A. The Increased Importance of Intangible Capital

A building block in the model we present later is the technology entrepreneurs use to produce the single final good

where A is a measure of productivity, k represents physical capital, z represents intangible capital and l stands for labor. Several papers have employed similar versions of this specification to analyze the effects of intangible capital.

McGrattan and Prescott (2007) adopt an identical technology to evaluate the ability of the otherwise conventional neoclassical growth model to account for the business cycle in the U.S. economy during the 1990s. The authors show that including intangibles significantly improves the performance of the calibrated model. Hall (2000) introduces the concept of E-capital as an input for final production, which for many purposes is equivalent to our notion of intangibles. Relying on a non-parametric approach to calculate the stock of E-capital, Hall claims it might be an important common element behind events in the U.S. stock and labor markets in the 1990s.

In competitive equilibria the exponents of the Cobb-Douglas production function represent the share of total income that accrues to each factor. This is a measure of the importance of each factor for production, and it is the one we use in the paper. Hence, the notion that the importance of intangible capital has increased over time relative to physical capital translates, in our paper, in an increase of β relative to α. Note that this assumption is not equivalent to saying that the stock of z has increased more than the stock of k, because such a movement could result from differences in initial conditions and general equilibrium effects, even without any change in the structure of the production function. The statement that β has increased relative to α is essentially an assumption about a change in the nature of the production process in developed countries. Interestingly, we find strong evidence suggesting this is actually the case.

Corrado et al. (2006) carefully measure the share of output that is earned by the owners of intangibles in the United States. Their definition of investment in intangible capital includes expenses on firm brand, firm-specific resources, scientific and non-scientific R&D and computerized information. Using data on national accounts and estimates presented in Corrado et al. (2005), the authors show that investment on intangibles has increased substantially in the recent decades, much more so than investment in physical capital.

More importantly, their calculations indicate that the fraction of non-farm business output in the United States accruing to intangible capital has increased from 9.4 percent on average in the 1973-1995 period to approximately 14 percent in the 1995-2003 period. They also estimate that, if one focuses exclusively on the years 2000-2003, the share of income earned by the owners of intangible capital reaches 15 percent, while the owners of physical capital obtain 25 percent and the remaining 60 percent is absorbed by labor. Since there is no evidence that the share of income earned by physical capital has increased, when translated to our model, these results confirm the notion that the $\frac{\beta }{\alpha }$ ratio has increased in the United States in the last decades. At the same time, the data also point to a general shift of production from labor to capital (an increase in α + β). Table 1 summarizes this information.

Table 1:

Shares of income to factor, non-farm business sector, 2000-2003, including and excluding intangible capital (from Corrado et al. 2006, Table 4).

Table 1:

Shares of income to factor, non-farm business sector, 2000-2003, including and excluding intangible capital (from Corrado et al. 2006, Table 4).

	Conventional, without intangibles	Including intangibles
Share of total income
Labor	70.4%	60%
Tangible capital	29.6%	25%
Intangible capital		15%

View Table

Table 1:

Shares of income to factor, non-farm business sector, 2000-2003, including and excluding intangible capital (from Corrado et al. 2006, Table 4).

	Conventional, without intangibles	Including intangibles
Share of total income
Labor	70.4%	60%
Tangible capital	29.6%	25%
Intangible capital		15%

Belhocine (2008) reproduces Corrado et al. (2006) and finds qualitatively similar evidence for Canada, where estimated investment in intangibles has become as large as the investment in physical capital. Fukao et al. (2007) and Marrano and Haskel (2006) present similar findings for Japan and the United Kingdom respectively.

B. Intangible Capital, Collateral and Financing Constraints

Collateral is an important instrument for borrowers to raise funds in imperfect capital markets. Moral hazard and adverse selection reduce lenders willingness to allocate capital to borrowers whose quality or actions are not completely observable. Collateral plays an important role in overcoming this failure since it allows lenders to seize assets in the event of default. Moreover, it increases the incentives of borrowers to be diligent and perform well.

As emphasized by Shleifer and Vishny (1992), what turns an asset into good collateral is its redeployability in the contingency of liquidation. In other words, good collateral is an asset which can be costlessly seized and sold in the market for a price similar to its value in the first best use. There is ample empirical evidence associating redeployability of collateral and access to financing. Benmelech et al. (2005) show that increased redeployability of commercial real estate increases the duration of mortgages, reduces the cost of borrowing and increases the loan-to-value ratio. Benmelech (2009) shows that leverage was associated with redeployability of assets for U.S. railroads in the 19th century, while Benmelech and Bergman (2008) estimate the impact of collateral redeployability on airline companies’ bargaining power during episodes of financial duress.

Intangible capital is, in general, a worse source of collateral compared to physical capital. First and foremost, intangible capital is, by its very nature, hard if not impossible to seize. Furthermore, it is fundamentally intrinsic to the firm or business that produced it. Even if in some cases lenders can obtain control over the intangibles of defaulting borrowers, the redeployability of these assets tend to be small since their value in alternative uses is reduced by the lack of specific knowledge or competences. Obviously, different sorts of physical capital suffer, to some extent, from the same problems.⁸ However, plants, equipment and machines are generally accepted by financial intermediaries as collateral for lending.

The second fundamental assumption in our paper builds on this reasoning to argue that, other things equal, the more firms rely on intangible capital the more the financing constraints bind. Several studies have documented this fact. Aghion et al. (2007) show that R&D investment correlates with financing constraints in a large sample of French firms. Carpenter and Petersen (2002) find that small high-tech U.S. firms suffer more from financing constraints. Canepa and Stoneman (2008) present similar evidence for firms in the United Kingdom.

A. The OLG Structure

The economy is represented by an OLG model with one final good (y) and three inputs: labor (l), physical capital (k), and intangible capital (z). At every period t, a new unit-measure generation is born with an endowment of 1 unit of time and no endowment of goods. Each generation is composed by a fraction π of entrepreneurs and a fraction 1 - π of households. Individuals live for three periods: young, middle age, and old. They only consume when old, and their utility is a linear function of end-of-life consumption. All individuals will try to use the best production and investment opportunities available during their lifetime to convert their initial endowment of time into end-of-life consumption. For simplicity, we assume there is no uncertainty in the economy.

Figure 1 illustrates the model by describing the evolution of the generation born at time t. When young, both households and entrepreneurs supply labor inelastically to the existing firms run by the middle-aged entrepreneurs of generation t - 1 in a competitive labor market. The labor services are paid only at the beginning of period t + 1, at a wage rate w(t + 1). The labor supply of the young will represent the aggregate labor supply of the economy in each period.

OLG Model

Citation: IMF Working Papers 2011, 271; 10.5089/9781463925260.001.A001

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OLG Model

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OLG Model

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At the beginning of period t + 1, households have no option but to lend their accumulated labor income at a gross rate R(t + 2) -- to be paid at date t + 2 -- to the contemporaneous entrepreneurs in order to transfer funds to the last period of life. Hence, at date t + 2, a household of generation t consumes her total wealth, represented by w(t + 1), and then dies.

Entrepreneurs, on the other hand, have different investment opportunities. They use their own funds and borrow additional resources from the contemporaneous households to accumulate physical and intangible capital. They combine accumulated capital with labor services from the newborn of generation t + 1 to produce final goods. Production by generation t takes place between periods t + 1 and t + 2 and generates output of the final good at the beginning of time t + 2, denoted y_t(t + 2). Note that y_t(t + 2) refers to the output produced by a single entrepreneur.

Finally, at date t + 2, the total output produced by generation t is distributed among the owners of the factors. Part of it goes to the old households of generation t who lent funds to the entrepreneurs of generation t + 1. Another part remunerates the labor services provided by generation t + 1. The remaining output is retained by the old entrepreneurs of generation t, who consume and die.

B. Production and Financial Frictions

Each middle-aged entrepreneur of generation t has access to a Cobb-Douglas production technology that combines physical capital k, intangible capital z and labor l as follows

where A is a productivity parameter that grows at the constant rate n and l_t(t + 1) represents the individual demand for labor in period t + 1. At the beginning of t + 1, a typical middle-aged entrepreneur uses her own funds, and borrows additional resources d_t(t + 1) to accumulate z_t(t + 1) and k_t(t + 1). This is the first stage of production. In a second and final stage, she hires labor from the newborn of generation t + 1.⁹ After production is completed, both k and z fully depreciate. The marginal rate of transformation of final goods into k and z equals 1.

We introduce a friction in the lending process by allowing entrepreneurs to default on their debt. In particular, we make the following assumptions. First, in the initial stage of production, an entrepreneur can default on her debt, close the firm, and take with her all intangible capital z she has accumulated.¹⁰ In this case, the lender will seize the physical capital that has been pledged for the loan. Once she defaults on the debt, the entrepreneur can open a new firm, borrow funds from another lender, and start production (of course, she might choose to default again). Second, after production is completed, the lender can seize output y, provided the original firm has not been closed. This ensures repayment of the loan once production is completed.¹¹ Third, we assume that during the first stage of production, the lender has the ability to determine how the funds will be utilized by the borrower. This captures the fact that lenders frequently monitor the use borrowers make of loaned funds, and covenants specify the allocation of the resources they provide.¹²

Given these assumptions, lenders will always require loaned funds to be completely invested in or backed by tangible capital. Suppose this is not the case, that is, assume the amount of physical capital owned by an entrepreneur in the first stage of production is smaller than the value of her debt. In this case, the entrepreneur would have the incentive to close the firm, keeping only the intangible capital accumulated. Naturally, she would lose completely whatever physical capital she had accumulated, but she would also get rid of the higher level of debt. She could, then, open a new firm, utilizing the same amount of physical capital as before by borrowing from an alternative lender who would impose the restriction that the new funds are only invested in k. Hence, by defaulting on the original loans and starting fresh, the entrepreneur can achieve the same scale of operation as under the original contract but with smaller indebtedness.

When the entrepreneur receives a loan which is specifically allocated to tangible capital she has no incentive to default before production, because the tangible capital would be seized by the lender. Nor has she incentive to default after production occurs, because the lender can seize the output of production. Finally, note that competition among lenders implies that the return on these loans equals the marginal productivity of tangible capital.

Under those assumptions, we have:

Hence, the problem of a middle-aged entrepreneur of generation t can be written as

subject to

Since the first constraint always binds in equilibrium, the borrowing restriction can be rewritten as

Simply put, the possibility of default together with the lack of pledgeability implies that intangible capital is financed with internal funds only.

In our model, we assume that entrepreneurs cannot issue equity in order to invest, but are constrained to issue collateralized debt only.¹³ The crucial aspect here is not the distinction between debt and equity financing, but rather the asymmetry between inside and outside investors. In this respect, physical capital works as a mechanism through which managers commit not to expropriate investors, a property that cannot be mimicked by intangible capital.

It is useful to compare the assumptions above with the environment of other papers. Martin and Ventura (2011), for example, prevent borrowing by the agents with better investment opportunities relative to their peers due to an “unspecified market imperfection”. Lorenzoni (2008) introduces limited risk-sharing by assuming that entrepreneurs can default after they produce, losing part of the output and remaining capital. Because we assume limited pledgeability of only one type of capital, in our model the default decision must be allowed to happen when K and Z are still differentiated. This is in fact similar to Kiyotaki and Moore (1997), where the borrower can close the firm and take away his human capital at any time, leaving behind the land (the tangible part). In their model, if the borrower defaults before production is completed, output is not produced and the unutilized land is seized and sold by the lenders on the market; therefore, creditors will make sure that the total amount to be repaid is at most as high as the future value of the land. In our model, lenders can seize physical capital and transfer it to another producer if default occurs, so the outside value of the collateral is k_t(t + 1), while the present value of promised repayment is d_t(t + 1) resulting in Lemma 1.¹⁴

A. The Bubbleless Economy

A bubbleless competitive equilibrium is defined as follows:

Solving in closed-form for the equilibrium is possible because of the simplifying assumptions we have made. When young, agents supply labor inelastically. When middle-aged, households supply their savings to the entrepreneurs inelastically as well.¹⁵ Only entrepreneurs take active decisions, and they coincide with profit maximization for the firm.

Therefore, the equilibrium wage and interest rate are determined by the first order conditions (F.O.C.) of the firm:

where K_t(t + 1) = πk_t(t + 1).

Two markets are open in period t + 1. One is the market for labor, in which the young of generation t + 1 supply labor to generation t’s firms. We have already imposed its equilibrium condition above. The other market is the one for savings which achieves equilibrium when:

The borrowing constraint (whose Lagrange multiplier we label λ_t referring to generation t)

coupled with the optimal choice of intangible capital by entrepreneurs implies:

These conditions show that, whenever the financing constraint binds, the marginal productivity of intangible capital is higher than the interest rate prevailing in the market for funds, which in equilibrium equals the marginal productivity of physical capital. We call that shadow rate of return IRR, to stress that it is the return an additional dollar could generate if freely transferred to an entrepreneur. Note that

so λ_t captures the spread between the return on a unit of internal funds and the return on a unit of funds raised through collateralized debt.

While analyzing equilibrium allocations, we focus exclusively on the aggregate variables (Y,K,Z) and on the prices (w,R). There is no loss of generality in doing so. Knowing the path of these aggregates is enough to fully describe the economy, since all other variables are immediately derived from them.

In order to characterize the behavior of the economy along an equilibrium path, it is necessary to determine under what conditions the borrowing constraint binds. Fortunately, the model features an interesting separation of the parameter space into two mutually exclusive and complementary regions, one where the constraint always binds and another where it never does. The next definition sets the stage for the formal proof of this result.

Note that Θ = B ∪ NB and B ∩ NB = ϕ, that is, the parameter space is distinctly separable into these two sets.

Proposition 1 builds on this definition to analyze the existence of a dynamic bubbleless equilibrium. It is straightforward to show that, if a steady-state exists, all real variables must grow at the gross rate (1 + g), such that:

According to Proposition 1, for a given value π, as the ratio $\frac{\beta }{\alpha }$ increases, the economy moves from equilibria in which financing constraints are irrelevant to ones in which they are binding. This result is intuitive. Other things equal, the demand for intangible capital is increasing in β. The higher the ratio $\frac{\beta }{\alpha }$ the more entrepreneurs want to accumulate intangible capital instead of physical capital. After a certain point, all internal funds are invested in Z, and financing constraints become binding.

As will become clear in the following sections, Proposition 1 contains the central result behind our connection between intangible capital accumulation, relative asset shortages, and bubbles. The increased importance of intangible capital due to technological change, represented by increases in the ratio $\frac{\beta }{\alpha }$, moves the economy towards a constrained region of the parameter space.¹⁶ At this point, the relative supply of stores of value is distorted because households’ savings can only be allocated to physical capital despite the higher productivity of intangibles. The over-accumulation of K reduces its marginal productivity and the rate of interest in the market for funds, which is key for rational bubbles to exist.

B. The Bubbly Economy

Let us now analyze the existence of bubbly equilibria.

The presence of a bubble changes some of the equilibrium conditions outlined above. In particular, the savings market equilibrium requires that, at each period, there are enough savings to finance physical and intangible capital accumulation as well as the acquisition of the bubbly security:

Because any individual can trade the bubbly security in the market, its rate of return (the rate at which its price evolves over time) is equalized to the return on the worst investment opportunities available to investors. This is always given by the return R that households obtain on their savings.¹⁷ Therefore, in any dynamic equilibrium the value of the rational bubble evolves according to:

In case financial frictions are binding, entrepreneurs will never be interested in the bubble, since their funds are invested at a rate

Hence, in any constrained equilibrium, the bubble is held in its entirety by the households of each generation.

At this stage, we establish some notation to facilitate the derivation of the equilibrium.

The first three definitions formalize the usual concept of a variable per efficiency unit, while the last one defines the bubble per unit of output. Note that we are comparing the price of the bubble that is exchanged at time t + 1 with the output generated at time t + 1 (by generation t - 1).

If the economy is in the NB region, the financing constraints are slack along any equilibrium path. The case where the economy belongs to the B region is more involved. Intuitively, for small enough bubbles, the financing constraint is binding, since the bubble does not drain too many funds, which means that the stock of capital will be high, inducing entrepreneurs to fully invest in intangibles. On the other hand, if the bubble is large enough, households will finance a small amount of physical capital. In this case, it is in the best interest of entrepreneurs to invest in K as well, which means that the borrowing constraint becomes slack.

Based on that, we can characterize in detail the bubbly dynamic equilibria. Let us start with Proposition 2, which restates Tirole’s (1985) result.

Whenever the average rate of return of the economy is below its growth rate in the bubbleless steady-state, rational bubbles can exist. This well known condition has been traditionally interpreted as a sign of dynamic inefficiency, since a low rate of return results from excessive accumulation of capital. It is important to note that the above Proposition defines the necessary and sufficient conditions for the existence of bubbly equilibria in the NB region -where R is equalized to IRR. In the B region, on the other hand, it is possible for bubbly equilibria to survive even if the bubbleless steady-state features πIRR + (1 − π) R ≥ (1 + g) as shown by Farhi and Tirole (2008).

It turns out that in the bubbleless steady-state, the comparison between the average rate of return and the growth rate of the economy is equivalent to the AMSZ test about the productivity of the corporate sector. AMSZ compare, at every period, the amount of funds absorbed by the corporate sector in the form of capital expenditures versus the excess of its output over labor costs. The authors present strong evidence that the corporate sectors of OECD economies are highly productive since they generate more funds than what they absorb. We require our model to satisfy this empirical benchmark.¹⁸

In what follows, we characterize the bubbly equilibria in the B region subject to the AMSZ empirical benchmark, which, for our purposes, defines the relevant region of the parameter space. We acknowledge that this restriction implies a counterfactually small share of labor income relative to output. However, this results from some of the stark assumptions we have imposed in order to make the model tractable and are not directly linked to the most important aspects of our analysis.¹⁹ We omit the presentation of the dynamic equilibria in the other regions of the parameter space, since they can be derived in similar ways.

Figure 2 represents graphically the relevant regions of the parameter space in terms of α and β, for given π and n. The B/NB line has slope $\frac{\pi }{1-\pi }$, and divides the space into the B and NB regions.

Characterization of the parameter space

Citation: IMF Working Papers 2011, 271; 10.5089/9781463925260.001.A001

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Characterization of the parameter space

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Characterization of the parameter space

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Proposition 3 nicely extends and qualifies the result of Proposition 1 -- about bubbleless equilibria -- to bubbly equilibrium paths, by identifying an additional division of the parameter space, related to the AMSZ line. This line separates economies in which the AMSZ benchmark is satisfied $\left(\alpha +\beta \ge \frac{1}{2}\right)$ from those in which it is not.

The dynamics of the bubbly equilibrium paths can be described as follows. In the region where the economy’s average rate of return is low relative to its growth rate, i.e. below the AMSZ line, all of Tirole’s (1985) standard results apply, whether the constraint binds or not. There can be bubbles that disappear asymptotically and bubbles that grow forever together with the economy. We do not focus on this area in our study, since it contradicts the empirical evidence.

In the area above the AMSZ line, the separation between NB and B defines the dynamics of the bubbly equilibria, just like in the case of the bubbleless equilibria. In the light gray area of the B region, the bubbleless steady-state features R < (1+g). This condition guarantees that there exist equilibrium paths in which the bubble-to-output ratio is positive but disappears asymptotically, as well as equilibrium paths that converge to a steady-state with a positive bubble-to-output ratio. In all bubbly paths the financing constraints are binding. In the rest of the B region and in the whole NB region, instead, no rational bubbles can be sustained in equilibrium.

Figures 3 represents the dynamics of the economy in the B region, above the AMSZ line, for B^* > 0. The lighter line represents the bubbleless equilibrium path that converges to the bubbleless steady-state (lighter dot). All the other equilibrium paths have a positive bubble. They either converge to the bubbleless steady-state (the bubble-to-output ratio disappears asymptotically as indicated by the arrows converging to the lighter dot), or they reach the bubbly steady-state through a saddle path, where the bubble is constant as a fraction of output. The saddle-path trajectory is represented by the horizontal arrows converging to the darker dot in the middle of Figure 3.

B region

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B region

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B region

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C. Stochastic Bubbles

The previous results can be extended to the case of stochastic bubbles. In order to prove that, we consider an environment in which, at every period, the rational bubble survives with probability p or bursts with probability 1 - p as in the classic model of Blanchard and Watson (1982). Naturally, as long as the bubble exists, it has to grow at a rate $\frac{R}{p}$ so its expected return is equal to R.

Let us start with the definition of a conditional steady-state.

Naturally, unless the economy starts at the conditional steady-state, it will almost surely not reach it, since the probability of that event happening converges to zero. The conditional steady-state is, notwithstanding, a useful benchmark because it allows us to determine the trajectory of a bubbly economy while the bubble lasts. In what follows, we characterize, through a series of propositions, the dynamics of the economy when bubbles are stochastic. We will focus on the case where the stochastic bubble has not burst until the most recent period, since the dynamics of the economy after the bubble bursts have been already explored in Section 4.1.

Lemma 4 highlights that the dynamics of the economy are not materially affected by the stochastic nature of bubbles. Whether the continuation is binding or not depends only on the bubble-to-output ratio and the region of the parameter space where the economy lies.

As before, economies that are productive in the AMSZ sense cannot sustain stochastic bubbly equilibria if they are in the NB region of the parameter space.

These results show that economies that satisfy the AMSZ criterion can sustain stochastic bubbles. Obviously, the conditions for the existence of stochastic bubbles are more stringent relative to the deterministic case. As mentioned before, conditional on not bursting, stochastic bubbles grow at the rate $\frac{R(t+1)}{p}>R(t+1)$ . Hence, for these bubbles to not outgrow the economy in finite time, interest rates in the bubbleless equilibrium have to be even lower compared to the economy’s growth rate in the long-run (the more so the higher the chance the bubble will burst). This can be achieved by having a ratio high enough and binding financing constraints.

It is interesting to consider the macroeconomic consequences of a bursting bubble. The crash immediately reduces the consumption of the currently alive old households whose portfolios were exposed to the bubble. Because the corresponding old entrepreneurs did not buy the bubbly security, there are unaffected. Hence, aggregate consumption is immediately reduced. Future output and investment, however, will expand as a result of the shock. In the absence of a bubble, the middle-aged households of following generations have no option but to make collateralized loans to the contemporaneous entrepreneurs, raising the stock of physical capital and increasing production. This predicted positive impact is at odds with the historical experience, since asset market crashes tend to be followed by prolonged slumps in real activity. The model’s inability to generate such effect results mainly from the lack of a direct link between the price of the bubbly asset and the balance sheet of levered agents (entrepreneurs), as is the case in Kiyotaki and Moore (1997) for example.

This limitation can be solved by extending the baseline specification and considering an environment where entrepreneurs’ capacity to raise funds is directly affected by the bubble. More specifically, instead of taking the total amount of the bubble as given, we can assume that, at each period, the middle-aged entrepreneurs are able to issue an extra quantity of bubbly assets to be sold to the contemporaneous households. This process continues as long as the bubble hasn’t burst, but stops permanently after a crash. Clearly, the continuing issuance of bubbles alleviates financial frictions, allowing entrepreneurs to increase the amount of resources invested in intangible capital. The bursting of a bubble shuts down this channel, exacerbating financial frictions and increasing distortions in real investments. Depending on the strength of this effect, a crash in bubbly securities might actually reduce output and investment in the long-run. This possibility is investigated in the appendix where it is confirmed that, for suitable choices of the parameters, bursting bubbles may actually produce long-term economic contractions.

A. The Importance of Intangible Capital and Bubbles

The results in Section 4 show that economies which satisfy the AMSZ condition can sustain bubbles only if intangible capital is sufficiently important in the production process. With a high β relative to α, financing constraints are active, and the equilibrium interest rate is low --which allows rational bubbles to be sustained in equilibrium.

To better illustrate the point, we calibrate the model and calculate the steady-state rates of return and growth rate of the bubbleless economy for different values of α and β.²¹ In this exercise, the fraction of income that goes to intangible capital β is increased by 15 percentage points, while α is decreased by 12 percentage points, the remaining 3 percentage points being taken from labor in order to preserve constant returns to scale. This calibration mimics the change in the share of output accruing to the different factors documented for the United States in the last 30 years by Corrado et al. (2006). Figure 4 shows how the bubbleless steady-state rates R and IRR shift as β gradually increases and at the same time α decreases -- the horizontal axis representing the change in β. It also compares the growth rate of the economy 1 + g with the equilibrium average rate of return, πIRR + (1 − π)R.²²

Steady states and technological transition

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Steady states and technological transition

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Steady states and technological transition

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The vertical threshold separates the NB region (left) from the B region (right). For small values of $\frac{\beta }{\alpha }$, the economy is in the NB region. The returns on tangible and intangible capital are equalized, so that R = IRR. Since the AMSZ condition is satisfied, R > 1 + g in the steady-state, and no bubbles can exist. For larger values of $\frac{\beta }{\alpha }$, on the other hand, the economy is in the B region. The wedge between R and IRR increases with $\frac{\beta }{\alpha }$ and might drive R below 1 + g in steady-state if $\frac{\beta }{\alpha }$ is sufficiently high. After this point, bubbly equilibria are possible.

The change in $\frac{\beta }{\alpha }$ we calibrate here represents a simple comparative statics exercise, but its implications are identical to an unexpected and exogenous technological shift. In particular, suppose that the economy is originally in the NB region of the parameter space, and satisfies the AMSZ condition. No bubble can exist at this time, and the economy is on a bubbleless path. Suppose all agents originally expect technology to be constant forever, but, at a certain time T, (α, β) suddenly shifts to (α′, β′) such that (α′ + β′) > 0.5 - ensuring the new economy satisfies the AMSZ condition as well. Finally, suppose no more changes are expected. The arrow in Figure 5 represents this shift in the space of (α, β). Under these assumptions, we can prove the following:

Technological Change

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Technological Change

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Technological Change

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This Proposition shows that bubbly equilibria can arise upon technological innovation. In the proof, we consider the case in which the entrepreneurs of generation T-1 are the ones who issue the bubbly security, selling it to the households of the same generation at the beginning of date T. The dynamics of the economy change, and (depending on the size of the bubble that is created) the economy will jump to one of the bubbly paths depicted in Figure 3.

The issuance of a bubble by the entrepreneurs has three effects. First, it immediately alleviates the financing constraints of the existing entrepreneurs, reducing distortions in capital allocation and directly benefiting all individuals from generation T - 1.²³ Second, there is a positive spillover impact on future generations since the increased production raises labor income, relaxing future financing constraints as well. Because of decreasing returns to capital, though, its importance dies out over time. Finally, there is a competition effect: as the bubble grows, it competes for funds with the entrepreneurs, reducing production. The lower level of output results in lower labor income and savings, diminishing the accumulation of both physical and intangible capital.²⁴

B. Stochastic Technological Progress and Bubbles

Proposition 6 has shown how unexpected technological shifts create the conditions for the emergence of bubbly equilibria. In this subsection, we extend the analysis above and investigate the case in which agents correctly factor in the possibility that a technological change may occur in the future. In particular, we demonstrate that even the possibility of experiencing a technological shift which increases β relative to α is already sufficient to allow for bubbly equilibria. Interestingly, bubbles can arise even if the economy is originally in the NB region and if it satisfies the AMSZ at all times.

Consider an economy that, as long as technological change has not occurred yet, belongs to the NB region of the parameter space and also satisfies the AMSZ condition. At every period, there is a probability q > 0 that technological progress will take place, changing the original parameters of the production function (α, β) to new values (α′, β′). Once this happens, no further changes occur and the economy becomes completely deterministic. The new specification (α′, β′) belongs to the B region and also satisfies the AMSZ criterion --(α′ + β′)>0.5 Additionally, we assume that entrepreneurs know exactly their technology at the time they make investment decisions. Hence, if technological change has not taken place until the beginning of period t + 1, for example, the middle-aged entrepreneurs of generation t will produce with the old technology and uncertainty might only affect the entrepreneurs of future generations. Under these assumptions, we can prove the following:

In the situation described above, the probability of the economy experiencing technological progress is positive and constant at all times until the change takes place. As a consequence, technological progress occurs with probability 1 in the long-run. If $\frac{\beta \prime }{\alpha \prime }$ is large enough, thus, the steady-state will feature low interest rates almost surely. We can show, however, that bubbles may arise even if there is uncertainty about interest rates in the long-run. To see that, let us assume a similar model as before, but suppose that the permanent technological progress occurs with probability q > 0 at the very beginning of time T only, where T is a future period. For any other date, q = 0 and there is no uncertainty. Under those assumptions, we can prove the following:

Proposition 8 illustrates how the stochastic nature of bubbles might be purely linked to the randomness of technological progress. If there is a technological “disappointment” and the production structure is unchanged at time T, the bubble immediately bursts.

Propositions 7 and 8 consider two polar cases. In the first, a technological shift can occur in any period with positive probability (until it takes place) whereas in the second scenario it might only happen at a specific point in time. Naturally, many intermediate cases where technological shifts may occur in a window of time would also be able to sustain bubbly equilibria. The rationale for these intermediate possibilities would be the same as the intuition behind propositions 6-8. If technological progress shifts the production function to be heavily biased towards intangibles relative to physical capital, it will move the economy to a region of the parameter space where the bubbleless steady-state features binding financing constraints and very low interest rates, despite high IRR - as illustrated by the red arrow in Figure 5. This change allows for the co-existence of bubbles and efficiency. Interestingly, the very possibility of such a change is already sufficient to generate paths with a positive bubble even before the technological shift takes place.

C. Relation to the Savings Glut Explanation

One recent alternative explanation for asset price bubbles is based on the so called “savings glut” phenomenon -- see Bernanke (2005). It points to the fast development of emerging markets like China and oil-exporting countries, which have large amounts of savings but are unable to supply sound stores of value due to underdeveloped domestic financial markets. The excess funds from these countries have flooded the capital markets of developed nations, putting downward pressure on interest rates and stimulating the appearance of speculative bubbles. Besides identifying a separate underlying story for the emergence of bubbles in developed economies, our model differs from the savings glut explanation along two main dimensions.

First, our model hinges on a relative imbalance of high-yield versus low-yield assets in developed economies -- a surplus of K and a shortage of Z -- rather than on an absolute lack of sound stores of value. While the latter is certainly reasonable for developing countries, it seems less relevant for advanced nations. If anything, one would expect that, as economies develop, their capacity to create stores of value increases. In this case, more securities can be issued and more accumulation of capital would be possible.

Second, most of the models that attribute the emergence of rational bubbles to an increase in the supply of funds feature a single-factor economy (physical capital) with a linear technology (AK) and limited pledgeability of output. The linearity assumption, however, is not innocuous. With limited pledgeability of output, as new funds enter the economy capital keeps accumulating, albeit at a lower level than it would be possible without frictions. In an economy where physical capital has decreasing marginal returns, this accumulation would in fact lower the internal rate of return of funds. This means that, at some point, the economy would have a low enough productivity of capital to fail the AMSZ test, thereby rendering it inconsistent with the available evidence. The assumption of linear technology shuts down this channel and keeps the internal return on funds artificially high.

In our framework, however, the savings glut explanation reinforces the technology-based approach. In the latter, the separation between the two types of capital allows the accumulation of physical capital to actually increase the wedge between R and IRR: the marginal productivity of physical capital decreases, while the marginal productivity of intangible capital increases. To the extent that foreign investors cannot overcome the moral hazard frictions in the recipient economy, an inflow of funds from abroad increases the stock of K and pushes the IRR up. This allows the AMSZ condition to be satisfied even for very low interest rates. We formalize this intuition through a stylized example of an open economy presented in the Appendix.

A. Traditional Tests of Dynamic Efficiency

As discussed in Section 4, bubbly equilibria can exist even if the economy satisfies the AMSZ empirical benchmark. In AMSZ’s original paper as well as in Tirole (1985), this restriction corresponds to the condition that the equilibrium interest rate is not below the growth rate of the economy in the long-run, which has been traditionally interpreted as a sign of dynamic efficiency. In particular, in that setting the following three statements are equivalent:

1. The economy satisfies the AMSZ condition.

2. The interest rate is higher than the growth rate of the economy in the long-run.

3. The economy is dynamically efficient.

In the presence of frictions, however, the equivalence of these statements is broken. For instance, in a similar environment, Farhi and Tirole (2008) have shown that statements 1 and 2 are delinked. We extend their analysis by proving that, in our model, statements 1 and 3 are not equivalent either. Moreover, we qualify the relation between 1 and 2. While our results are derived under a stylized setting, they allow us to make the general point that, by creating heterogeneity in investment opportunities, financial constraints can decisively influence the validity of traditional tests of dynamic efficiency.²⁵

Lemma 3 formalizes the connection between the AMSZ restriction and rates of return. Note that as long as α + β > 0.5 the AMSZ test will be satisfied. This does not, however, guarantee that the interest rate condition will be met at all times, since Lemma 3 only holds in the steady-state. Nonetheless, analyzing steady-states is central since they contain important information about the efficiency properties of allocations.

We now formally prove that the AMSZ test is not sufficient to guarantee the efficiency of allocations once financial frictions are taken into account. For this purpose, it is important to discuss the dimensions on which a planner could act in order to look for potential improvements over market allocations.

A benevolent planner can operate on at least two margins. First, whenever the borrowing constraints bind, the planner can make intra-generational transfers from unconstrained households to constrained entrepreneurs. This, however, has no major implications for the topic of dynamic efficiency, since it does not address the issue of over or under-accumulation of capital. Much more interesting, though, is to consider improvements in the intertemporal margin while respecting existing borrowing constraints.

One could expect that the constrained efficiency requirement would coincide with the AMSZ test. The following proposition proves that this is not the case.

The reason why the AMSZ criterion does not guarantee efficiency in our model is related to the heterogeneity of investment opportunities created by financial frictions. In the binding region of the parameter space, there is a wedge between the interest rate obtained by the creditors and the rate of return earned by entrepreneurs. Looking at the average rate of return in the economy -- as in the AMSZ test -- is not sufficient to judge the efficiency property of market allocations, because there might be over-accumulation among those who cannot park funds in the productive assets. The social planner can improve an allocation by transferring funds across households from different generations, compensating all entrepreneurs along the way. This can be done even if the growth rate of the economy is lower than the average rate of return, as long as it is greater than R.

One could argue, though, that our suggested transfer scheme does not do justice to the AMSZ test. On the one hand, we do not allow the planner to overcome the inefficiency resulting from moral hazard, but on the other hand, she is allowed to identify the different groups according to their investment opportunities and target them for reallocation purposes. To address this criticism, we analyze the efficiency properties of allocations that satisfy the AMSZ test in a setting where the planner cannot target specific groups.

Interestingly, even when we consider only blind reallocations, it is still the case that the AMSZ benchmark does not coincide with efficiency, this time for the opposite reason: there exist allocations that do not satisfy the AMSZ benchmark, but cannot be improved by a blind social planner.

The intuition behind this result is straightforward. The AMSZ test compares the average rate of return on the economy with its growth rate in the long-run. However, a planner trying to improve market allocations without targeting specific groups in society is required to transfer resources over time at the highest rate of return every period. This is the only way to ensure that those who face the best investment opportunities (entrepreneurs) are not harmed by reallocations. Hence, the fact that the economy fails the AMSZ test does not imply there are feasible improvements for blind planners, unless IRR < (1+g) in steady-state.

B. Normative and Positive Implications in a Bubbly Economy

The previous subsections have shown that it is possible for the economy to satisfy the AMSZ benchmark while being inefficient. Moreover, Sections 4 and 5 demonstrated that the economy can sustain bubbly equilibria even if we impose the AMSZ condition. A natural question, therefore, is whether rational bubbles are capable of restoring efficiency. As in some of the previous literature(Grossman and Yanagawa, 1993, and Farhi and Tirole, 2008), we argue this is not the case in general, as the crowding out effect of bubbles on investment has a negative externality on some agents.

To see that, consider an economy where borrowing constraints bind. In this context, the introduction of a rational bubble has redistributive effects that negatively affect some individuals. On the one hand, bubbles improve the investment opportunities of savers. On the other hand, as the bubble grows, it competes for funds with entrepreneurs who borrow resources to make real investments in future periods. Therefore, typical rational bubbles are not Pareto improving in our model.

From a practical perspective, the present paper suggests one natural approach to mitigate the relative shortage of assets and prevent the emergence of bubbles: to improve the pledgeability of future cash flows, particularly those stemming from activities that are intensive in intangible capital. The increased pledgeability of future output allows funds to be allocated to their most productive ends, increasing the productivity in the overall economy and raising the rate of return faced by outside investors. This is crucial to preempt the emergence of new bubbles. Policies targeted to strengthen property rights and the enforcement of contracts are obvious candidates to achieve this goal. So are measures that facilitate investments in education and human capital which tend to be productive, but in general are subject to important financing constraints.²⁶