Lecture 2: Consumption

Transcription

1 Lecture 2: Consumption Professor Eric Sims University of Notre Dame Fall 2009 Sims (Notre Dame) Consumption Fall / 45

2 Environment Assume there exists a representative household who lives for two periods the present and the future This is an endowment economy we abstract from production for now This means that household income is exogenous Helpful to think about income and consumption as being in the same units Call these units fruit Households derive utility from their consumption of fruit For now, partial equilibrium Sims (Notre Dame) Consumption Fall / 45

3 Utility Utility is how much happiness the household derives from its consumption The utility function is a function which maps the amount of consumption into the amount of utility ( happiness ). Denote this function u(c). c denotes consumption in the present, and c 0 denotes future consumption (similar notation for income) Utility function is the same in either period Sims (Notre Dame) Consumption Fall / 45

4 Properties of Utility Increasing: marginal utility ( rst derivative) is positive More is better Concave: second deriviative is negative Utility is increasing at a decreasing rate Diminishing returns: rst unit of fruit increase utility by more than the second unit of fruit Draw example utility function and marginal utility schedule Sims (Notre Dame) Consumption Fall / 45

5 Example Utility Functions Log: Linear: General iso-elastic form: u(c) = ln(c) u(c) = c σ u(c) = c1 1 σ Log and linear special cases of general form: σ = 0 is linear, σ = 1 is log (use L Hopital s rule to show that) Sims (Notre Dame) Consumption Fall / 45

6 Marginal Utility First derivative of utility function Basic rules for derivatives Sims (Notre Dame) Consumption Fall / 45

7 Lifetime Utility Lifetime utility is equal to a weighted sum of utilities in both periods of life U = u(c) + βu(c 0 ) β 1 and is called the discount factor People put less weight on future utility. Why? The objective of the household is to maximize its lifetime utility to make itself as happy as possible Can t just consume as much as it wants consumption decisions are constrained by income (how much fruit falls from the tree) Sims (Notre Dame) Consumption Fall / 45

8 Endowments and Prices Let y and y 0 denote how much fruit falls from the household s tree in the present and future, respectively Households can buy other households fruit or sell their own fruit to other households at prices, p and p 0. Households are price-takers Households also have a savings vehicle risk-free nominal bonds, which we denote by b. Bonds are IOUs which pay an interest rate, i Sims (Notre Dame) Consumption Fall / 45

9 Within Period Budget Constraints In the rst period, dollar value of consumption plus saving must equal dollar value of endowment (i.e. income): In period 1, dollar value of consumption must equal dollar value of endowment plus accrued interest: Why is there no bond-holding in the future? Sims (Notre Dame) Consumption Fall / 45

10 Intertemporal Budget Constraint What links the two within period budget constraints? Nominal saving Solve for b from either within period constraint and plug into the other. Show derivation. Simpli cation yields: c + c i p 0 1 = y + y 0 p 1 + i p 0 p Sims (Notre Dame) Consumption Fall / 45

11 Real Rate of Return i is the nominal rate of return: it tells you how many dollars you get next period if you save $1 today r is the real rate of return: it tells you how many fruits you get tomorrow if you save one fruit today What is the real rate of return? 1 + r = (1 + i) p p 0 r i π r is the intertemporal price of consumption: summarizes the tradeo between present and future Sims (Notre Dame) Consumption Fall / 45

12 Real Intertemporal Budget Constraint Plug the Fisher equation into the intertemporal budget constraint: c + What does this say in words? c0 1 + r = y + y r The present discounted stream of consumption is equal to the present discount stream of income (endowment) Present discounted value: the amount today that makes you indi erent between that amount and some future amount Sims (Notre Dame) Consumption Fall / 45

13 Household Problem The household s objective is to maximize its lifetime utility subject to the real intertemporal budget constraint: max U = u(c) + βu(c 0 ) c + c0 1 + r s.t. = y + y r We can characterize the solution to the household s problem graphically using an indi erence curve/budget line diagram Sims (Notre Dame) Consumption Fall / 45

14 Indi erence Curve What is an indi erence curve? It s a plot showing di erent consumption bundles which hold total utility constant In intertemporal setting, just think of consumption in each period as di erent goods What is the slope of an indi erence curve? This has a special name the marginal rate of substitution (MRS) Sims (Notre Dame) Consumption Fall / 45

15 Graphical Representation Graph sample indi erence curves Why is the curve steep near the origin and atter as we move further to the right? Sims (Notre Dame) Consumption Fall / 45

16 Budget Line Budget line: a line showing the feasible set of consumption bundles, taking the endowment and interest rate as given Derivation Connect these with a straight line and that s the budget line. is the slope? (1 + r) Why? What Sims (Notre Dame) Consumption Fall / 45

17 Budget Line Graphically Graph budget line What does budget line show us? It shows us the attainable set of consumption bundles. All points on or inside the line are feasible. All points beyond the line are not Budget line must pass through the endowment point it is always feasible to simply consume the endowment and save or borrow at all It is helpful to show this when graphing budget lines Sims (Notre Dame) Consumption Fall / 45

18 Optimal Consumption Bundle The household optimizes where it achieves the highest utility level subject to the budget constraint This occurs where the indi erence curve just touches the budget line i.e. where the two curves are tangent (slopes are equal) Show graphically Mathematically, we know the two slopes. Optimality condition (or tangency condition or Euler equation ) is: MRS = u0 (c) βu 0 (c 0 ) = 1 + r Sims (Notre Dame) Consumption Fall / 45

19 Comparative Statics Exogenous variables: y, y 0, and r (for now). Endogenous variables: c and c 0 β is a parameter Change the exogenous variables, and see how the values of the endogenous variables change Show graphically Sims (Notre Dame) Consumption Fall / 45

20 Discussion Increasing either current or future income leads to higher consumption both today and in the future This is important current consumption depends not only on current income, but on expectations about future income An increase in the interest rate increases future consumption, has an ambiguous e ect on current consumption Substitution e ect Income e ect Sims (Notre Dame) Consumption Fall / 45

21 Algebraic Discussion Indi erence curve/budget line diagram is useful, but it is di cult to say much speci c We can algebraically solve the household problem using a little bit of calculus. Assume the within period utility is log, and that the discount factor is 1. The household problem is then: max U = ln(c) + ln(c 0 ) c + c0 1 + r How do we solve this problem? s.t. = y + y r Sims (Notre Dame) Consumption Fall / 45

22 The Consumption Function The Euler equation/tangency condition is a condition characterizing the optimal consumption pro le But we would like to derive a consumption function optimal consumption in each period as a function of things the household takes as given How do we do this? Take the tangency condition, solve for either c or c 0, plug back into the budget constraint, and simplify Show derivation Sims (Notre Dame) Consumption Fall / 45

23 Savings Sometimes it is useful to think about a savings function (Real) Savings is just de ned as current income minus current consumption: s = y c Savings is increasing in current income, decreasing in future income, and ambiguously related to the real interest rate, but probably increasing in r Sims (Notre Dame) Consumption Fall / 45

24 Marginal Propensity to Consume The marginal propensity to consume (MPC) is the rst derivative of the consumption function with respect to current income it tells us how much consumption today responds to an increase in income today, holding everything else xed For the example utility function, we see that the MPC is 1 2. MPC is always going to be bound between 0 and 1 The This means that people like to smooth consumption out over time an increase in current income causes people to increase both current consumption and future consumption (savings) The MPC is not a deep parameter it doesn t show up in the utility maximization problem. Rather, it depends on the form of the utility function, how long households live, how households discount the future, etc. Sims (Notre Dame) Consumption Fall / 45

25 Application: Permanent and Transitory Tax Cuts Let s introduce a government to this framework Each period, the government con scates some of the household s fruit. Denote these taxes by T and T 0 For now, assume that the government is completely wasteful it does nothing with the fruit it con scates, and simply allows it to spoil The only change to the household s problem is the intertemporal budget constraint, which now re ects the net endowment Sims (Notre Dame) Consumption Fall / 45

26 Application: Permanent and Transitory Tax Cuts Consider two stimulus plans: one cuts taxes today, with no expected cut in taxes tomorrow. The other cuts taxes in both periods. Which has the bigger e ect on desired household consumption? Why? Is a policy of targeted, timely, and temporary tax cuts a good one to stimulate? No, but there are caveats We ll come back to the targeted part Sims (Notre Dame) Consumption Fall / 45

27 Application: Ricardian Equivalence Instead of assuming that the government does nothing with the fruit it con scates, now assume that the government consumes it (G and G 0 ) Household IBC is the same Now the government has an intertemporal budget constraint of the same form: G + G r = T + T r Sims (Notre Dame) Consumption Fall / 45

28 Application: Ricardian Equivalence Household IBC can be simpli ed using the government budget constraint (show): What is missing? Taxes. Taxes dissappear from the household budget constraint Ricardian equivalence: the timing of taxes does not matter for household decisions. Only the time path of government purchases does What happens if government temporarily cuts current taxes through de cit nance? Consumers don t adjust their consumption at all. Since disposable income is higher, they just save the whole tax cut They do this because they know they ll face higher taxes in the future Sims (Notre Dame) Consumption Fall / 45

29 Permanent Income/Life Cycle Hypothesis Assume now that households live for a number of periods, not just two Household dies at some known date, works for a known number of periods with a known income stream until retirement Denote time periods by t, t + 1, t + 2,... etc. Sims (Notre Dame) Consumption Fall / 45

30 The Generalized Intertemporal Budget Constraint IBC: PDV of consumption equals PDV of income. expanded to any number of periods: Summation notation (explain and show): This can be Sims (Notre Dame) Consumption Fall / 45

31 The Permanent Income Hypothesis There will be an Euler equation / MRS = price ratio / tangency condition for each two adjacent periods of time Simplifying assumptions: β(1 + r) = 1 ( subjective rate of time preference = objective intertemporal price ). This implies perfect consumption smoothing Log utility tangency condition: c t+1 c t = β(1 + r) c t+1 c t = 1 ) c t+1 = c t = c This means that consumption is constant and the same in each period of life Sims (Notre Dame) Consumption Fall / 45

32 Solving for Consumption Make these substitutions into the IBC to solve for the amount of consumption in each period of life: Loosely speaking, consumption in each period is equal to the average income over life Basic result: Households save during years of work, draw down savings during retirement, fully depleting savings at death Sims (Notre Dame) Consumption Fall / 45

33 Picture: Life-Cycle Model Draw in consumption/income pro le Also do it for more realistic scenario where income pro le is growing over time How does saving change throughout the life cycle? If income pro le is su ciently steep, you borrow during early years of work, save during late years of work Sims (Notre Dame) Consumption Fall / 45

34 Permanent vs. Transitory Income Milton Friedman (1957): current income comprised of permanent and transitory components: Y = Y P + Y T Permanent income: the annuity value of wealth What does that mean? Loosely, average income. With non-zero interest rate, take a stream of income and gure out the constant payout during each period that depletes the stream by the end of life Friedman: c = Y P. Consumption equal to permanent income. Transitory uctuations in income are saved Sims (Notre Dame) Consumption Fall / 45

35 Permanent vs. Transitory Income Simple example: go back to life-cycle setup Suppose income goes up in just one period of life. this have on average income? What e ect does Now suppose income goes up by the same amount both in the current period and in every future period. What happens to consumption? MPC out of permanent income should be close to 1, MPC out of transitory income should be close to zero Sims (Notre Dame) Consumption Fall / 45

36 Precautionary Saving What is the impact of uncertainty on optimal household behavior? MRS: equate ratio of expected marginal utilities of consumption to one plus interest rate Discussion of expected value In general, expected marginal utility is greater than marginal utility of expected future consumption Bad state of the world hurts consumers more than the good state helps them, so they save for the bad state. More uncertainty more savings, less consumption Is this empirically relevant? Sims (Notre Dame) Consumption Fall / 45

37 Random Walk Hypothesis (Hall (1978)) Assumes quadratic utility, so uncertainty has no e ect on consumption Basic idea: changes in consumption should be unpredictable Why? Consumption is forward-looking, and is a su cient statistic for all information about future wealth Example: you know that your salary is going to go up a year from now. Should your consumption in a year move when the salary goes up? No, consumption should go up today when you get the news Best predictor of future consumption is current consumption: c 0 = c + ε 0 ε 0 represents news about lifetime wealth revealed in the future and unknown in the present. Its mean is zero. Sims (Notre Dame) Consumption Fall / 45

38 Summary of Predictions of Model of Consumption Consumption depends on current and future income E ect of changes in interest rate on consumption is technically ambiguous, probably negative Consumption should respond little to temporary changes in income, and roughly one for one with permanent changes in income People should save during their working years Consumption should not fall at retirement Changes in consumption should be unpredictable Sims (Notre Dame) Consumption Fall / 45

39 What Does the Evidence Say? A theory is only good if it can help explain the data How well does our theory of consumption explain the data? Sims (Notre Dame) Consumption Fall / 45

40 Evidence: Tax Cuts Shapiro and Slemrod (2003): 2001 tax rebates of $300 or $600. First installment of a 10 year reduction, so more or less permanent Few people planned to spend it. MPC should be close to 1 if it s permanent Ricardian motivations? Maybe they didn t believe it to be permanent? Shapiro and Slemrod (2008): 2008 tax rebates as part of stimulus plan. Most people planned to save the rebates or pay down debt (e ectively saving) Tax rebates were clearly transitory, so this is in accord with our theory Sims (Notre Dame) Consumption Fall / 45

41 Evidence: Retirement Fact: consumption of food drops at retirement (also during unemployment spells) Shouldn t happen in our model unless retirment is unexpected How do we measure consumption? By expenditure But what happens when people retire (or become unemployed)? They have more time more to shop for bargains, more time to prepare food at home Hurst and Aguilar (2005): consumption of food measured by calories does not change at retirement or drop by much during unemployment Sims (Notre Dame) Consumption Fall / 45

42 Evidence: Anticipated Tax Changes Parker (1999): looks at consumption responses to predictable changes in take home pay from Social Security taxes Only pay social security taxes on rst $50,000 or so of income at the time, now $108,600 Once yearly income has hit threshold, take home pay goes up by about 7 percent (now 7.65%) Consumption should not change when take home pay does go up But it does Sims (Notre Dame) Consumption Fall / 45

43 Predictable Changes in Income Shea (1995): unionized workers know something about the future time path of their earnings Pre-speci ed wage increases, etc. Consumption should not respond to this it should be set at the moment workers get knowledge of future time path of earnings But consumption does respond Consumption is excessively sensitive to predictable changes in income Sims (Notre Dame) Consumption Fall / 45

44 Reconciling Theory and Evidence Some instances where PIH seems to t the data; others where it fails How can we reconcile theory and evidence? Possibility 1: liquidity constraints e.g. consumers are restricted from borrowing, so c y (or more generally there is a limit on how much they can borrow) Draw in indi erence curve/budget line With liquidity constraints, consumption will be more tied to current income Application: transitory and targeted tax cuts Sims (Notre Dame) Consumption Fall / 45

45 Rule of Thumb Consumers Campbell and Mankiw (1989): propose idea of rule of thumb consumers These consumers just set consumption equal to current income (or equal to some fraction of income) Why? Liquidity constraints. Costs of optimization Utility cost of failing to optimize is pretty small; so this makes sense Evidence suggests that there is little or no excess sensitive to large predictable changes in income; gives some credence to this story Sims (Notre Dame) Consumption Fall / 45