Why Have Interest Rates Fallen Far Below the Return on Capitalademu-project.eu/wp-content/uploads/2017/06/Magali-Marx... · 2020-04-23 · The decrease of real interest rates 1970

Why Have Interest Rates FallenFar Below the Return on Capital

Magali MarxBanque de France

Benoıt MojonBanque de France

Francois R. VeldeFederal Reserve Bank of Chicago

Macroeconomic and Financial Imbalances and Spillovers, June 9

Why Have Interest Rates Fallen Far Below the Return on Capital 1

mailto:[email protected]

http://www.chicagofed.org/economic_research_and_data/index.cfm

The decrease of real interest rates

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Rea

l ra

tes

(%)

-2

0

2

4

6

8

10

12

Hamilton et al 10yr MA, US

King Low (indexed bonds)

US 10yr ex-ante real rate

Hamilton et al 10yr MA, France

Laubach-Williams r*


Which does not reflect the evolution of capital return

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Rea

l re

turn

on c

apit

al (

%)

-2

0

2

4

6

8

10

12

14

16

18

US

EA large 4 countries

EA

Japan


The usual suspects

Low rates have been loosely tied to “secular stagnation”

A number of potential explanations have been cited:

I productivity slowdownI changing demographics (population slowdown, increased longevity)I change in the price of investment goodsI tightening of borrowing constraintI shortage of safe assetsI rising inequality


Our goal

quantitative assessment of the various factors cited

embed them in a single, tractable model

explain both the evolution of capital return and risk-free rateI this means having risk, and attitudes toward risk, in the model


Related literature

low rates: King and Low (2014); Hamilton et al. (2016); Holston et al. (2016)

safe assets: Coeurdacier et al. (2015); Caballero et al. (2008); Caballero and Farhi(2014)

deleveraging: Eggertsson and Krugman (2012); Korinek and Simsek (2016); Farhiand Werning (2013)

secular stagnation: Bean et al. (2015); Rachel and Smith (2015)

demographics: Carvalho et al. (2016); Gagnon et al. (2016)

risk: Kozlowski et al. (2015); Hall (2016)

return on capital: Caballero et al. (2017)


The Model

add risk to Eggertsson and Mehrotra (2014) and Coeurdacier et al. (2015).

time is discrete, infinite

3-period OLG structure (y , m, o)I population Nt , growth rate gL

recursive preferences with Epstein-Zin-Weil utility function

capital and labor (supplied inelastically), age-specific productivities (ey , 1, 0)

output Y = Kα(AL)1−α

I productivity A: trend growth gA + shock with variance σ (only source of risk)I growth in price of investment gI


Preferences

Epstein and Zin (1989)–Weil (1990) recursive preferences:

Vt = U(ct ,EtVt+1) =(c1−ρt + β ( (EtVt+1

1−γ)1

1−γ )1−ρ) 1

1−ρ


Preferences

Epstein and Zin (1989)–Weil (1990) recursive preferences:

Vt = U(ct ,EtVt+1) =(c1−ρt + β ( (EtVt+1

1−γ)1

1−γ )1−ρ) 1

1−ρ

CES functional form applied to

time:(c1−ρt + β (·t+1)1−ρ) 1

1−ρ

I ρ: inverse of intertemporal elasticity of substitution

risk: (EtVt+11−γ)

11−γ

I γ: risk aversion

when ρ = γI standard time-additive preferencesI tension between

F high γ required to match asset pricingF low ρ required to match consumption growth with interest rates


Budget constraints

young borrow from middle-aged up to a fraction θ of their t + 1 labor incomeI we focus on equilibria where this bindsI no other frictions (e.g., price stickiness)

middle-aged lend to young, buy capital from old, invest

old collect returns, sell depreciated capital

cyt = byt+1 + wte

yt

byt+1 ≤ θtEt(wt+1/Rt+1)

cmt+1 − bmt+2 + pk

t+1kmt+2 = wt+1 − Rt+1b

yt+1

cot+2 = (pkt+2(1− δ) + r kt+2)km

t+2 − Rt+2bmt+2

market-clearing:

gL,tbyt+1 + bm

t+1 = 0

gL,t+1byt+2 + bm

t+2 = 0


Production

Yt = (Nt−2kmt )α [At(e

yt Nt + Nt−1)]1−α

Nt−2kmt : capital (chosen by current old in the previous period)

eyt Nt + Nt−1: labor (of young and middle-aged)

Competitive factor markets:

wt = (1− α)A1−αt kαt

r kt = αA1−αt kα−1

t

both written in terms of the capital/labor ratio kt defined as

kt ≡Nt−2k

mt

eyt Nt + Nt−1=

kmt

gL,t−1(1 + eyt gL,t).


Solution strategy

only the middle-aged have an intertemporal problemI how much to saveI in what form: bonds or capital

write the middle-aged’s Euler equation and substitute equilibrium quantitiesI quantity of bonds determined by young’s constraintI Euler equation also relates risk-free rate R and return to capital Rk

we derive a law of motion expressed in terms of R or equivalently k


Solution strategy (2)

Middle-aged FOCs:

(cmt )−ρ = β[Et(c

ot+1)1−γ

] γ−ρ1−γ

Et

[(cot+1)−γRk

t+1

](cmt )−ρ = β

[Et(c

ot+1)1−γ

] γ−ρ1−γ

Et

[(cot+1)−γ

]Rt+1.

Define Rmt+1 = αtR

kt + (1− αt)Rt+1 and express budget constraints as

Wt = Yt − cmt

cot+1 = Rmt+1Wt .

Portfolio choice: set αt so that

Et(Rmt+1

−γ)Rt+1 = Et

(Rmt+1

−γRkt+1

)Saving decision:

Yt =(

1 + (βφtR1−ρt+1 )−

1ρ

)Wt

Then use market clearing to express Yt , Wt , Rmt+1 in term of the aggregate capital stock


Law of motion

(1 + (βφt)

−1/ρR1−1/ρt+1

)−1

︸︷︷︸saving rate

(1− θt−1

at)(1− α)(

Rkt+1

gI− 1 + δ)kt︸︷︷︸

income

= gL,t

α(1 + eygL,t+1)︸︷︷︸capital

+

(1

ξt

)(1− α)θt(1− gI

1− δRt+1

)︸︷︷︸bonds

kt+1

︸︷︷︸investments


Law of motion

(1 + (βφt)

−1/ρR1−1/ρt+1

)−1

︸︷︷︸saving rate

(1− θt−1

at)(1− α)(

Rkt+1

gI− 1 + δ)kt︸︷︷︸

income

= gL,t

α(1 + eygL,t+1)︸︷︷︸capital

+

(1

ξt

)(1− α)θt(1− gI

1− δRt+1

)︸︷︷︸bonds

kt+1

︸︷︷︸investments

overlapping generationsI saving only done out of labor income

borrowing constraintI disappears if θ = 0, ey = 0

riskI φt : precautionary saving, acts like discount factor distortion (≶ 1)I 1/ξt : portfolio choice


Risk terms

The factors φt and ξt are

ξt =Et(ut+1

−γ at+1)

Et(ut+1−γ)

φt =[Etut+1

1−γ](γ−ρ)/(1−γ)

Etut+1−γvt

ρ

with

ut+1 ≡ α(1 + eygL,t+1)at+1 + (1− α)θt

at+1 ≡A1−α

t+1

EtA1−αt+1

.

only functions of (moments of) the exogenous process At+1

when δ 6= 1, φt involves Rt+1 as well


Risky steady state

to account for risk in a tractable way, we appeal to the concept of “risky steady state”:

exogenous trends as in the data

productivity shock is assumed i.i.d.

in the law of motion, at set at its mean, at+1 is stochastic

agents take into account the uncertainty


Risk and borrowing constraint

When δ = 1, ρ < 1, θ = 0, ey = 0 (no young):

φt ' 1 +1

2γ(1− ρ)σ2

1

ξt' 1



When δ = 1, ρ < 1:

φt ' 1 +1

2γ(1− ρ)

α2(1 + eygL,t+1)2

(α(1 + eygL,t+1) + (1− α)θt)2σ2

1

ξt' 1+γ

α(1 + eygL,t+1)

α(1 + eygL,t+1) + (1− α)θtσ2



When δ = 1, ρ < 1:

φt ' 1 +1

2γ(1− ρ)

α2(1 + eygL,t+1)2

(α(1 + eygL,t+1) + (1− α)θt)2σ2

1

ξt' 1+γ

α(1 + eygL,t+1)

α(1 + eygL,t+1) + (1− α)θtσ2

law of motion: 1 + (βφtθ,σ−+

)−1/ρR1−1/ρt+1

−1

(1− θt−1

at)(1− α)

Rkt+1

gIkt

= gL,t

α(1 + eygL,t+1) +

(1

ξt

)θ,σ−+

(1− α)θt

kt+1


Long run determinantsof the bond interest rate r and the return on capital rK

δ = 1, ρ = 1:

Observable factorsI productivity growth gAI evolution of working age population gLI trend in investment price gI

Unobservable factorsI borrowing constraint θI variance of the shock on the trend of productivity σ.


Long run determinantsof the bond interest rate r and the return on capital rK

δ = 1, ρ = 1:

Observable factorsI productivity growth gAI evolution of working age population gLI trend in investment price gI

Unobservable factorsI borrowing constraint θI variance of the shock on the trend of productivity σ.

r = r + (gL − 1) + (gA − 1)− 1

2(gI − 1) + cθ + γu(θ|σ

+

, σ2

−)

rK = r + γv(θ|σ−, σ2

+)

The wedge between r and rK is only affected by θ and σ


Empirical strategy

our targets are the risk-free rate and the return on capitalwe segregate the usual suspects into

I the observables: productivity, demographics, price of investmentI the “less observables”: borrowing constraint, productivity risk

Three steps:1 input the observables, set θ and σ constant to match the levels of the targets2 input the observables, compute θ to match the risk-free rate, keep σ constant3 input the observables, compute σ to match the risk-free rate, keep θ constant4 input the observables, compute θ and σ to match both targets

repeat for US and Euro area (and the world)then stare at the pictures. . .caveats

I we interpret the generations loosely (10-year averages)I risk-free rates before the 1980s are less meaningful (financial repression etc), so we focus on

1990s to present


Model calibration and data sources

ParametersT length of period (years) 10β discount factor 0.98T

α capital share 0.28γ risk aversion 100ρ inverse of IES 0.8δ capital depreciation rate 0.1 ∗ Tey relative productivity of young 0.3FactorsgL,t growth rate of population 20-64 US, EA (France), China, Japan: OECDgI ,t investment price growth DiCecio (2009)gA,t productivity growth US: Fernald (2012), Euro: NAWM modelRt real interest rate US: Hamilton et al. (2016), FranceRkt return on capital US, EA: our calculations a la

Gomme et al. (2015)at productivity shock ln(a) is a i.i.d. N(−σ2/2, σ2)Free parametersθ borrowing constraint on youngσ2 variance of at


The inputs

1970 1980 1990 2000 20100

1

2

3

4

Productivity growth (gA − 1)

1970 1980 1990 2000 20100

1

2

3

Working age pop. growth (gL − 1)

1970 1980 1990 2000 2010

-2

0

2

4

Risk free rate (r)

1970 1980 1990 2000 2010

6

8

10

12

Capital return (rK)

US

EA

World


Impact of observable factors, in the USObservable factors explain about 1.4% from 1992 to 2014

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Productivity Risk (std) σ

1970 1980 1990 2000 2010

-5

0

5

10

15

Observed and simulated rates

1970 1980 1990 2000 2010

2

4

6

8

10

12

Observed and simulated risk premium


Impact of observable factors, in the EAObservable factors explain about 1.8% from 1992 to 2014

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Impact of the borrowing constraint, in the US.A tighter constraint can account for the fall in the risk-free rate and 0.8% increase of the risk premium

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Impact of the borrowing constraint, in the EA.A tighter constraint can account for the fall in the risk-free rate and 0.7% increase of the risk premium

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Impact of risk, in the US.A higher risk perception can account for the fall in the risk-free rate and the increase in the risk premium

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Impact of risk, in the EA.A higher risk perception can account for the fall in the risk-free rate and the increase in the risk premium

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Impact of risk and the borrowing constraint, in the US.With higher risk perception data are consistent with non decreasing debts

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15

Observed rates

1970 1980 1990 2000 2010

2

4

6

8

10

12

Observed risk premium


Impact of risk and the borrowing constraint, in the EA.With higher risk perception data are consistent with non decreasing debts

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

Observed rates

1970 1980 1990 2000 2010

2

4

6

8

10

12



Borrowing constraint and risk, in the US.

1970 1980 1990 2000 2010

0

0.1

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.1

0.2

Productivity Risk (std) σ1970 1980 1990 2000 2010

0

5

10


1970 1980 1990 2000 2010

2

6

10


1970 1980 1990 2000 2010

0.1

0.2

0.3

Debt/income


Borrowing constraint and risk, in the EA.

1970 1980 1990 2000 2010

0

0.1

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.1

0.2

Productivity Risk (std) σ1970 1980 1990 2000 2010

0

5

10


1970 1980 1990 2000 2010

2

6

10


1970 1980 1990 2000 2010

0

0.2

0.4

Debt/income


Global perspectiveImpact of observable factors

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Global perspectiveImpact of the borrowing constraint

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Global perspectiveImpact of risk

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15


1970 1980 1990 2000 2010

2

4

6

8

10

12



Global perspectiveImpact of risk and the borrowing constraint

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio θ

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2


1970 1980 1990 2000 2010

-5

0

5

10

15

Observed rates

1970 1980 1990 2000 2010

2

4

6

8

10

12



Conclusion

usual suspects aren’t enoughI deleveraging story

increased (perception of) risk can account for the patternsI but it’s a residual

more work to be done onI longevityI inequality (through a bequest motive)I exogenous supply of safe assets


Additional slides


Sensitivity to γ (US)

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2

Borrowing ratio

1970 1980 1990 2000 2010

0

0.05

0.1

0.15

0.2Productivity Risk (std)

γ = 100

γ = 50

1970 1980 1990 2000 2010

-5

0

5

10

15Observed and simulated rates

1970 1980 1990 2000 2010

2

4

6

8

10

12Observed and simulated risk premium


Sensitivity to γ (EA)

1980 1990 2000 2010 2020

0

0.05

0.1

0.15Borrowing ratio

1980 1990 2000 2010 2020

0

0.05

0.1

0.15

0.2Productivity Risk (std)

γ = 100

γ = 50

1980 1990 2000 2010 2020

0

5

10Observed and simulated rates

1980 1990 2000 2010 2020

0

5

10Observed and simulated risk premium


The inputs for the US

1970 1980 1990 2000 20100

1

2

3Productivity growth

1970 1980 1990 2000 20100

0.5

1

1.5

2Working Age Population

1970 1980 1990 2000 2010-5

-4

-3

-2

-1

0Relative investment price

1970 1980 1990 2000 20100

100

200

300Debt/GDP (%)

safe assets

private

total


The inputs for the EA

1980 2000 20200

1

2

3Productivity growth

1980 2000 20200

0.5

1

1.5

2Working Age Population

1980 2000 2020-5

-4

-3

-2

-1

0Relative investment price

1980 2000 202050

100

150

200

250Debt/GDP (%)

safe assets

private

total


Measures of risk


References I

Bean, S. C., C. Broda, T. Ito, and R. Kroszner (2015): “Low for Long? Causes andConsequences of Persistently Low Interest Rates,” Geneva Reports on the WorldEconomy 17, International Center for Monetary and Banking Studies.

Caballero, R. J. and E. Farhi (2014): “The Safety Trap,” Working Paper 19927, NBER.

Caballero, R. J., E. Farhi, and P.-O. Gourinchas (2008): “An Equilibrium Model of“Global Imbalances” and Low Interest Rates,” American Economic Review, 98, 358–93.

——— (2017): “Rents, Technical Change, and Risk Premia: Accounting for SecularTrends in Interest Rates, Returns on Capital, Earning Yields, and Factor Shares,”NBER Working Papers 23127, National Bureau of Economic Research, Inc.

Carvalho, C., A. Ferrero, and F. Nechio (2016): “Demographics and Real Interest Rates:Inspecting the Mechanism,” European Economic Review, 88, 208 – 226, sI: ThePost-Crisis Slump.

Coeurdacier, N., S. Guibaud, and K. Jin (2015): “Credit Constraints and Growth in aGlobal Economy,” American Economic Review, 105, 2838–81.

DiCecio, R. (2009): “Sticky wages and sectoral labor comovement,” Journal of EconomicDynamics and Control, 33, 538–553.

Eggertsson, G. B. and P. Krugman (2012): “Debt, Deleveraging, and the Liquidity Trap:A Fisher-Minsky-Koo Approach,”The Quarterly Journal of Economics, 127, 1469–1513.


References II

Eggertsson, G. B. and N. R. Mehrotra (2014): “A Model of Secular Stagnation,” WorkingPaper 20574, NBER.

Epstein, L. G. and S. E. Zin (1989): “Substitution, Risk Aversion, and the TemporalBehavior of Consumption and Asset Returns: A Theoretical Framework,”Econometrica, 57, 937–969.

Farhi, E. and I. Werning (2013): “A Theory of Macroprudential Policies in the Presenceof Nominal Rigidities,” Tech. rep., MIT.

Fernald, J. G. (2012): “A quarterly, utilization-adjusted series on total factorproductivity,” Working Paper Series 2012-19, Federal Reserve Bank of San Francisco.

Gagnon, E., B. K. Johannsen, and D. Lopez-Salido (2016): “Understanding the NewNormal: The Role of Demographics,” Finance and Economics Discussion Series2016-080, Board of Governors of the Federal Reserve System.

Gomme, P., B. Ravikumar, and P. Rupert (2015): “Secular Stagnation and Returns onCapital,” Federal Reserve Bank of St Louis Economic Synopsis, 19.

Hall, R. E. (2016): “Understanding the Decline in the Safe Real Interest Rate,” Workingpaper 22196, NBER.

Hamilton, J. D., E. S. Harris, J. Hatzius, and K. D. West (2016): “The Equilibrium RealFunds Rate: Past, Present and Future,” IMF Economic Review, 64, 660–707.


References III

Holston, K., T. Laubach, and J. C. Williams (2016): “Measuring the Natural Rate ofInterest: International Trends and Determinants,” Working paper 2016-11, FederalReserve Bank of San Francisco.

King, M. and D. Low (2014): “Measuring the “World” Real Interest Rate,” working paper19887, NBER.

Korinek, A. and A. Simsek (2016): “Liquidity Trap and Excessive Leverage,” AmericanEconomic Review, 106, 699–738.

Kozlowski, J., L. Veldkamp, and V. Venkateswaran (2015): “The Tail that Wags theEconomy: Belief-Driven Business Cycles and Persistent Stagnation,” Working Paper21719, NBER.

Rachel, L. and T. D. Smith (2015): “Secular Drivers of the Global Real Interest Rate,”Staff Working Paper 571, Bank of England.

Weil, P. (1990): “Nonexpected Utility in Macroeconomics,” The Quarterly Journal ofEconomics, 105, 29–42.


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Why Have Interest Rates Fallen Far Below the Return on Capitalademu-project.eu/wp-content/uploads/2017/06/Magali-Marx... · 2020-04-23 · The decrease of real interest rates 1970