BIS Working Papers
No 794
Why have interest rates fallen far below the return on capital?
by Magali Marx, Benoît Mojon and François R Velde
Monetary and Economic Department
July 2019
JEL classification: D22, D84, E31
Keywords: inflation expectations, firms' survey, new information.
This publication is available on the BIS website (www.bis.org).
© Bank for International Settlements 2019. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.
ISSN 1020-0959 (print)
ISSN 1682-7678 (online)
Why have interest rates fallen far below the return on capital?
by Magali Marx, Benoît Mojon and François R Velde
Abstract
Risk-free rates have been falling since the 1980s while the return on capital has not. We analyze these trends in a calibrated OLG model with recursive preferences, designed to encompass many of the “usual suspects” cited in the debate on secular stagnation. Deleveraging cannot account for the joint decline in the risk free rate and increase in the risk premium, and declining labor force and productivity growth imply only a limited decline in real interest rates. If we allow for a change in the (perceived) risk to productivity growth to fit the data, we find that the decline in the risk-free rate requires an increase in the borrowing capacity of the indebted agents in the model, consistent with the increase in the sum of public and private debt since the crisis.
JEL classification: D22, D84, E31
Keywords: inflation expectations, firms' survey, new information.
The "Global Financial Crisis" began ten years ago and within two years short-term nominal interest rates were driven to near-zero levels in the large advanced economies (U.S., Euro-area, UK) where they haved stayed since. While the Fed raised interest rates at a relatively slow pace since December 2015, policy rates in Japan and the Euro area are still below zero. With low and (relatively) steady inflation, real rates have been negative for a while, and not just short-term rates but also rates at the 5-year and 10-year horizon (Hamilton et al., 2016). The decline is persistent and substantial, on the order of 4 to 5%
Much of the macroeconomic research responding to the financial crisis has taken place within the New Keynesian DSGE paradigm. Understanding the reasons for reaching the lower bound (the reason for low interest rate) was less urgent than understanding the proper responses to the situation. Also, the methodology relies on some approximation around a steady state, whether linear or nonlinear (Fernandez-Villaverde et al., 2012; Gust et al., 2012). Hence the low (real) interest rates are modeled as the result of an exogenous shock, for example to the discount rate or to a borrowing constraint (Eggertsson and Woodford, 2003; Eggertsson and Krugman, 2012), that shifts the supply of funds outwards and induces deviations from a steady state whose dynamics (modified as needed by policy) are the core prediction of the model.
After a decade of low interest rates, the shock paradigm becomes less attractive because of the strains it places on the assumption of independent Gaussian shocks (Aruoba
et al., 2013). At the same time observers are focusing increasingly on the "secular stagnation" hypothesis (Summers, 2014): low interest rates may not be temporary deviations but a now-permanent state of affairs. Are low rates the "new normal"? If so, why, and what can be done about it?
These are important policy questions (Fischer, 2016a,b). Three recent policy-oriented publications (Teulings and Baldwin, 2014; Bean et al., 2015; Gourinchas et al., 2016) have collected the possible explanations for such a permanent decline in interest rates. These include aging pressure on savings, income inequality, a slower pace of productivity, deleveraging, a collapse in the relative price of investment, a shortage of safe assets and an increase in the perception of risks. But so far there has been little quantitative evaluation of the competing explanations (but see Rachel and Smith, 2015) and little of it is model- based. Our question is simple: can we account for current low interest rates in a model that encompasses the most likely factors?
To answer it we develop a framework that combines Coeurdacier et al. (2015) and Eg- gertsson and Mehrotra (2014) to encompass most of the current qualitative explanations for low interest rates. We extend the model to include risk. There are two reasons for this. One is to make contact with the literature on the shortage of safe assets (Caballero et al., 2008; Caballero and Farhi, 2014) because safe assets only make sense in a context with risk. Second, we want to address a fact on which we elaborate in the next section, namely the divergence between rates on (government) bonds, which have fallen, and the return on capital, which has not (Figure 2).
Our paper relates to three literatures that investigate secular stagnation, the shortage of safe assets and long-term risk.
Several empirical papers document the role of demographics in explaining low interest rates. Ferrero et al. (2017) who find an effect of 0.5 percent in the last decade from a dynamic VAR on a panel of OECD countries (resulting from the population growth and the change in dependency ratio) and Busetti and Caivano (2017) who estimate such effects for eight advanced economies at low frequencies since the 1980s. Aksoy et al. (2016), however, in a broader analysis of demographic trends and the macro-economy, do not find significant effects on interest rates. Del Negro et al. (2017) who decompose changes in the US natural rate with either a DSGE or an identified VAR estimate that the slowdown of productivity can account for as much as 60bp in the decline of the natural rate since the mid 1990s, but attribute more of the decline (around 1 percent) to an increase in the convenience yield of safe assets (Treasury bonds). Favero et al. (2016) explore the role of demographic factors in an affine term structure model of interest rates. XXX How does this interact with our model?
A growing number of papers use OLG structures to assess how the aging of the baby- boomers explains either an increase in desired savings or the decelerating productivity or both. Eggertsson and Mehrotra (2014) provide a qualitative assessment in a closed economy set-up. Our results concur with the results of Gagnon et al. (2016), who use a rich OLG structure and find that aging can account for only as much at 1.2 percent decline in real interest rates. This accord with Carvalho et al. (2016), who find a 1.5 percent effect since 1990 in a simpler model with workers and retirees.
In our simulations, the contribution of ageing and the slowdown of productivity are consistent with these findings. They explain that the risk free rate has declined by 1 to 1.5 percent since 1990, i.e., much less than the declined observed in real rates. More importantly, these papers all consider a single asset class as a vehicle for savings. They do not account for the fact that, in the data, the return on capital, which equates the real interest rate in OLG models with production, has not declined. Instead we give households the choice either to own the capital used in production or to lend to the next generation. Hence we can use our model to replicate the evolution of the interest rate and the risk premium paid to own a capital stock of which the return is risky. Eggertsson et al. (2017) also differentiate the return on capital from the real interest rate. Their gap is due to a mark-up while ours reflect risk premium. They find that the fall in mortality, fertility and productivity since 1970 each explain nearly 2 percent drop in the real interest rate, a total effect of nearly -6 percent, which is compensated by a 2 percent increase due to the rise in public debt. The much larger effects that paper finds for demographic factors and productivity is largely due to the choice of 1970 as their starting point, as a major part of the decline in these factors take place between 1970 and 1990, a period when real rates actually increased.
Our paper is also related to the literature on safe assets and their "shortage." In a seminal paper, Caballero et al. (2008) associated global imbalances to a growing demand of economic agents in emerging economies for "safe assets" that are typically issued by the US and other large OECD countries. Coeurdacier et al. (2015) use an OLG structure to estimate the effects of opening capital flows to China where severe credit constraints push down the equilibrium interest rate. As a result, the "world" interest rate can decline substantially. Coeurdacier et al. (2015) estimate that the equilibrium interest rate could fall by as much as 6 percent, however in a set up where the level of the steady state risk free interest rate is not consistent with the data. More recently Caballero et al. (2016) stress how the shortage of safe assets can slow economic growth, a force that would in turn push the risk free interest rate further down. Their qualitative exercise cannot be used to quantify the role of each of the forces that influence the equilibrium interest rate. Caballero et al. (2017) introduce an accounting approach to jointly explain the decline in the risk free rate, the stability of the return of capital and the decline in labor share. They show that even with a set of parameters that maximise the effects of increasing mark-ups on the gap between the interest rate and the return on capital, a large share of the increase in the risk premium remains unexplained. Hall (2016) models the decline of the risk free rate as resulting from a change in the composition of savers, with an increase in the weight of risk adverse savers in the economy. Our contribution with respect to this literature is that we offer a quantitative analysis of the role of risks in an OLG model where the other forces of secular stagnation can also have a role.
Third, we relate to the asset pricing literature on long term risk. Bansal and Yaron (2004) show that Epstein-Zin-Weil preferences combined with persistent growth rate of consumption and small uncertainty on its fluctuation can explain both a low risk free interest rate and a high risk premium. However, somewhat surprisingly, this finance literature has not investigated whether long term risks have changed over time. Our contribution is twofold: to put Epstein-Zin-Weil preferences in an OLG model and use such a model to compute the low frequency changes in long term risk that are consistent with the data.
Finally our paper contributes to the small literature that investigate whether inequality can explain the level of interest rates. Auclert and Rognlie (2016) present a new-Keynesian model with wage rigidities and agents facing uninsurable idiosyncratic risks. Calibrating to the present, their model shows that a rise in inequality similar to that observed in the US since the 1980s would induce a further drop of 0.90 percent in real interest rates in the context of a permanently binding ZLB. We allow inequality to impact savings and interest rates in our OLG model. We show that it plays no role in the increase of the risk premium.
First, real interest rates have declined steadily over the last 2 decades (Figure 1). This downward trend is observed across OECD countries, for short-term and long-term interest rates as well as estimates of the natural rate of interest, and whatever the approach taken to approximate inflation expectations to define ex ante real interest rates (King and Low, 2014; Hamilton et al., 2016; Rachel and Smith, 2015; Laubach and Williams, 2016; Holston et al., 2016; Fries et al., 2016; Fischer, 2016a,b). Since the 1970s correspond to a period of financial repression with limited openness of euro area financial markets, we focus this paper on understanding the decline in real rates since the 1980s.
Second, the return to capital as measured from national accounts has remained flat. Gomme et al. (2011) build the return to productive capital as the net operating surplus, which is equal to value added minus depreciation and payments to labor, divided by the capital stock. Gomme et al. (2015) and Caballero et al. (2017) stress that, in the US, return to productive capital has no trend. It fluctuates with the cycle around 10 to 11 percent before tax and around 7 percent after tax. In Figure 2 we report similar indicators of the return on productive capital for the Euro area, Japan and the US from the AMECO database. Again, we see no downward trend in this measure of the return on investment.
Using a single framework that encompasses the broad range of proposed explanations is like placing all the "usual suspects" in the same lineup. Many of the factors cited in Bean et al. (2015), Rachel and Smith (2015) and Gourinchas et al. (2016) can be embedded in the single OLG model we present here, which nests Eggertsson and Mehrotra (2014) and Coeurdacier et al. (2015), and adds risk. This comes at a cost if we want the model to remain tractable: there are only three generations and only one source of risk.
The determination of the interest rate in those models comes down to the Euler equation of savers, within which the constraints faced by borrower agents and other determinants enter through market-clearing. In the presence of risk, the savers also face a portfolio choice.
In each discrete time period t a generation is born that lives 3 periods y, m, o. The size of the generation born at t is Nt = gL,tNt-1. Preferences are of the Epstein and Zin (1989) - Weil (1990) form:
with в the discount factor, 7 > 0 the coefficient of relative risk aversion, 1 /p > 1 the intertemporal elasticity of substitution.
The factors of production are capital, which depreciates at a rate 5, and labor, supplied inelastically by young and middle-aged agents. The labor productivity of a member of generation t is ey when young and em = 1 when middle-aged. The only source of risk comes from the aggregate productivity of labor over time, At = gA,tAt-1 which is stochastic. A neo-classical constant-returns production function combines capital (with share a) and labor (with share 1 — a) to produce output, one unit of which can become either one unit of consumption or 1/p\ units of investment; the relative price of investment goods is exogenous, deterministic and follows = g1tp(L 1. Markets are competitive and prices are flexible. Labor earns a wage wt while capital earns a return r^.
Agents can purchase investment goods, and can also borrow from and lend to each other at a gross rate Rt+1, but they cannot owe more (principal and interest) than a fraction 6t of next period's expected labor income. We will focus on situations in which the young borrow from middle-aged, and the middle-aged lend to the young and invest in physical capital by buying the depreciated stock in the hands of the old and purchasing investment goods. Notice that we assume that the young cannot borrow to invest for the sake of simplicity. This is also in line with the data: ? how that young US adults hold less that 10% of their debt outstanding in the form of stocks. In the euro area, the debt issued by younger households also dwarfs their the holding of stocks (see the European Household Finance and Consumption Network).
The following equations summarize the above. Agents of generation t choose jcy, ct+1, c°+2, km2, Ь+1, 2} to maximize (1-3) subject to three budget constraints and one
1.2. Equilibrium conditions
The solution proceeds as follows. Following Giovannini and Weil (1989) we first express the middle-aged agent's first-order conditions in terms of a total return on their portfolio, and derive the demand for the two available assets, capital and loans to the young, as well as a relation between the two returns. We then use market clearing: the demand for capital must equal the aggregate stock of capital, while the young's borrowing constraint, expressed in terms of their wages, determines the supply of the other asset. This allows us to derive the law of motion for the capital stock or, equivalently, the risk-free rate. We assume here that 6 =1; the general case is treated in the appendix.
To see it as a portfolio problem, express the budget constraints (12)—(13) in terms of income It — wt — 9t-1Et-1wt and total savings Wt invested in capitalplk±1k'm±1 with return R+i and loans — 6(+1 with return Rt+1. Letting Tt be the portfolio weight on capital, the total return on the middle-aged agent's portfolio is Rmi — TtRkk + (1 — rt)Rt+1 and the budget constraints become
The law of motion (26) is the core of the model and the basis for our simulations. It rewrites the middle-aged agent's optimal choice of saving (18) with market-clearing imposed on the quantities to formulate a law of motion for aggregates. The left-hand side is the savers' income, while the right-hand side is (the inverse of) the saving rate multiplying savings. Equation (26) is an implicit relation, we show in Appendix that it defines properly Rt+1. To develop more intuition we first examine the form it takes in a deterministic steady state, then examine the role of risk.
1.3. Deterministic steady state
In steady state (16) implies that (At/kt)a/p^ is constant, hence the growth rate of capital must be gk = Qa/qi 1/(1-a), as it would be in an infinitely-lived representative agent model. Capital grows at the same rate as labor productivity; the trend in the price of investment goods acts in this respect like an additional form of technological change (gI < 1 leading to growth in the capital stock).
From (19) it also follows that, in steady state, the income of the middle-aged It (and, by (18), their consumption as well as aggregate consumption) grows at the rate gIgk, that is, the growth rate of capital priced as investment goods. The only determinants of these steady state rates are the technological parameters qa and gI. The other parameters affect R and the allocation across generations.
The equation determining the steady state interest rate can be expressed as
The structure of the equation remains a modified Euler equation. On the left-hand side the term qagI- T=a is the steady state rate of growth of capital, which depends only on productivity growth (including the effect of the price in investment goods). This growth rate is unaffected by the various other features of the model. On the right-hand side are three terms. The first is the savings rate. The second term in square brackets represents the "pure" OLG component, specifically the fact that those who save do so out of labor income only; capital income is used by the old to finance their consumption. The last term captures the effect of the borrowing constraint: this can be seen by setting ey =0 and в = 0, which deprives the young of income and prevents them from borrowing, effectively eliminating them. Then that last term reduces to 1, and the model is isomorphic to a two- period overlapping generations model with no borrowing constraint.
Risky steady state
We assume that uncertainty on the productivity can be modelled as an i.i.d process.
Assumption 1 (Distribution of the productivity shock). Assume that the productivity shock is i.i.d, and log-normal with mean 1 and variance a.
To account for the impact of risk while retaining tractability, we appeal to the concept of risky steady state (Juillard, 2011; Coeurdacier et al., 2011). As explained in Coeurdacier et al. (2011), the risky steady-state is the point where agents choose to stay at a given date if they expect future risk and if the realization of shocks is at its mean before this date. Thus, we set the current period at to its mean of 1 but maintain at+1 as a stochastic variable, assuming that it is In effect, we assume that in every period the current realization of the shock is at its mean but agents take into account the risk in the future.
The following result describes the behavior of the risky steady-state.
Compared to the deterministic case, the presence of risk adds two channels, captured by the terms ф and £, functions of the exogenous factors only. .
The first channel £ relates to the risk premium and its impact on the portfolio choice of the agent. This can be seen in two ways. First, from (17) the risk premium Rk/R is at+i/£t. Second, the share of the agent's savings invested in capital (the risky asset) is a(1 + eygL,t+i)£t/vt, which is proportional to the term £t/vt = £t/(a( 1 + eygL)£t + (1 - а)в.
The following result describes the properties of the risky steady-state, when a follows a log-normal law.
The risk premium obviously increases with risk aversion 7, but also with a tightening of the borrowing constraint (lower 9) or a fall in population growth gL, both of which reduce the supply of bonds.
The second channel, фг, acts like a distortion to the discount rate, and is familiar from the literature on recursive preferences.
There are two things to note. One is that the sign of the sensitivity of ф4 to risk depends on whether the intertemporal elasticity p is high or low relative to 1. The discussion in Weil (1990, 38) applies here: a high IES (p < 1) means that the income effect is small relative to the substitution effect, and the "effective" discount factor фв rises with risk: the agent behaves as if she were more patient, and a higher interest rate R is required in equilibrium. The IES determines the sign, but the magnitude of the effect is determined by the risk aversion 7.
The second point is that the relative strength of the two channels (the ratio of дф/да2 to д£/да2) is (1 — p)a( 1 + eygL)/2(a(1 + eygL) + (1 — a)9) which, for low 9, is close to (1 — p)/2. For a IES close to 1, the effect of risk through the precautionary channel will be much smaller (in our calibration of p = 0.8, one order of magnitude smaller) than through the portfolio channel. For log utility (IES=1) there is only the portfolio channel. In Appendix, figure 21 represents the evolution of the share allocated to capital for the US.
Thus, even for p = 1 there is room for risk to affect interest rates, through the portfolio channel. The return to capital, however, does not depend on risk. Increasing risk raises the risk premium and compresses the risk-free rate, which is (roughly) what we see in the data. Indeed, risk is the only one of our "suspects" that affects only the risk-free rate.
1.4. Modelling the risk
We choose to adopt a very simple process for the evolution of the productivity shock. The effect on the risk premium thus comes from the standard deviation of the shock. Testing for other processes do not point to features linked to the skewness or general asymmetries of the process. What differs in our exercise from the study of Bansal and Yaron (2004) is the fact that consumption in our work behaves endogenously while it is exogenous in that paper. This explains that the behavior of the risk premium is slightly different.
In this section we use the model to match the data on the risk free real interest rate and the return of capital since 1970.
5.1 Calibration of the model
The spirit of the exercise is as follows.
We distinguish between (a) the structural parameters в, Y, P a, 5, ey characterizing preferences and technology, held fixed throughout and (b) the factors gj, gA, gL that vary over time but are readily observable (Table 1). The structural parameters are calibrated in line with the literature: the discount factor в, the capital share a, the inter-temporal elasticity of substitution 1/p, the capital depreciation rate 5 and the relative productivity of young ey have standard values. It is well known that risk aversion y is required to be very high to match the risk premium. n the one hand, using grid search, Rudebusch and Swanson (2012) estimate values of risk aversion lying between 75 and 88 in a small DSGE model with three shocks. These values are of similar magnitude to those found in other quantitative studies, see Darracq Paries and Loublier (2010) and references therein. This leaves us with в (the borrowing constraint) and a (the amount of risk) which we approach flexibly because we see them as less easily observable. We proceed in several steps. First, we fix both в = 0.07 and a = 0.09 for the US and the EA. These calibrations are chosen roughly in terms of the estimated values on the whole period. Given that a period last 10 years, в = 0.07 is approximately equal to 70 percent of a yearly GDP. Turning a, we hence assume that gA has 95 percent chances to be between its mean value plus/minus 0.18. Such values for these two parameters impact the level of the interest rates, but they impact marginally their evolution over time.
With the leverage constraint and the degree of uncertainty fixed, we can evaluate the effects of demography, trend productivity and the relative price of investment on the risk free rate and the risk premium. Then we keep in turn в and a fixed and compute a time series for the other in order to match the path of the risk-free rate R. Finally let both vary and we back out time series of et and at that will result in sequences of risky steady states R and Rk matching the observations. Ultimately, of course, we will need to confront these time series to data in order to assess the model's success or failure at accounting for the decline in interest rates.
Our model periods last 10 years. In the figures that follow, each year N on the x- axis corresponds to the average 10-year lagging average (years N — 9 to N), both for data and simulations.Our reasoning is that deciding when our 10-year periods start and end
is somewhat arbitrary, and can lead to suspicions of cherry-picking. Presenting the data and simulations in this manner allows the readers to make that decision, as long as they keep in mind that we are not representing annual time series, but sequences of {t, t + 10} pairs.
1.2. Results
The impact of observable factors
To measure this impact we fix в and a and analyze the model-based interest rates, when we use as inputs the growth rate of productivity, the change in demography and in relative investment price that we observe in the data. These factors are represented in Figure 3. We observe a steady productivity slowdown in the Euro area. Its year on year growth rate falls from 3 percent in 1980 to less than 1 percent since 2008. In the US, there is no such slowdown for the entire period. Productivity accelerate from 1995 to 2003 and then decelerates since. Productivity growth measured at the world level remains stable as the weight of China in world output increases. The decline in the population growth rate is more homogenous. It declines by about 1 percent in the three economies we consider from 1980 to today. .
Figure 4 shows the effects of these observable factors on the risk free rate (in the left panel) and the risk premium (in the right panel) for the US. In the US, interest rates decline essentially since 2003, due to the deceleration of productivity. Their total decline since 1990 amounts to 0.7 percent. These estimates are comparable to the ones of Gagnon et al. (2016) and Carvalho et al. (2016). The former estimate that demography account for
a decline of the US equilibrium real rate by 1.25 percent from 1980 to 2015, and the latter estimate a decline by 1.5 percent between 1990 and 2014. The larger effects (compared to our results) comes from a different calibration of inter-temporal elasticity of substitution, but mostly from their assumption that retirees turn over their assets to a mutual fund and earn a return augmented for the probability of death (Blanchard, 1985). Turning to the euro area (Figure 5), the simulated decline in interest rates is much larger. Both the risk free rate and the return on capital decline by 2.3 percent. This reflects the steadier deceleration of productivity throughout the last three decades. However, neither productivity nor demography have any effect on the risk premium. In our simulation, it remains flat both in the US and the euro area.
Figure 3: The inputs
The borrowing constraint
To measure the explanatory power of the borrowing constraint, we fix a and compute the parameter в which is consistent with the risk-free rate, and the observable inputs. The implied borrowing constraints and the model-based return on capital, and risk-premium are represented in Figures 6, for the US and 7, for the EA. This exercise shows that for both areas, the decline in the risk-free rate requires a tightening in the borrowing constraint, from 0.15 in 1990 to 0.05 in the US (from 0.12 to 0.08 in the EA). This evolution of the borrowing constraint hardly replicates the increase in the risk-premium in both areas, 1.1 percent in the US between 1990 and 2014 (0.2 percent in the EA on the same period).
Risk
We now consider the borrowing constraint as fixed over time, and assess the evolution of the variability in productivity that is required to reproduce the decline in the risk-free rate. The implied changes in variability and the model-based return on capital are represented in Figure 8, for the US and Figure 9, for the EA. The variance of productivity has to increase from about 0.04 percent in 1990 to 0.14 today, for the US (from 0.08 to 0.12 for the EA). For both areas, this evolution since 1990 replicates quite well the evolution in the return on capital, and risk-premium. Such increases in the uncertainty of the growth rate of productivity are fairly small. For instance, if the mean growth rate of productivity is 2 percent per annum, the increase in risk premium observed in the data could reflect a mere change in the distribution of productivity risk: had 95 percent to be between 1.92 and 2.08 in 1990 and 95 percent to be between 1.80 and 2.20 in 2014. Given that we have gone through the Great Recession in 2008, such a broadening of uncertainty does seem so large. In the euro area (Figure 9), the increase in risk perception required to match the reduction in the risk free rate since 1990, from a = 0.08 to a = 0.12, is even smaller than in the US.
Risk and the borrowing constraint
In Figure 10, for the US and Figure 11, for the EA, we let both в and a change over time so that we can replicate perfectly both the risk free rate and the return on capital.
In particular, the trend decline in the risk free rate since 1990 is due exclusively to an increase in the risk of productivity, from 0.08 to 0.19 in the US (0.09 to 0.16 in the EA).
Moreover, the evolution of the risk-free rate and the return on capital is consistent with a non decreasing pattern of the borrowing constraint. This shows that, according to the model, the drop in real interest rate does not necessarily reflect deleveraging headwinds. What evidence do we have that uncertainty has effectively increased over the last 25 years? Baker et al. (2016) indicates that there may be an upward trend of economic uncertainty from the 1985 to 2012 and a clear acceleration of political uncertainty over the last fifteen years. In particular the so called "great moderation" period, usually dated from 1985 to 2007 does not correspond to a decline in uncertainty as measured by Baker et al. (2016). Altogether, that our simulation point to an increase in perceived risk as suggested by the steady increase in the risk premium from 1990 to 2016 is not inconsistent with these other measures that show uncertainty trending up, at least from 1990 to 2012.
The evolution of the borrowing constraint ?
As shown by Buttiglione et al. (2014) there has been hardly any overall deleveraging since the crisis. Private debt in advanced economies adds up to 178 percent of GDP in 2016, i.e. the same level than in 2010, while public debt increased from 75 to 87 percent of GDP over the same period. Deleveraging of the private sector has been very large in Spain and in the United Kingdom, but it increased in France and Canada. And public debt increased in all G7 countries but Germany.
We thus use the model to infer the borrowing constraint consistent with the evolution of debt and the risk-free rate in both areas, depicted in Figure 12, for the US and Figure 13, for the EA. In this exercise, we pin down the evolution of 9, the borrowing constraint, with the ratio of debt/income observed in the data. . We simulate the level of a that matches the risk-free rate. Two results are striking. First, в is not increasing in spite of the increase in the debt/income ratio. This is because of the decline in interest rates which implies that a the borrowing constraint of the young binds at higher levels of debt. Second, the increase in uncertainty required to replicate the decline in the risk free rate strikingly explains the increase of the risk premium, both in the US and in the euro area.
1.3. A global perspective
A fair criticism of our calculations is that, by focusing on the US and the Euro area, we neglect the phenomenon described as "savings glut" or "global imbalances" of the 2000s, namely the increase in savings from emerging economies. We repeat our calculations for the a world economy which we define as the aggregate of the US, the euro area, Japan and China. The "world" population aged 20 to 64 is the ones of these four economies, investment price evolves as the American one, productivity is an aggregate of four economies productivities weighted by their GDP. The target rates, both for return on capital and for the risk-free rate, are an average of the US and Euro area, considering that world capital markets are integrated.
The results at the world level are broadly similar to the ones we found for the US and the EA: risk is the main factor that can account for the behavior of the risk premium since 1990. The picture is similar: we see в rising since the mid-1980s, suggestive of the global savings glut. Then the borrowing ratio stops rising in the late 1990s and barely falls after that. Hence, deleveraging does not seem to be at play since the financial crisis.
1.4 Extensions
Longevity
We introduce longevity as a survival probability. Specifically, we replace equation (2) with
The data is taken from Bell and Miller (2005). We compute At as the probability of surviving from age 60 to age 70 at different points in time. This probability rises steadily through the sample from about 0.8 to about 0.9. In our model this is equivalent to a time- varying (and rising) discount factor.
The quantitative impact is to lower both the risk-free rate and the risky rate, by nearly identical amounts: 65bp for the risk-free rate and 69bp for the risky rate. The risk premium shrinks slightly.
Figure 18: Effect of longevity, in the model
Labor Share
The decline of the labor share has been documented in the US (Elsby et al., 2013) and elsewhere (Karabarbounis and Neiman, 2014). The corresponding pattern for the capital share is a rise from around 33 percent in the early 1970s to 38 percent by 2014 (Koh et al., 2016).
In our model an increase in the capital share pushes up the risk-free rate, and increases slightly the risk-premium. Indeed, equation (38) can be rewritten as on the left hand, the function is increasing in R. Since gl < 1, the right hand side is an increasing function in a.
Thus, interest rate is an increasing function of the capital share a. If we compare to our baseline calibration for which a is the sample average, the interest rate is lowered at the beginning of the sample by 0.6 percent but raised at the end by 1.1 percent. The risk premium is raised 20bp at the beginning and lowered 23bp at the end.
This factor, therefore, does not help in accounting for the patterns in the data. Figure 19 illustrates the role changes in the labor share.
Markups
The decline of the labor share has been interpreted as reflecting growing monopoly power (Jones and Philippon, 2016; Barkai, 2017; De Loecker and Eeckhout, 2017). We introduce market power in a standard way.
There is a continuum of intermediate goods |у*}*е[о,1] produced using a neo-classical constant-returns production function that combines capital (with share a) and labor (with share 1 — a). The final good, produced with the production function Yt = /0, can become either one unit of consumption or 1/p’k units of investment as before.
Market power over intermediate goods introduces a distortion between marginal products on one hand, and the wage rate wt and capital rental rate on the other:
The agents' budget constraints are slightly modified: instead of investing pk+1k™2 in capital, the middle-aged agent is now investing sJ+1 in equity. The funds are used by equity in the intermediate goods producers, and to purchase used capital and investment goodspk+1 k™2. The capital is rented out on competitive markets. The total returns to equity at t + 2 are Rt+2s'm+1 and now include, in addition to the market value of the depreciated capital and its rental income, a third term representing the profits of the intermediate goods firms. These profits, in aggregate, are (1 — ^ )Yt. Per unit of capital, the profit rate is nt = (1 — ^)At1~akOL~1. The total return to capital is
The law of motion (26) is modified as:
but, when expressed in terms of R, nothing is changed. Moreover equations (17) and (22) are unchanged, and p does not enter the auxiliary variables vt and ut+1. Therefore, introducing markups changes the value of the capital/output ratio k, but it does not affect the risk-free rate or the equity premium (as noted by Farhi and Gourio 2018).
Extension to the case S = 1
The general case S =1 is presented in Appendix, the dependence of R and Rk is less obvious, since they appear as a fixed point of a more complex function. However, when R, Rk and all the inputs are observable, the estimation of the associated в and a is not more difficult. The corresponding parameters for different values of S is represented in Figure 20, it exhibits that the global pattern is similar, when the depreciation is not total, the corresponding borrowing constraint is slightly smaller.
Inequality
Our final extension is to allow for changes in inequality.
We model inequality by introducing heterogeneity in middle-age productivity: agents of type I have productivity ej, and changes in inequality arise from mean-preserving changes in the distribution {ej} j. There is no additional uncertainty: young agents know which productivity they will have in the next period, and their borrowing constraint reflects their known productivity.
Introducing bequests
In our model consumption and savings in the second period are linear in lifetime wealth, hence mean-preserving spreads in inequality will have no aggregate effects. To break this linearity we introduce a bequest motive in the last period (De Nardi, 2004; Benhabib et al., 2011). The utility from leaving a bequest B is assumed to be of the form hB1-£ with h
measuring the strength of the bequest motive. The elasticity e, however, must be different from p.
This can be seen in a simple two-period planning problem, with an added bequest motive h(B):
For a steady-state with bequests to be well-defined, we need to make an adjustment to the formulation of preferences, as follows. Preferences are then represented as
We explore the quantitative effects of a rise in inequality as follows. We assume that the productivity distribution in middle age is binary with half of middle age agents having a high productivity , and half with low productivity
Then, we study the effect of a change in the level of e. When e varies from 0.1, the effect on the risk premium is around 10—5 percent (for standard calibration of all the other parameters). This simple exercise shows that, inside the model, the impact of an increase of inequality on the evolution of the risk-premium remains limited.
Risk-free rates have been falling since the 1980s while the return on capital has not. We analyze these trends in a calibrated overlapping generations model designed to encompass many of the "usual suspects" cited in the debate on secular stagnation. Declining labor force and productivity growth imply a limited decline in real interest rates and deleveraging cannot account for the joint decline in the risk free rate and increase in the risk premium. If we allow for a change in the (perceived) risk to productivity growth to fit the data, we find that the decline in the risk-free rate requires an increase in the borrowing capacity of the indebted agents in the model, consistent with the increase in the sum of public and private debt since the crisis but at odds with a deleveraging-based explanation put forth in Eggertsson and Krugman (2012).
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