1. L4: Consumption and Saving1 Lecture 4: Consumption and Saving The following topics will be covered: –Consumption and Saving under Certainty Aversion to consumption fluctuation over time Relative fluctuation aversion measure –Uncertainty and Precautionary Saving Precautionary premiums –Risky savings and Precautionary Demand –Time Consistency (Basically, this chapter is on how consumers make their consumption and investment decisions given a concave utility function)

2. L4: Consumption and Saving2 Consumption and Saving under Certainty An agent lives for a known number of periods Endowed with a flow of sure income y t Risk free interest rate r z t is the cash transferred from data t-1 to t, i.e., accumulated saving in t c t is the consumption in t The agent selects consumption plan c=(c 0, c 1, …, c n-1 ) to maxU(c 0, c 1, …, c n-1 ) Subject to the dynamic budget constraint: z t+1 =(1+r)[z t +y t -c t ] PV(z n )≥0. This can be rewritten as:

3. L4: Consumption and Saving3 More on Solutions Note: The optimal condition implies Fisher’s separation theorem 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, and services; car is durable goods, house is too. but CPI does not count housing price, only rental price

4. L4: Consumption and Saving4 Given Independence in Consumption

5. L4: Consumption and Saving5 Tendency to Smooth Consumptions If П t =1 for all t (i.e., r=0), then FOC: u’(c t )=ξ in each period The optimal consumption path does not exhibit any fluctuation in consumption from period to period: c t =w 0 /n Note: even revenue flow y t is known, they may not be stable over time. Thus borrowing and lending is required.

6. L4: Consumption and Saving6 Relative Fluctuation Aversion Measure If one is asked to sacrifice 1 unit of consumption today, he is demand more than one unit of consumption tomorrow as a compensation If there is no credit market: c t =y t Suppose y 0 ≤y 1 U’(y 0 )=(1+k)u’(y 1 ); k can be considered as the aversion to give up current consumption, i.e., the resistance to intertemporal substitutions A first-order Taylor expansion of u’(y 0 ) around y 1 gives rise to:

7. L4: Consumption and Saving7 Optimal Consumption Growth In general, the real interest rate is not zero and agents are impatient Assuming consumers use exponential discounting: P t =β t –β =(1+δ) -1 – multiplying u(c t ) by β t is equivalent to discounting felicity at a constant rate δ 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)=c 1-γ /(1-γ), where is the constant degree of fluctuation aversion. We have c t =c 0 a t. Where,

8. L4: Consumption and Saving8 More on Consumption Growth Rate The optimal growth rate of consumption is positive when r is large δ A greater γ reduces consumption growth Business cycles cause fluctuation in consumption and affect the growth rate But the impact is trival

9. L4: Consumption and Saving9 Income Uncertainty and Precautionary Saving Now y t is no longer certain Two period model to decide how much to save at date 0 in order to maximize their expected lifetime utility

10. L4: Consumption and Saving10 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

11. L4: Consumption and Saving11 An Example Lifetime utility is U(c 0, c 1 )=u(c 0 )+u(c 1 ) Assuming E(y 1 )=y 0 If y 1 is not risky. I.e., y 1 =y 0 Then u’(y 0 -s)=u’(y 0 +s), then s*=0 If y 1 is risky, FOC is:

12. L4: Consumption and Saving12 Risky Saving and Precautionary Demand Saving is no longer risk free now Let w0 denote the wealth, the consumer’s objective is:

13. L4: Consumption and Saving13 What if there are more than two dates? The issue is if the consumer will follow his consumption plan when date t occurs? This is called a time-consistency problem

14. L4: Consumption and Saving14 Exercises Derive (6.14) on page 97 EGS: 6.1; 6.4; 6.5