Topic 2: Consumption

Topic 2: Consumption Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 1 / 48

Reading and Lecture Plan Reading 1 SWJ Ch. 16 and Bernheim (1987) in NBER Macro Annual. Plan 1 Review of the Keynesian consumption function 2 Intertemporal Consumption Problem 3 Interest Rates 4 Government Policy and Ricardian Equivalence Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 2 / 48

Keynes Three Conjectures on the Consumption Function 1 Marginal propensity to consume is between zero and one. The impact of fiscal policy is determined from the feedback between income and consumption. 2 Average propensity to consume rises as income falls. Idea being that saving was a luxury and so rich people save more than poor. 3 Income is the primary determinant of consumption, not the interest rate. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 3 / 48

Keynesian Cross Planned and actual expenditures: E p C (Y, T, ω) + I p (r, A) + G }{{}}{{} (i) (ii) Y is total income, T is taxation, ω is household wealth, A is autonomous investment and G is government purchases. (i) The consumption function is based on household behavior. Consumption rises with income, so dc /dy > 0. (ii) Investment demand is based on firm behavior. Planned investment, I p, falls with the interest rate, so di p /dr < 0. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 4 / 48

Keynesian Cross Analysis In equilibrium, E p = Y. On a diagram, the equilibrium, in (E p, Y )-space, will be on the 45-degree line. Suppose there is a shock to the economy: r > 0. That implies E p < Y as I p has fallen. There is disequilibrium in the goods market; planned expenditure is less than total output. Over time, E p and Y fall, until we reach a new equilibrium (i.e. E p = Y ), where both E p and Y are lower than before. This can be represented as a shift in the planned expenditure line. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 5 / 48

Keynesian Cross Equilibrium: planned expenditure=spending (income): E p = Y. E p, Planned Expenditure Y = E p E p (r 0 ) E p (r 1 ) ΔI Y 1 Y 0 Y, Output Higher r, lowers I, which lowers E p s.t. E P 1 < Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 6 / 48

IS Curve: Mathematical Derivation of Slope The IS curve shows combinations of real output Y and the real interest rate r such that planned and actual expenditures are equal. Totally differentiating Y = E p w.r.t. Y and r (holding G, T, A, ω fixed) yields: Y = C Y Y + I r r or, ( Y ) /( r) IS = I r /(1 C Y ) < 0 This is the slope of the IS Curve, where 0 < C Y < 1 is the MPC and I r < 0. IS is steep if the interest sensitivity of planned expenditure (I r ) is high or the marginal propensity to consume out of disposable income (C Y ) is large. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 7 / 48

Dudley Cooke At(Trinity points College Dublin) where ETopic p 2: Consumption Y the goods market is 8 / 48 IS Curve Diagram: IS Curve r, Interest Rate r 1 E p <Y r 0 E p >Y Y 1 Y 0 IS Y, Output

Shifting the IS Curve The IS curve shifts with changes in fiscal policy. The standard multipliers: 1 Government purchases multiplier: 0 < C Y MPC < 1 so 1/(1 C Y ) > 1. Larger than one because G Y C by G C Y, which Y etc... 2 Tax multiplier: C Y /(1 C Y ) < 0. 3 Balanced budget multiplier (i.e. when G = T ): unity. 1 1 Since only part of the money taken away from households would have actually been used in the economy, the change in consumer expenditure will be smaller than the change in taxes. Therefore the money which would have been saved by households is instead injected into the economy, itself becoming part of the multiplier process. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 9 / 48

Functional Forms We usually assume C ( ) and I p ( ) are linear functions: C (Y, T, ω) = δy d + ω I p (r, A) = A ar where Y d Y T is disposable income. This gives clear implications for the slope of the IS and the multipliers. 1 The slope: ( Y ) /( r) IS = a/(1 δ) < 0 2 The multipliers: ( Y ) /( G ) r = 1/(1 δ) > 1 ( Y ) /( T ) r = δ/(1 δ) < 0 Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 10 / 48

Keynesian Consumption Function Some criticisms of the Keynesian Consumption Function (KCF): 1 C t = δ(y t T t ) + ω is OK. But what about C t s relation to Y t+1, Y t+2, etc.? Agents ought to be forward-looking; consumption today should account for changes in future income to maximize utility. 2 Why doesn t C t depend on r t in the same way investment does? And what about r t+1, r t+2, etc.? Again, agents ought to be forward-looking. 3 There is no differentiation between permanent and temporary changes in policy. Conclusion: We need a more developed model to deal with this. Given our criticisms, it ought to include agents who consider the future. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 11 / 48

Permanent Income and Life Cycle Hypotheses The Permanent Income Hypothesis, proposed by Milton Friedman, loosely states that households base consumption on average income levels, i.e. their permanent income and not merely year-to-year income. This implies that households need to know what their income is over a long time horizon. This leads to consumption smoothing. The Life Cycle Hypothesis, proposed by Franco Modigliani, suggests that households consume a constant percentage of the present value of lifetime income. The average propensity to consume is high in households that are young or old. Young households borrow against future income, while old households spend down lifetime savings. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 12 / 48

A Two-Period Endowment Economy Households are alive for two periods, t = 1, 2. tomorrow. There is a single good, Y t. Call these today and Households borrow and lend at the same market interest rate, r. (There are perfect capital markets.) There is no uncertainty about the future. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 13 / 48

Consumer preferences Household maximizes lifetime (t = 1, 2) utility, U. U = u(c 1 ) + βu(c 2 ) (1) β (0, 1) is a fixed subjective discount factor, sometimes called the private discount factor. It measures the household s impatience to consume. Lower β means agents are more impatient. C is consumption of the good. u is the period utility function, with u (C ) > 0 and u (C ) < 0. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 14 / 48

The Budget Constraint The following describes the household s lifetime budget constraint. C 1 + C 2 1 + r = (Y 1 1 T 1 ) + (Y 2 T 2 ) 1 + r where 1/ (1 + r) is the market discount factor for future consumption. Equation (??) tells us that the present value of lifetime consumption is equal to the present value of lifetime income, minus taxes. 2 2 We get here by combining the t = 1 and t = 2 period constraints. That is, W t = W t 1 (1 + r) = Y t C t, for t = 1, 2, with W 0 = W 2 = 0, assumed. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 15 / 48

The Household Utility Optimization Problem Maximize (1) subject to the budget constraint. s.t. max C 1 U = u(c 1 ) + βu(c 2 ) C 2 = (1 + r)(y 1 C 1 T 1 ) + (Y 2 T 2 ) Take derivatives to get the first order condition. The Consumption Euler equation. u (C 1 ) β(1 + r)u (C 2 ) = 0 βu (C 2 ) u (C 1 ) = 1 1 + r This tells us that the substitution of consumption across time (i.e., the intertemporal marginal rate of substitution) should be equal to the intertemporal relative price. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 16 / 48

Equilibrium Suppose taxes are zero. That is, let T 1 = T 2 = 0. Recall, the budget constraint is, Taking the derivative, we have, C 1 + C 2 1 + r = Y 1 + Y 2 1 + r C 2 / C 1 = (1 + r) Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 17 / 48

Relation to Microeconomics In micro, we solve the following problem: such that, maxu(c a, C b ) C a,c b P a C a + P b C b = I The relative price between the two goods (intratemporal price), P a /P b, determines the slope of the budget constraint. In our example, 1/ (1 + r) is the intertemporal price, that is, the relative price of consumption (of a single good) over time. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 18 / 48

Intertemporal Consumption Function (ICF) We make some functional form assumptions on utility, for the same reasons we cited when we looked at the Keynesian consumption function. Suppose: u(c t ) = C 1 1 σ t 1 1 ; σ > 0, σ = 1 σ = ln C t ; σ = 1 This is sometimes called power utility. The parameter σ determines the slope of the indifference curves and how consumption is substituted across time. Recall that r determines the slope of the budget constraint. High σ will be consistent with high substitution. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 19 / 48

Intertemporal Consumption Function The consumption Euler equation is now: C 2 = [β(1 + r)] σ C 1 Using the budget constraint, we get the consumption function: C 1 = (Y 1 T 1 ) + (Y 2 T 2 ) 1 1+r 1 + (1 + r) σ 1 β σ We ll call this the Intertemporal Consumption Function (ICF). We can compute C 2 by plugging the two equations together. Recall, the KCF: C t = δ(y t T t ) + ω ; for t = 1, 2, say Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 20 / 48

Comparing the KCF and ICF Important point: in the ICF, consumption depends on r. doesn t matter in the KCF. But r 1 Y 2 and T 2 also matter for C 1 in the ICF. 2 Expectations about the future matter for today s consumption, unlike in the KCF. This is more intuitive. 3 This has policy consequences. For example, there will be differences between permanent and temporary taxation policies. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 21 / 48

Comparing ICF and KCF Let s compare marginal propensities to consume in the Intertemporal and Keynesian consumption functions. Let 1/θ 1 + (1 + r) σ 1 β σ and Y t T t = Yt d. Consumption function: ( ) C 1 = θ Y1 d + Y2 d 1 1 + r where θ (0, 1) measures the MPC of wealth (here, disposable income in both periods) However, δ in the KCF measured the MPC out of current disposable income alone. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 22 / 48

Comparing ICF and KCF δ versus θ is not a good comparison of the consumption functions. 1 δ is MPC of current disposable income. 2 θ is MPC of wealth. Let (1 + g e ) Y d 2 /Y d 1. Then: C 1 = θy d 1 θ = θ [ 1 + ( )] 1 + g 1 + r Now θ is the MPC of current disposable income and is directly comparable to δ. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 23 / 48

Comparing ICF and KCF The KCF is: The ICF is: C 1 = δy d 1 + ω C 1 = θy 1 d [ ( )] 1 + g e 1 θ = 1 + 1 + r 1 + (1 + r) σ 1 β In essence, we have micro-founded δ. We have a solid idea of where θ comes from. It depends on preferences and: g e : expected future disposable income r : the market rate of return β : impatience to consume Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 24 / 48

Comparing ICF and KCF If Y d 1, from the KCF, C 1. The ICF suggests two effects, one implies a direct C 1 and the other implies g e C 1. Overall, C 1. Clearly, C 2 also changes. 3 This is an example of how intertemporal analysis lets us investigate subtle effects that may not be immediately intuitive. 3 Why? Consumption smoothing. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 25 / 48

Comparing ICF and KCF: Does ICF Perform Better? Our new model is more complicated. That is no good if it is not better than the old model. We need to check the data. One idea: KCF: Richer people have higher average savings than poor people - assumed. ( ) ICF: Suppose ghigh e Y2 d > Y 1 d, like students high MPC from ( ) current income, i.e. a high θ. Now suppose glow e Y2 d < Y 1 d, like pensioners low MPC. The models are consistent. However, ICF is better. It implies we have different marginal propensities to consume over different parts of the lifecycle. That is, θ changes over time whereas δ does not. This is far more empirically relevant. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 26 / 48

The Interest Rate The interest rate play a significant role in the intertemporal consumption function. A change in r changes the MPC of wealth, θ. θ/ r > 0 if σ < 1 θ/ r < 0 if σ > 1 This is a result of income, substitution and wealth effects. When σ = 1, these cancel out and do not affect the MPC of wealth, i.e. θ/ r = 0. However, even if σ = 1, θ/ r < 0. That is, higher interest rates lower the MPC of current disposable income. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 27 / 48

The Interest Rate: Changes To r Consider the effect of an r on C 1. 1 Substitution (negative) effect: we swap C 2 for C 1. We increase saving when the incentive to save increases. 2 Income (positive) effect: r allows C 2, for given income. This increase in feasible consumption implies C 1 (lower savings). Note the opposite effects. 1 When σ > 1, the substitution effect dominates the income effect. When σ < 1, things go the other way. 2 When σ = 1, the fraction of lifetime income spent on C 1 does not depend on r. The income and substitution effects are offsetting. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 28 / 48

The Interest Rate: Changes To r However, when σ = 1, r still affects C 1, via a wealth effect. Wealth (negative) effect: The change in r changes lifetime income, not just the fraction of lifetime income devoted to present consumption. That is, r lowers the present discounted value of income, reinforcing the substitution effect. Which direction is more plausible? Let s look at the data. It suggests σ = 2. effect dominates. We increase saving. That is, the substitution Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 29 / 48

The Interest Rate: Mathematical Details Quick references on income and substitution effects: (i) Romer 3rd ed. pp. 363-365 [macro] and (ii) Varian ch.8 [micro]. In our example, we can calculate the following (you can verify this at home): dc 1 dr = (Y 1 C 1 ) σc 2 1 1+r (1 + r) + C 2 /C 1 0 When σ = 1: C 2 = [β(1 + r)] σ C 1 C 1 = Y 1 + Y 2 1 1+r 1 + β dc 1 dr < 0 There is no ambiguity in dc 1 dr as income and substitution effects cancel out when σ = 1. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 30 / 48

Government Policy We have yet to consider policy options. change in taxation, T. In the KCF case, consider a ( Y ) /( T ) r = C Y /(1 C Y ) < 0. All else equal, an increase in taxation reduces output. Points we have missed out: 1 Does the timing of taxation matter? 2 Does the financing of a change in taxation matter? That is, does moving from a situation in which T = G to deficit financing make a difference? Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 31 / 48

Tax Policies: Temporary and Permanent Recall the basic ICF: C 1 = θ [ ] 1 (Y 1 T 1 ) + (Y 2 T 2 ) 1 + r A temporary policy here is a T 1 with T 2 = T, fixed. A permanent policy is T 1 = T 2 = T, T. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 32 / 48

Tax Policies: Temporary and Permanent Temporary Policy (T 1 = T 2 ): C 1 / T 1 = θ Permanent Policy (T 1 = T 2 = T ): ( C 1 / T = θ 1 + 1 ) 1 + r In the first case, T 1 C 1, but as θ < 1, C 2 also. Savings rise as households engage in consumption smoothing. In the second case, the effects are stronger. When β = (1 + r), consumers want perfectly smooth consumption over time C 1 / T = 1 Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 33 / 48

Tax Policies: Financing a Tax Cut The funds to finance the tax cut have to come from somewhere. To analyze this, we need to know the government s lifetime budget constraint, in the same way that we need to know the household s budget constraint. Government constraint: D 0 + G 1 + G 2 1 1 + r }{{} govt. consumption = T 1 + T 2 1 1 + r }{{} govt. income where D 0 is the initial (given) level of government debt. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 34 / 48

Constant Government Spending Suppose G 1 = G 2 = G. So government spending is constant. This has immediate implications: dt 1 = dt 2 1 1 + r If taxes fall today, they have to rise tomorrow. But surely consumers know this? So how will they react? Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 35 / 48

Implications of the Lifetime Budget Constraint If in t = 1, the government lowers T 1 without a change in G 1, then D 1 must rise, i.e. the government issues more debt. If in t = 2, the government does not subsequently change G 2, then it has to raise T 2 to pay for the principal and interest on the extra debt it bore. Returning to forward-looking households. What if you knew this was the case? (i.e. you took Economics) Well, you know that if T 1 without G 1 or G 2, this implies a rise in the present value of of total taxation by an amount equal and opposite to the T 1. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 36 / 48

Implications of the Lifetime Budget Constraint Formally: implies, [ ] 1 C 1 = θ (Y 1 T 1 ) + (Y 2 T 2 ) 1 + r [ ] 1 dc 1 = θ dt 1 + dt 2 = 0 1 + r So, T 1 without G 1 or G 2 must imply no change in consumption. That is, the financing of the deficit has no implication for consumption. We can choose either tax or deficit financing. The present value of the tax burden is the same. This is known as Ricardian Equivalence. 4 4 In contrast, recall that the balanced-budget multiplier in the KCF was non-zero. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 37 / 48

Ricardian Equivalence Ricardian Equivalence is a benchmark result. It breaks down if we change our model only slightly. In particular, consider any of the following alternative assumptions: 1 Governments and households live for a different no. of periods. 2 Non-lump-sum taxes. There are no associated distortions with lump-sum taxes, but they are very rare in real life. Distortionary taxation: Anything that affects markets, like VAT or income tax. 3 We don t borrow and lend at the same rate. Some people are credit constrained. 4 However, Barro (1974) provides an interesting analysis when we make bequests. 5 5 See his paper in the Journal of Political Economy on Government Bonds and Net Wealth. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 38 / 48

Ricardian Equivalence: Empirical Evidence Keynesian view: Deficit-financed tax cuts increase aggregate demand. Ricardian view: Taxpayers understand that the present discounted value of taxes depends on government spending. Tax cuts will have no effect on aggregate demand. Context: The mid 1980 s saw a high US government deficit. Main Finding: There is a short-run relationship between deficits and aggregate consumption, both in cross-country data and time series tests of the consumption function. So the strict Ricardian view doesn t hold. Problem: We may not really be seeing a test of Ricardian equivalence in this paper. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 39 / 48

What the data say, Bernheim (1987) Keynesian view: deficit financed tax cuts increase aggregate demand. Ricardian view: taxpayers understand that the p.d.v. of taxes simply depends on government spending and tax cuts will have no effect on aggregate demand. Context: The mid 1980 s saw a high US government deficit. Main Finding: There is a short-run relationship between deficits and aggregate consumption both in cross country data and times series tests of the consumption function. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 40 / 48

Testing the Consumption Function Two main types of tests, which are basically equivalent (old papers from 1970-1986). Test of tax discounting hypothesis: C t = α 0 + α 1 (Y t T t ) + α 2 G t + α 3 D t + α 4 W t + α }{{} 5 H 0 :α 5 =0 (T t G t R t D t ) + X t α + ε t }{{} govt. deficit Test of Pure Ricardian equivalence: C t = β 0 + β 1 Y t + β 2 G t + β 3 D t + β 4 W t + β 5 }{{} (T t G t R t D t ) + X t β + η t H 0 :β 5 =0 Above, H 0 is the null hypothesis. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 41 / 48

Interpretation Interpreting regression results: 1 α 5 = 0 (β 5 = β 1 ): tax discount hyp. of Ricaridan view. 2 α 5 = α 1 (β 5 = 0): pure Ricardian view. 3 α 1 α 5 (β 5 ): measures the effect on current consumption of a 1$ tax deficit swap. The estimation method is typically Ordinary Least Squares (OLS). Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 42 / 48

Problems If we take some data and run a regression are we actually testing what we want? We definitely test something that looks like an implication of our model. However, the conditions above look like KCF s. testing the KCF view. Thus we are really They are not exactly tests of Ricardian Equivalence as we have formulated it. Our formulation used a Consumption Euler equation. Thus if we run this regression we will get some biased estimates. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 43 / 48

Round-up Keynesian versus intertemporal consumption function Ricardian equivalence and tax cuts Confronting the ICF with data Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 44 / 48

Other Points 1 (not crucial - but FYI): the role of the elas y of substitution Everything comes down to 2 equations. Consumption Euler: βu (C 2 )/u (C 1 ) = 1/ (1 + r) and intertemporal budget constraint: C 1 (1 + r) + C 2 = Y 1 (1 + r) + Y 2. We then suppose u(c ) = C 1 σ 1 / ( 1 1 ) σ so, u (C ) = C σ 1. Without the functional form assumption, taking logs, ln β + ln [u (C 2 )] ln [u (C 1 )] = ln (1 + r). The [ total ] deriv. is, [ ] u (C 2 ) [u (C 2 )] 1 dc 2 u (C 1 ) [u (C 1 )] 1 dc 1 = d [ln (1 + r)], but noting dc 1 /C 1 = d [{ln [u ](C 2 )]}, and defining, σ (C ) [u (C )] /C u (C ), the former expression reduces to, ( ) C2 d ln = σd [ln (1 + r)] C 1 such that a high σ implies a strong reaction of relative consumption to changes in the real interest rate. That is, a high substitution effect, and gently sloped indifference curve. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 45 / 48

Other Points 2: Income, Substitution, and Wealth Effects Given the utility form, consumption is the following. [ 1 C 1 = 1 + (1 + r) σ 1 Y 1 + Y ] 2 β σ 1 + r C 2 = [β(1 + r)] σ C 1 If σ = 1 things simplify alot. Thus, we might suspect σ is critical. In determining how consumption reacts to changes in the interest rate. Income: positive; substitution: negative; wealth: negative. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 46 / 48

Income, Substitution, and Wealth Effects... Call R (1 + r) 1 and W Y 1 + Y 2 / (1 + r). Thus, C 1 = C 1 (R, W ), which is the Marshallian demand function, as it depends on wealth. NB: Hicksian demands depend on utility and show the pure substitiution effects because we are accounting for the income and wealth effects by holding utility constant. Obviously, dw /dr = Y 2 > 0. However, it is not too difficult to show the following result: dc 1 dr = (σ 1) C 1 (β/r) σ 1 + R 1 σ β }{{ σ + } incm and sub fx Y 2 1 + R 1 σ β σ } {{ } wealth fx Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 47 / 48

Income, Substitution, and Wealth Effects... Now we can why σ 1 is important. There is an alternate expression (similar to that derived in the main text) that also helps to clarify things: dc 1 dr = σ C 1 (β/r) σ 1 + R 1 σ β }{{ σ } sub fx + Y 2 C 2 1 + R 1 σ β σ }{{} incm vs. wealth fx The idea of (Y 2 C 2 ) 0 is important here (above it was Y 1 vs. C 1 ). Suppose r rises. and you are a second period borrower (i.e., a first period lender). If you are a lender, you get a utility gain. Thus, (Y 2 C 2 ) > 0 (that is, (Y 1 C 1 ) < 0) reinforces the sub. fx., that an increase in r reduces C 1. Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 48 / 48