Consumption ECON 30020: Intermediate Macroeconomics Prof. Eric Sims University of Notre Dame Fall 2016 1 / 36
Microeconomics of Macro We now move from the long run (decades and longer) to the medium run (several years) and short run (months up to several years) In long run, we did not explicitly model most economic decision-making just assumed rules (e.g. consume a constant fraction of income) Building blocks of the remainder of the course are decision rules of optimizing agents and a concept of equilibrium Will be studying optimal decision rules first Framework is dynamic but only two periods (t, the present, and t + 1, the future) Consider representative agents: one household and one firm Unrealistic but useful abstraction and can be motivated in world with heterogeneity through insurance markets 2 / 36
Consumption Consumption the largest expenditure category in GDP (60-70 percent) Study problem of representative household Household receives exogenous amount of income in periods t and t + 1 Must decide how to divide its income in t between consumption and saving/borrowing Everything real think about one good as fruit 3 / 36
Basics Representative household earns income of Y t and Y t+1. Future income known with certainty Consumes C t and C t+1 Begins life with no wealth, and can save S t = Y t C t (can be negative, which is borrowing) Earns/pays real interest rate r t on saving/borrowing Household a price-taker: takes r t as given Do not model a financial intermediary (i.e. bank), but assume existence of option to borrow/save through this intermediary 4 / 36
Budget Constraints Two flow budget constraints in each period: C t + S t Y t C t+1 + S t+1 S t Y t+1 + r t S t Saving vs. Savings: saving is a flow and savings is a stock. Saving is the change in the stock As written, S t and S t+1 are stocks In period t, no distinction between stock and flow because no initial stock S t+1 S t is flow saving in period t + 1; S t is the stock of savings household takes from t to t + 1, and S t+1 is the stock it takes from t + 1 to t + 2 r t S t : income earned on the stock of savings brought into t + 1 5 / 36
Terminal Condition and the IBC Household would not want S t+1 > 0. Why? There is no t + 2. Don t want to die with positive assets Household would like S t+1 < 0 die in debt. Lender would not allow that Hence, S t+1 = 0 is a terminal condition (sometimes no Ponzi ) Assume budget constraints hold with equality (otherwise leaving income on the table), and eliminate S t, leaving: C t + C t+1 1 + r t = Y t + Y t+1 1 + r t This is called the intertemporal budget constraint (IBC). Says that present discounted value of stream of consumption equals present discounted value of stream of income. 6 / 36
Preferences Household gets utility from how much it consumes Utility function: u(c t ). Maps consumption into utils Assume: u (C t ) > 0 (positive marginal utility) and u (C t ) < 0 (diminishing marginal utility) More is better, but at a decreasing rate Example utility function: u(c t ) = ln C t u (C t ) = 1 C t > 0 u (C t ) = C 2 t < 0 Utility is completely ordinal no meaning to magnitude of utility (it can be negative). Only useful to compare alternatives 7 / 36
Lifetime Utility Lifetime utility is a weighted sum of utility from period t and t + 1 consumption: U = u(c t ) + βu(c t+1 ) 0 < β < 1 is the discount factor it is a measure of how impatient the household is. 8 / 36
Household Problem Technically, household chooses C t and S t in first period. This effectively determines C t+1 Think instead about choosing C t and C t+1 in period t max U = u(c t ) + βu(c t+1 ) C t,c t+1 s.t. C t + C t+1 1 + r t = Y t + Y t+1 1 + r t 9 / 36
Euler Equation First order optimality condition is famous in economics the Euler equation (pronounced oiler ) u (C t ) = β(1 + r t )u (C t+1 ) Intuition and example with log utility Necessary but not sufficient for optimality Doesn t determine level of consumption. To do that need to combine with IBC 10 / 36
Indifference Curve Think of C t and C t+1 as different goods (different in time dimension) Indifference curve: combinations of C t and C t+1 yielding fixed overall level of lifetime utility Different indifference curve for each different level of lifetime utility. Direction of increasing preference is northeast Slope of indifference curve at a point is the negative ratio of marginal utilities: slope = u (C t ) βu (C t+1 ) Given assumption of u ( ) < 0, steep near origin and flat away from it 11 / 36
Budget Line Graphical representation of IBC Shows combinations of C t and C t+1 consistent with IBC holding, given Y t, Y t+1, and r t Points inside budget line: do not exhaust resources Points outside budget line: infeasible By construction, must pass through point C t = Y t and C t+1 = Y t+1 ( endowment point ) Slope of budget line is negative gross real interest rate: slope = (1 + r t ) 12 / 36
Optimality Graphically Objective is to choose a consumption bundle on highest possible indifference curve At this point, indifference curve and budget line are tangent (which is same condition as Euler equation) CC tt+1 (1 + rr tt )YY tt + YY tt+1 CC 2,tt+1 (2) YY tt+1 CC 3,tt+1 CC 0,tt+1 (3) UU = UU 2 CC 1,tt+1 (0) (1) UU = UU 0 UU = UU 1 CC tt YY tt CC 0,tt CC 3,tt CC 2,tt CC 1,tt YY tt + YY tt+1 1 + rr tt 13 / 36
Consumption Function What we want is a decision rule that determines C t as a function of things which the household takes as given Y t, Y t+1, and r t Consumption function: C t = C d (Y t, Y t+1, r t ) Can use indifference curve - budget line diagram to qualitatively figure out how changes in Y t, Y t+1, and r t affect C t 14 / 36
Increases in Y t and Y t+1 An increase in Y t or Y t+1 causes the budget line to shift out horizontally to the right In new optimum, household will locate on a higher indifference curve with higher C t and C t+1 Important result: wants to increase consumption in both periods when income increases in either period Wants its consumption to be smooth relative to its income Achieves smoothing its consumption relative to income by adjusting saving behavior: increases S t when Y t goes up, reduces S t when Y t+1 goes up Can conclude that C d Y t > 0 and C d Y t+1 > 0 Further, C d Y t < 1. Call this the marginal propensity to consume, MPC 15 / 36
Increase in r t A little trickier Causes budget line to become steeper, pivoting through endowment point Competing income and substitution effects: Substitution effect: how would consumption bundle change when r t increases and income is adjusted so that household would locate on unchanged indifference curve? Income effect: how does change in r t allow household to locate on a higher/lower indifference curve? Substitution effect always to reduce C t, increase S t Income effect depends on whether initially a borrower (C t > Y t, income effect to reduce C t ) or saver (C t < Y t, income effect to increase C t ) 16 / 36
Borrower CC tt+1 Hypothetical bundle with new rr tt on same indifference curve YY tt+1 h CC 0,tt+1 CC 1,tt+1 CC 0,tt+1 Original bundle New bundle CC tt YY tt CC 1,tt h CC 0,tt CC 0,tt Sub effect: C t. Income effect: C t Total effect: C t 17 / 36
Saver CC tt+1 New bundle CC 1,tt+1 h CC 0,tt+1 Hypothetical bundle with new rr tt on same indifference curve Original bundle CC 0,tt+1 YY tt+1 h CC 0,tt CC 0,tt YY tt CC tt CC 1,tt Sub effect: C t. Income effect: C t Total effect: ambiguous 18 / 36
The Consumption Function We will assume that the substitution effect always dominates for the interest rate Qualitative consumption function (with signs of partial derivatives) C t = C (Y t, Y t+1, r t ). + Technically, partial derivative itself is a function However, we will mostly treat the partial with respect to first argument as a parameter we call the MPC + 19 / 36
Algebraic Example with Log Utility Suppose u(c t ) = ln C t Euler equation is: C t+1 = β(1 + r t )C t Consumption function is: C t = 1 [ Y t + Y ] t+1 1 + β 1 + r t 1 MPC: 1+β. Go through other partials 20 / 36
Permanent Income Hypothesis (PIH) Our analysis consistent with Friedman (1957) and the PIH Consumption ought to be a function of permanent income Permanent income: present value of lifetime income Special case: r t = 0 and β = 1: consumption equal to average lifetime income Implications: 1. Consumption forward-looking. Consumption should not react to changes in income that were predictable in the past 2. MPC less than 1 3. Longer you live, the lower is the MPC Important empirical implications for econometric practice of the day. Regression of C t on Y t will not identify MPC (which is relevant for things like fiscal multiplier) if in historical data changes in Y t are persistent 21 / 36
Applications and Extensions We will consider several applications / extensions: 1. Wealth 2. Permanent vs. transitory changes in income 3. Uncertainty 4. Random walk 22 / 36
Wealth Allow household to begin life with stock of wealth H t 1. Real price of this asset in t is Q t Household can accumulate more of this asset or sell it Period t constraint: C t + S t + Q t (H t H t 1 ) Y t Period t + 1 constraint: C t+1 + S t+1 + Q t+1 (H t+1 H t ) Y t + (1 + r t )S t Imposing terminal conditions, IBC is: C t + C t+1 1 + r t + Q t H t = Y t + Y t+1 1 + r t + Q t H t 1 + Q t+1h t 1 + r t 23 / 36
Simplifying Assumptions and the Consumption Function First, assume household must choose H t = 0. It simply sells off the asset in period t at price Q t : C t + C t+1 1 + r t = Y t + Y t+1 1 + r t + Q t H t 1 Increase in Q t then functions just like increase in Y t C t = C d (Y t, Y t+1, r t, Q t + + + ) Empirical application: stock market boom of 1990s (increase in Q t ) 24 / 36
Alternative Simplifying Assumption Assume H t 1 = 0, and assume that household must purchase an exogenous amount of the asset, H t (e.g. has to buy a house) IBC: C t + C t+1 = Y t + Y ( ) t+1 Qt+1 + H t Q t 1 + r t 1 + r t 1 + r t Increase in Q t+1 : functions like increase in Y t+1 : C t = C d (Y t, Y t+1, r t, Q t, Q t+1) + + + Empirical applications: house price boom of 2000s (anticipated increase in Q t+1 ) 25 / 36
Permanent vs. Transitory Changes in Income Go back to standard consumption function: C t = C d (Y t, Y t+1, r t ) Take total derivative (differs from partial derivative in allowing everything to change): dc t = C d ( ) dy t + C d ( ) dy t+1 + C d ( ) dr t Y t Y t+1 r t If just dy t = 0, then dc t dy t is equal to partial C d ( ) Y t But if changes in income are persistent (i.e. dy t > 0 dy t+1 > 0), then dc t dy t > C d ( ) Y t Implication: consumption reacts more to a change in income the more persistent is that change in income 26 / 36
Application: Tax Cuts Suppose household pays taxes, T t and T t+1, to government each period, so net income is Y t T t and Y t+1 T t+1. Consumption function is: C t = C d (Y t T t, Y t+1 T t+1, r t ) A cut in taxes is equivalent to an increase in income Implication: tax cuts will have bigger stimulative effects on consumption the more persistent the tax cuts are Empirical studies: Shaprio and Slemrod (2003) and Shapiro and Slemrod (2009) Initial installment of Bush tax cuts in 2001 was close to permanent (ten years). Theory predicts consumption ought to react a lot. It didn t. Tax rebates 2008: known to be only one time. Theory predicts consumption should react comparatively little. It did. 27 / 36
Uncertainty Suppose that future income is not known with certainty Say it can take on two values: Yt+1 h with probability p, and with probability 1 p. Expected future income is: Y l t+1 E (Y t+1 ) = py h t+1 + (1 p)y l t+1 Household will want to maximize expected lifetime utility. Utility from future consumption is not known, because future consumption isn t known given uncertainty about income Euler equation looks the same, but has an expectation operator: u (C t ) = β(1 + r t )E [ u (C t+1 ) ] Key insight: expected value of a non-linear function is not equal to the function of the expected value. 28 / 36
Expected Marginal Utility vs. Marginal Utility of Expected Consumption Assume u ( ) > 0. Then E [u (C t+1 )] > u (E [C t+1 ]) uu (CC tt+1 ) uu ll (CC tt+1 ) EE[uu (CC tt+1 )] uu (EE[CC tt+1 ]) uu (CC h tt+1 ) ll CC tt+1 EE[CC tt+1 ] h CC tt+1 CC tt+1 29 / 36
Increase in Uncertainty: Mean-Preserving Spread Raises E [u (C t+1 )] uu (CC tt+1 ) uu ll (CC 1,tt+1 ) uncertainty EE[uu (CC tt+1 )] uu ll (CC 0,tt+1 ) EE[uu (CC 1,tt+1 )] EE[uu (CC 0,tt+1 )] uu (EE[CC tt+1 ]) uu h (CC 0,tt+1 ) uu h (CC 1,tt+1 ) ll ll h h CC 1,tt+1 CC 0,tt+1 EE[CC tt+1 ] CC 0,tt+1 CC 1,tt+1 CC tt+1 30 / 36
Precautionary Saving Increase in uncertainty raises expected marginal utility of consumption E [u (C t+1 )] For the Euler equation to hold, need to adjust current consumption to raise current marginal utility, u (C t ) This requires increasing saving consume less holding current income fixed Intuition: bad state of the world hurts you more than the good state helps you, so you adjust behavior in present to effectively self-insure against bad future state Empirical application: high uncertainty and weak consumption during and in wake of Great Recession 31 / 36
Random Walk Hypothesis Suppose that β(1 + r) = 1 Euler equation is then implies u (C t ) = E [u (C t+1 )] Suppose that u ( ) = 0 (so no precautionary saving). Then this implies that E [C t+1 ] = C t In expectation, future consumption ought to equal current consumption. This is the random walk hypothesis Doesn t mean that future consumption always equals current consumption But it does imply future changes in consumption ought to not be predictable, because in expectation future consumption should equal current consumption Random walk model due to Hall (1978) 32 / 36
Empirical Tests Random walk hypothesis one of the most tested macroeconomic theories Generally fails: Parker (1999): exploits facts about social security withholding and predictable changes over course of year. Consumption reacts to predictable changes in take home pay Evans and Moore (2012): look at relationship between receipt of paycheck (which is predictable) and within-month mortality cycle 33 / 36
Borrowing Constraints Empirical failures can potentially be accounted for by borrowing constraints Simplest form of a borrowing constraint: you can t. S t 0. Introduces kink into budget line CC tt+1 (1 + rr tt )YY tt + YY tt+1 YY tt+1 Infeasible if SS tt 0 CC tt YY tt YY tt + YY tt+1 1 + rr tt 34 / 36
Binding Borrowing Constraint If borrowing constraint binds you locate at the kink in the budget line (i.e. Euler equation does not hold) CC tt+1 CC 0,tt+1 = YY 0,tt+1 CC 0,dd,tt+1 CC tt CC 0,tt = YY 0,tt CC 0,dd,tt 35 / 36
Implications of a Binding Borrowing Constraint Current consumption equals current income Means that if household gets more income, will spend all of it Further means that if household expects more income in future, can t adjust consumption until the future consumption will react to anticipated change in income Potential resolution of some empirical failures of random walk / PIH model Also has policy implications. Makes sense to target taxes/transfers to households likely to be borrowing constrained if objective is to stimulate consumption 36 / 36