Lecture 9: 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. L9: Consumption, Saving, and Investments

2. 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: L9: Consumption, Saving, and Investments

3. 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 L9: Consumption, Saving, and Investments

4. Independence in Consumption L9: Consumption, Saving, and Investments

5. Objective Function Again L9: Consumption, Saving, and Investments

6. 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. L9: Consumption, Saving, and Investments

7. 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, L9: Consumption, Saving, and Investments

8. 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 L9: Consumption, Saving, and Investments

9. 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 L9: Consumption, Saving, and Investments

10. 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: L9: Consumption, Saving, and Investments

11. Risky Saving and Precautionary Demand • Saving is no longer risk free now • Let w0 denote the wealth, the consumer’s objective is: L9: Consumption, Saving, and Investments

12. 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. L9: Consumption, Saving, and Investments

13. 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. L9: Consumption, Saving, and Investments

14. 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. L9: Consumption, Saving, and Investments

15. 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 L9: Consumption, Saving, and Investments

16. Solution L9: Consumption, Saving, and Investments

17. 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. L9: Consumption, Saving, and Investments

18. 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: L9: Consumption, Saving, and Investments

19. Solution L9: Consumption, Saving, and Investments

20. 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? L9: Consumption, Saving, and Investments

21. 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)>0 L9: Consumption, Saving, and Investments

22. More on this Case L9: Consumption, Saving, and Investments

23. Exercises • Derive (6.14) on page 97 • EGS: 6.1; 6.4; 6.5 • EGS, 7.1; 7.3 L9: Consumption, Saving, and Investments