L9: Consumption, Saving, and Investments 1 Lecture 9: Consumption, Saving, and Investments The following topics will be covered: –Consumption and Saving

L9: Consumption, Saving, and Investments 1

Lecture 9: Consumption, Saving, and Investments

• The following topics will be covered:– Consumption and Saving under Certainty– Uncertainty and Precautionary Saving– Risky savings and Precautionary Demand– Dynamic Investment and Portfolio Management

Materials are from chapters 6 & 7, EGS.


Consumption and Saving under Certainty• An agent lives for a known number of periods

• Yt: income, or endowment (continuous)

• Risk free interest rate r

• zt is the cash transferred from data t-1 to t, i.e., accumulated saving in t

• ct is the consumption in t

• The agent selects consumption plan c=(c0, c1, …, cn-1) to

maxU(c0, c1, …, cn-1) • Subject to the dynamic budget constraint: zt+1=(1+r)[zt+yt-ct]

• PV(zn)≥0. This can be rewritten as:

0)1(

1

0

n

tttt

r

cy or 0

1

0

wcn

ttt

where tt r )1( and tt yw 0 is life time wealth


Solutions and Considerations• See Figure 6.1, page 90, EGS

• The optimal condition implies Fisher’s separation theorem– Every investor should choose the investment which maximizes NPV of

its cash flow

• Similar to the static decision problem of an agent consuming n different physical goods in the classical theory of demand.

• three components of consumption: – nondurables,

– durables,

– services

– car is durable goods, house is too. but CPI does not count housing price, only rental price


Independence in Consumption

1

0

)()(n

ttt cucU

-- independence axiom (which precludes consumption habit) ut(ct) is the intraperiod utility of consumption – felicity function of comsumption at date t. More specifically, we have ut(ct)=ptu(ct) where u is called a felicity function of consumption at date t. u is increasing and concave;

Note: pt here no longer represents price of the commodity. It instead is interpreted as the discount factor for felicity u(ct) occurring at date t. If pt is less than 1, it can be interpreted as a proportional loss of utility due to postponing consumption, i.e., a discount factor on felicity. p0 is normalized to 1.


Objective Function AgainAs a result, the objective function becomes:

1

0

)(maxn

ttt

ccup subject to 0

1

0

wcn

ttt

Solution: The first order condition is ttt cup )(' for t=0, 1, …, n-1

Notes: o In this problem, the revenue flow is certain. Then there is

no probability function involved in the analysis. o It is similar to analysis on AD securities, while that

involves probability distributions.


Tendency to Smooth Consumptions• If Пt=1 for all t (i.e., r=0), then FOC: u’(ct)=ξ in each period

• The optimal consumption path does not exhibit any fluctuation in consumption from period to period: ct=w0/n

• Note: even revenue flow yt is known, they may not be stable over time. Thus borrowing and lending is required.


Optimal Consumption Growth

• In general, the real interest rate is not zero and agents are impatient• Assuming consumers use exponential discounting: pt=βt

– β =(1+δ)-1 – multiplying u(ct) by βt is equivalent to discounting felicity at a constant rate δ (see page 94, EGS)

• Under this condition, there are two competing considerations driving consumption decisions:– Impatience induces agents to prefer consumption earlier in life

– High interest rate makes saving more attractive

• Suppose that u(c)=c1-γ/(1-γ), where is the constant degree of fluctuation aversion. We have ct=c0at, where,

/1)

1

1(

r

a

rr

g 1)1

1( /1


Income Uncertainty and Precautionary Saving• Now yt is no longer certain

• Two period model to decide how much to save at date 0 in order to maximize their expected lifetime utility

))1(()()(max~

1100 ysrEusyusVs

FOC for s* is: )*)1((')1(*)('~

1100 ysrEursyu

That is: the willingness to save is determined by the expected marginal utility of future consumption. The uncertainty affecting future incomes introduces a new motive for saving. (why??) This is the precautionary motive for saving. It can be shown that a condition that consumers become more prudent when u1’’’>0. Prudence corresponds to the positivity of the third derivative of the utility function. If we take a DARA function, implicitly we assume prudence. (page 96)


Precautionary Premium• Precautionary motive: the uncertainty affecting future incomes introduces

a new motive for saving. The intuition is that it induces consumers to raise their wealth accumulation in order to forearm themselves to face future risk

• Let ψ denote the precautionary premium

• Two period model

• Optimal saving s under uncertainty of income flow y, i.e. labor income risk

)(')(' 1111 EywuywEu


An Example

• Lifetime utility is U(c0, c1)=u(c0)+u(c1)• Assuming E(y1)=y0

• If y1 is not risky. I.e., y1=y0

• Then u’(y0-s)=u’(y0+s), then s*=0

• If y1 is risky, FOC is:

)(')(')(' 010 syusEyEusyu

we have s*=1/2ψ. If the consumer is prudent, there will be a precautionary demand for saving, s*>0. An interesting case is that if the felicity function is quadratic, we have ψ=0. there is no precautionary saving motive.


Risky Saving and Precautionary Demand• Saving is no longer risk free now

• Let w0 denote the wealth, the consumer’s objective is:

))1(()()(max 100 srEuswusVs

where E(r)>0.

FOC for s* is: ])1((')1[()(' 0 srurEswu

If the bond is risk free and the rate stays constant over time, we have

])1((')1[()(' 0 srurswu

Under certain conditions, we can show the optimal saving s* is one there is no fluctuation in consumption between dates: c0=c1

Introducing uncertainty in risk free rate makes the bond less attractive to consumers.


Dynamic Investments

• An investor endowed with wealth w0 lives for two periods. He will observe his loss or gain on the risk he took in the first period before deciding how much risk to take in the second period

• How would the opportunity to take risk in the second period (Period 1) affect the investor’s decision in the first period (period 0)?– In other words, would dynamic investment attract more risk taking?

• To solve this problem, we apply backward induction. That is, to solve the second period maximization first taking the first period investment decision as given. To be specific

x

α0 Period 0 α1 Period 1

• Note: this is not the general form– A close look at the example finds that α1 is about consumption, not an asset allocation issue.


Backward Induction

• Assuming the first period payoff is z(α0, x)

• The second objective function is

• Then solve for the first period Ev(z(α0, x))

• Good examples of the backward induction application:– Froot, K. A., David S. Scharfstein, and J. Stein. "Risk Management:

Coordinating Corporate Investment and Financing Policies."

Journal of Finance 48, no. 5 (December 1993): 1629-1658. – Froot, K. A., and J. Stein. "Risk Management, Capital Budgeting and Capital

Structure Policy for Financial Institutions: An Integrated Approach." Journal of Financial Economics 47, no. 1 (January 1998): 55-82.

),(max)( 11

zUzv

http://www.afajof.org/jofihome.shtml


Two-Period Investment Decision• Assume the investor has a DARA utility function.

– The investor would take less risk in t+1 if he suffered heavy losses in date t

• The investor makes two decisions• In period 1, the investor invests is an AD portfolio decision,

• In period 0, the investor invests in risky portfolio (selecting α0), which decides z. He attempts to optimize his expected utility which contingent on period 1 allocation.

s

s 1

1

0,...,

)(max)(10

S

sss

cccupzv

S

subject to zcS

sss

1

0

Among the portfolio set (c0, c1, …, cS-1), c0 is the consumption at the end of period 1; (c1, …, cS-1) = α1 is the consumption set at the end of period 2.

1

10

101 )()(),(

S

sss

S

sss

cupcz

upzU ; v(z)= ),(max 11

zU

),(max)( 11

zUzv


Implicit Assumptions

• Investment decision is made only in period 0

• Only two periods

• No return in risk-free assets

• The key is to compare the investment in risky asset, α0, for this long term investors with that of a short-lived investor

• This is to compare the concavity of these two utility functions

)),((maxarg 0

^

00

xzEu


Solution

Proposition 7.1 The value function for the Arrow-Debreu portfolio problem has a degree of absolute risk tolerance given by

1

0

)()(

)()(

S

sssv cT

zv

zvzT

where c* is the optimal solution to problem (7.3) and T(.)=-u’(.)/u’’(.) is the absolute risk tolerance for final consumption. -- whether to take more risk in a dynamic investment setting is to compare Tv(z) and Tu(z), where

))(()(''

)(')(

1

0

S

sssu cT

zu

zuzT


So,• It states that the absolute risk tolerance of the value function is a weighted

average of the degree of risk tolerance of final consumption.• If u exhibits hyperbolic absolute risk aversion (HARA), that is T is linear

in c (see HL chapter 1 for discussions on HARA), then v has the same degree of concavity as u – the option to take risk in the future has no effect on the optimal exposure to risk today

• If u exhibits a convex absolute risk tolerance, i.e., T is a convex function of z, or say T’’>0, then investors invest more in risky assets in period 0. Opposite result holds for T’’<0

• Proposition 7.2: Suppose that the risk-free rate is zero. In the dynamic Arrow-Debreu portfolio problem with serially independent returns, a longer time horizon raises the optimal exposure to risk in the short term if the absolute risk tolerance T is convex. In the case of HARA, the time horizon has no effect on the optimal portfolio.

• If investors can take risks at any time, investors risk taking would not change if HARA holds.


Time Diversification

• What would there are multiple consumption dates?

• This is completely different setting from the previous one

• The setup the problem is as following:

1

0

)(max)(n

ttt

ccupzv subject to nyzc

n

tt

1

0

where pt is the discount factor associated to date t and z+ny is the lifetime wealth the solve the problem of maxα0Ev(z(α0,x))


Solution

1

0

)()(

)()(

n

ttv cT

zv

zvzT

In words, the degree of tolerance to the risk on initial wealth equals the sum of the absolute tolerance to risk on consumption over the lifetime of the consumer. Assuming the consumer is not impatient, so that pt=1 for all t. Then it is optimal to smooth consumption completely: ct*=y+(z/n). Then we have

))(()(n

zynTzTv

I.e., the absolute tolerance to risk on wealth is proportional to the lifetime of the gambler. This is so call Time Diversification.


Liquidity Constraint

• Time diversification relies on the condition that consumers smooth their consumption over their life time

• The incentive to smooth consumption would be weakened if consumers are faced with liquidity constraints

• Conservative

• How about other considerations regarding saving and consumption decisions listed in Chapter 6?


Dynamic Investment with Predictable Returns

• What if the investment opportunity is stochastic with some predictability• Two period (0, 1); two risk (x0, x1), where x1 is correlated with x0

• Investors invest only for the wealth at the end of period 1. i.e., there is no intermediate consumption

• E(x0)>0In the second period, we have the following expected utility function

0

1

1)0 |

1

)(max,( x

xzExzv

Given the property that under constant relative risk aversion, the demand for stocks is proportional to wealthThis can be written as

1)(,(

1

0)0z

xhxzv

where 0

1100 |))(1[()( xxxaExh


More on this Case

In the first period, the investor chooses α0, which affects z, to achieve an optimal expected utility:

1

)()()(maxarg

1

000

0

*

0

xwxhEH

Without predictability, we have

1

)(maxarg

1

00

0

0

xwEm

When returns are predictable, the hedging demand to take on more risks is m0

*0 .

The hedging demand is positive if the derivative of H evaluated at m0

*0 is positive. The condition

can be set as below:

0]))(([)(' 000000 xwxHxEH mm

whenever 0])([ 0000 xwxE m


Exercises

• Derive (6.14) on page 97

• EGS: 6.1; 6.4; 6.5

• EGS, 7.1; 7.3

Documents

L9: Consumption, Saving, and Investments 1 Lecture 9: Consumption, Saving, and Investments The following topics will be covered: –Consumption and Saving