Gall, J. "Monetary Policy and Rational Asset Price Bubbles." American Economic Review, 104(3), 2014, 721-52.

Gall, J., and L. Gambetti. "The Effects of Monetary Policy on Stock Market Bubbles: Some Evidence." American Economic Journal: Macroeconomics, 7(1), 2015, 233-57.

Gordon, R. J. "Secular Stagnation: A Supply-Side View." American Economic Review, 105(5), 2015, 54-59.

Hirano, T., and N. Yanagawa. "Asset Bubbles, Endogenous Growth, and Financial Frictions." Review of Economic Studies, 84(1), 2016, 406-43.

Kiyotaki, N., and J. Moore. "Credit Cycles." Journal of Political Economy, 105(2), 1997, 211-48.

Lansing, K. "Speculative Bubbles and Overreaction to Technological Innovation." Journal of Financial Transformation, 26,2009,51-54.

Martin, A., and J. Ventura. "Economic Growth with Bubbles." American Economic Review, 102(6), 2012, 3033-58.

--. "Managing Credit Bubbles." Journal of the European Economic Association, 14(3), 2016, 753-89.

Miao, J., and P. Wang. 2011. "Asset Bubbles and Credit Constraints." Mimeo. Boston University.

Takao, K. "Asset Bubbles and Economic Growth under Endogenous Market Structure." Macroeconomic Dynamics, 2017, 1--22.

Tirole, J. "Asset Bubbles and Overlapping Generations." Econometrica, 53(6), 1985, 1499-528.

JONG KOOK SHIN and CHETAN SUBRAMANIAN *

* We are grateful to the Editor, Martin Gervais, and one anonymous referee for insightful comments and suggestions. The usual disclaimer applies.

Shin: Lecturer (Assistant Professor), Newcastle University Business School, Newcastle University, Newcastle upon Tyne, NE1 4SE, UK. Phone +44 (0) 191 208 1677, Fax +44(0) 191 208 1738, E-mailjong.shin@newcastle.ac.uk

Subramanian: Professor, Department of Economics and Social Sciences, Indian Institute of Management (IIM), Bangalore, 560076, India. Phone +91 (0) 80 2699 3345, Fax +91 (0) 80 2658 4050, E-mail chetan.s@iimb.emet.in

(1.) See also Farhi and Tirole (2011), Miao and Wang (2011), Martin and Ventura (2012, 2016), Hirano and Yanagawa (2016), and Takao (2017) on the role of bubbles in alleviating credit constraints.

(2.) Please also see Barlevy (2007) and Aghion, Howitt, and Mayer-Foulkes (2005). Consistent with our results, these papers document that R&D expenditures in the Unied States are procyclical.

(3.) More generally, one can analyze an economy in which both innovators and savers can initiate new bubbles and/or the new bubbles are large enough that R&D can be self-financed. The introduction of such productive bubbles will not alter the key predictions of our model.

(4.) When [z.sub.t] - [xi] = 0, any return within the [1, [R.sup.*]] range is consistent with the credit market equilibrium. Without loss of generality, we assign this nongeneric borderline case to the constrained economy.

(5.) Implicitly, we assume that the value of 5 is not too large so that the addition of new bubbles does not violate the upper bound in Corollary 1, nor is it sufficient for funding [N.sup.*], the optimal level of R&D.

(6.) For this exercise, we use the same set of parameters in Table 1.

Caption: FIGURE 1 Sequence of Events

Caption: FIGURE 2 Credit Market Equilibrium

Caption: FIGURE 3 Bubble Dynamics

Caption: FIGURE 4 Effect of New Bubbles

Caption: FIGURE 5 Comparison of Bubbles and Government Debt Policy

Caption: FIGURE 6 Constant Credit Supply versus LAW Policy
TABLE 1
Simulation Parameters

Parameter                            Value

q                                    0.5
[alpha]                              0.3
[epsilon]                            0.050
[gamma]                              5
[bar.[lambda]], [sigma]              0.5, 0.05

Pr([[lambda].sub.t] =                0.98
  [[lambda].sub.H]|[[lambda].sub.
  t-1] = [[lambda].sub.h] =
  Pr([[lambda].sub.t] =
  [[lambda].sub.L]|[[lambda].sub.
  t-1] = [[lambda].sub.L])
Pr([D.sup.s.sub.t] =                 0.90
  1|[D.sup.s.sub.t-1] = 1) =
  Pr([D.sup.s.sub.t] =
  0|[D.sup.s.sub.t-1] = 0)
Pr([D.sup.n.sub.t] =                 0.90
  1|[D.sup.n.sub.t-1] = 1) =
  Pr([D.sup.n.sub.t] =
  0|[D.sup.n.sub.t-1] = 0)

Parameter                            Remark

q                                    Loan-to-value ratio
[alpha]                              Capital share
[epsilon]                            Measure of innovators (a)
[gamma]                              Target productivity
[bar.[lambda]], [sigma]              Steady state R&D efficiency
                                       and the size of shock
Pr([[lambda].sub.t] =                Conditional probability
  [[lambda].sub.H]|[[lambda].sub.      concerning R&D efficiency
  t-1] = [[lambda].sub.h] =
  Pr([[lambda].sub.t] =
  [[lambda].sub.L]|[[lambda].sub.
  t-1] = [[lambda].sub.L])
Pr([D.sup.s.sub.t] =                 Conditional probability
  1|[D.sup.s.sub.t-1] = 1) =           concerning sentiment shock
  Pr([D.sup.s.sub.t] =
  0|[D.sup.s.sub.t-1] = 0)
Pr([D.sup.n.sub.t] =                 Conditional probability
  1|[D.sup.n.sub.t-1] = 1) =           concerning new bubble shock
  Pr([D.sup.n.sub.t] =
  0|[D.sup.n.sub.t-1] = 0)

(a) According to Global Entrepreneur Monitor (2016), 13% of the U.S.
population run their own business and 37% of them engage in some sort
of innovation. In our model, the fraction of innovator is [epsilon]/
(1 + [epsilon]) = 0.13 x 0.37.
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