Macroeconomics: Fluctuations and Growth Francesco Franco 1 1 Nova School of Business and Economics Fluctuations and Growth, 2011 Francesco Franco Macroeconomics: Fluctuations and Growth 1/54
Introduction Keynesian consumption function in which consumption depends on current income The marginal propensity to consume provides the crucial ingredient of the multiplier Microfoundation of behavior lead to PIH (Friedman) and LCH (Modigliani) Francesco Franco Macroeconomics: Fluctuations and Growth 2/54
Introduction What is consumption? Largest component of expenditure Consider first a Keynesian consumption function: C t = c 0 + cy d. aggregate consumption and personal income should be equally volatile no difference between the effect on consumption of temporary changes in personal income and permanent changes Francesco Franco Macroeconomics: Fluctuations and Growth 3/54
U = u(c 1 )+bu(c 2 ) u 0 (x) > 0, u 00 (x) < 0 u is the istantaneous felicity function, b is the discount factor Francesco Franco Macroeconomics: Fluctuations and Growth 4/54
The household has initial wealth A 1 > 0andexogenouslabor income Y 1 > 0, Y 2 > 0. perfect foresight: prediction are driven by its intrinsic logic The household can save or borrow at an exogenous constant interest rate r Francesco Franco Macroeconomics: Fluctuations and Growth 5/54
The dynamic budget constraint (DBC) for the first period and the second period are respectively: A 2 = (1 + r)(a 1 + Y 1 C 1 ) C 2 = A 2 + Y 2 A 1 given A 3 = 0 Francesco Franco Macroeconomics: Fluctuations and Growth 6/54
The intertemporal budget constraint (IBC) is: C 1 + C 2 1 + r = A 1 + Y 1 + Y 2 1 + r. We can write the IBC with an equality (why?) and the Lagrangian for her maximization problem is: L = u(c 1 )+bu(c 2 )+l A 1 + Y 1 + Y 2 C 2 C 1 1 + r 1 + r. Francesco Franco Macroeconomics: Fluctuations and Growth 7/54
let me rewrite Max L = u(c 1 )+bu(c 2 )+l A 1 + Y 1 + Y 2 1 + r C 1 C 2 1 + r. the first order conditions for C 1 is: u 0 (C 1 )=l, the first order condition for C 2 is: bu 0 (C 2 )= l 1 + r. Francesco Franco Macroeconomics: Fluctuations and Growth 8/54
combining the two first order conditions we obtain the intertemporal Euler-Ramsey-Keynes equation: u 0 (C 1 )=(1 + r)bu 0 (C 2 ). Intuition: at a utility maximum, the consumer cannot gain from feasible shifts in consumption between periods. A one unit reduction in first period lowers U by u 0 (C 1 ). The consumption unit thus saved can be converted by lending it into 1 + r units of second period consumption that raise U by (1 + r)bu 0 (C 2 ). The Euler condition states that at an optimum these two quantities are equal. Francesco Franco Macroeconomics: Fluctuations and Growth 9/54
An alternative interpretation is to rewrite the Euler condition: bu 0 (C 2 ) u 0 (C 1 ) = 1 1 + r where the LHS is the MRS between consumption in period 1 and period 2 while the RHS is the price of future consumption in terms of present consumption. Francesco Franco Macroeconomics: Fluctuations and Growth 10/54
We have used four variables to describe the budget of the household: A 1, Y 1, Y 2 and r but from the diagram it is clear that consumption in every period depends only on the slope and the position of the budget line. Changes in Y 1 or Y 2 affect consumption in a period only through their effect on consumer wealth: W = A 1 + Y 1 + Y 2 1 + r. Exercises: DY 1 > 0; DY 1 = DY 2 > 0; DW > 0. Francesco Franco Macroeconomics: Fluctuations and Growth 11/54
An increase in r has three effects: 1 it makes future consumption cheaper relative to present consumption (substitution effect). 2 allows to have more second period consumption with the same amount of resources saved (income effect). 3 if Y 2 is positive, it reduces the present value of W (life time effect). Theoretically the effect of the interest rate on consumption is not predictable. Francesco Franco Macroeconomics: Fluctuations and Growth 12/54
Extending the two periods model T periods: U = T Â t=1 b t 1 u(c t ) and the IBC at time 0 is: T Â t=1 C t (1 + r) t 1 = A 1 + T Â t=1 Y t (1 + r) t 1. Francesco Franco Macroeconomics: Fluctuations and Growth 13/54
the first order conditions for {C t } T t=1 : u0 (C 1 )=l, bu 0 (C 2 )= l 1+r, b2 u 0 (C 3 )= l,...,b (1+r) T 1 u 0 l (C 2 T )= (1+r) T 1 u 0 (C t )=(1 + r)bu 0 (C t+1 ). For simplicity consider the case where the discount rate is equal to the interest rate, (1 + r)b = 1. Francesco Franco Macroeconomics: Fluctuations and Growth 14/54
In this case it is easy to see C 1 = C 2 =... = C T substituting: T Â Y t! C = µ A 1 + t=1 (1 + r) t 1 {z } life time income permanent income =µw t for all t C = µw t where µ = 1 Â T 1 t=1 (1+r) t 1 is the propensity to consume out of wealth W Francesco Franco Macroeconomics: Fluctuations and Growth 15/54
The previous analysis implies that consumption in a given period depends on the income over the entire lifetime and not in the income in that period. Define Yt P µw t, permanent income and Yt T = Y t Yt P transitory income at time t. Francesco Franco Macroeconomics: Fluctuations and Growth 16/54
Assume that in period 1 the households receives from the government G. Her current income increases by G,but her consumption increases by µg. if T (the horizon of the Household) is relatively long the impact of G on today consumption is relatively small. Francesco Franco Macroeconomics: Fluctuations and Growth 17/54
Define the household saving in period t : S t = Y t C t = (Y t µ T Â t=1 Y t (1 + r) t 1 ) µa 1 saving is high when current income is high relative to permanent income, which is to say when transitory income is high...too stark? Consider the following: Francesco Franco Macroeconomics: Fluctuations and Growth 18/54
Fiscal Policy add a public sector that purchases goods G t collect taxes t t and issue bonds B G t. The government has to respect the following DBC: B G t+1 = (1 + r)(b G t + G t t t ) B G 1 > 0 B G T +1 = 0 Francesco Franco Macroeconomics: Fluctuations and Growth 19/54
Fiscal Policy The IBC of the Government is: T Â t=1 G t (1 + r) t + BG 1 = T Â t=1 t t (1 + r) t The household budget constraint becomes using the government budget constraint: T Â t=1 C t (1 + r) t 1 = A 1 + T Â t=1 Y t t t (1 + r) t 1. Francesco Franco Macroeconomics: Fluctuations and Growth 20/54
Fiscal Policy T Â t=1 C t (1 + r) t 1 = A 1 B G 1 + T Â t=1 Y t G t (1 + r) t 1 where you can see that taxes do not appear! This result is called the Ricardian equivalence and says that a variation in taxes does not affect consumption as long as the sequence of public expenditure is fixed and that government bonds are not net wealth Francesco Franco Macroeconomics: Fluctuations and Growth 21/54
Econometrics Consider the following consumption function suggested by theory: C t = c 1 Yt P c 1 = 1 + e t Assume that E (e t )=0andE (Y T t Cov(e t, Yt P )=Cov(Yt T )=0and, Yt P )=Cov(e t, Y T t )=0. Keynesian consumption function implies that we estimate the following equation: C t = b 0 + b 1 Y t + u t. Francesco Franco Macroeconomics: Fluctuations and Growth 22/54
Econometrics Let s look at the coefficient estimates: 1 T ˆb 1 = ÂT t=1(c t C )(Y t Ȳ ) 1 T ÂT t=1(y t Ȳ ) = Cov(C t, Y t ) Var (Y t ) Var (Y = P ) c 1 Var (Y P )+Var (Y T ) 1 = c 1 < c 1 + Var (Y T ) 1 Var (Y P ) Francesco Franco Macroeconomics: Fluctuations and Growth 23/54
Econometrics Then the OLS estimate of the propensity to consume is biased and not consistent. ˆb 0 = C ˆb1 Ȳ Then even if the true b 0 is equal to 0, it becomes positive. you are regressing current consumption on current income, and obtain a propensity to consume that basically says that if the variation of transitory income is high relative to the variation in permanent income consumption rises little with current income. Francesco Franco Macroeconomics: Fluctuations and Growth 24/54
Conclusions The theoretical implications developed above for consumption behavior contain much truth: we certainly think about future income in deciding how much to consume these same implications say that current income has no role in explaining consumption beyond its indirect effect on permanent income Truth is that the basic theory we have developed, contemplates aworldwithoutuncertaintyandimperfections:thehousehold was able to forecast without error her labor income Francesco Franco Macroeconomics: Fluctuations and Growth 25/54
Let us start by relaxing the assumption of Uncertainty and consider our two period model but this time the labor income of period 2 is stochastic. The household has to solve the following problem: max E 1 [u(c 1 )+bu(c 2 )] s.t A 2 = (1 + r)(a 1 + Y 1 C 1 ) C 2 = A 2 + Y 2 where the expectation has a subscript to indicate that the expectation is conditional on all the information the household possesses at the moment of the maximization. Francesco Franco Macroeconomics: Fluctuations and Growth 26/54
Once you combine the FOC you get a Euler condition: u 0 (C 1 )=(1 + r)be 1 u 0 (C 2 ) which says that at the optimum the household equates the marginal value of consuming one unit less today to the expected marginal value of consuming the same unit (plus the interest on that unit) tomorrow. Francesco Franco Macroeconomics: Fluctuations and Growth 27/54
Throughout the discussion assume 1) utility has the following quadratic specification: u(c ) = C a > 0 a 2 C 2 2) b(1 + r) =1(rememberwhataretheimplicationsofthis assumption). Plugging in this specification into the Euler condition we obtain Hall s (1978) result, namely that consumption is a random walk under the permanent income hypothesis: E 1 (C 2 )=C 1 or in words that the expectation of my consumption tomorrow is my consumption today. Francesco Franco Macroeconomics: Fluctuations and Growth 28/54
Taking expectations on both sides of the intertemporal budget constraint and plugging in it the previous optimal condition we obtain that consumption in period 1 is equal to: C 1 = 1 + r 2 + r apple A 1 + Y 1 + E 1Y 2 1 + r so that the solution is simply an expected value rendition of the permanent income consumption function. Francesco Franco Macroeconomics: Fluctuations and Growth 29/54
This result is called the certainty equivalence result and says that people make decisions under uncertainty by acting as if future stochastic variables were sure to turn out equal to their conditional means 1. To see our permanent income theory at work assume for simplicity that r = 0andthatcurrentincomeY t is the sum of a permanent component and a transitory component: Y t = Y P t + Y T t Y T t = e t where e t N(0, s e ) Y P t = Y P t 1 + u t where u t N(0, s u ). 1 The crucial assumption to obtain this result is the linearity of the marginal utility of consumption. Francesco Franco Macroeconomics: Fluctuations and Growth 30/54
Assume that in the first period both the transitory and the permanent components have a positive increase so that e 1 > 0and u 1 > 0(andY0 P = 0). The optimal consumption in period one is then: C 1 = 1 2 A 1 + 1 2 e 1 + u 1 where you can see that the consumer smooth between the two period of his life the transitory increase in income but immediately consume the all increase in his permanent income. Francesco Franco Macroeconomics: Fluctuations and Growth 31/54
Now assume that in period 2 there are no changes in both the income components so that e 2 = 0andu 2 = 0. In this case C 2 = C 1 = 1 2 A 1 + 1 2 e 1 + u 1 which means that consumption in period 2 is identical to consumption in period 1. What if in the second period u 2 or e 2 were different from zero? In this case consumption in period 2 would be equal to consumption in period one plus or minus the changes in the transitory or permanent component occurred in period 2, for instance with u 2 > 0and e 2 = 0: C 2 = C 1 + u 2 C 2 C 1 = u 2 Francesco Franco Macroeconomics: Fluctuations and Growth 32/54
Extending this reasoning to many periods is straightforward (do it) and leads to the result that changes in consumption between period t and period t 1areequaltothechangeinindividual estimates of his or her lifetime resources:! C t C t 1 = 1 T t T t T t  E t Y t+k  E t 1 Y t+k. k=0 k=0 Francesco Franco Macroeconomics: Fluctuations and Growth 33/54
In word this means that under the specified hypothesis, changes in consumption are unpredictable which gives us a testable implication of the permanent income hypothesis: the information of the household at t 1shouldnotbeabletoexplainhischangein consumption between t and t 1. Francesco Franco Macroeconomics: Fluctuations and Growth 34/54
-Campbell and Mankiw (1989) have proposed to test an alternative hypothesis using an instrumental-variable approach: assume that a fraction l of consumers follow a rule of thumb instead of optimizing and consume every period their current income and that afraction(1 l) behave according to the random walk hypothesis. The change in aggregate consumption is then: C t C t 1 = l(y t Y t 1 )+(1 l)u t DC t = ldy t + v t where u t is the change in consumer s estimate of their permanent income from t 1tot. DY t and v t are almost surely correlated (times when current income increases are usually also times when households receive favorable news about their total life-time incomes). Francesco Franco Macroeconomics: Fluctuations and Growth 35/54
This is a case where theory suggests many potential instruments: the residual n t reflects new information between t 1andt and therefore any variable that is known as of time t 1isuncorrelated with the residual. Instrumenting with past values of income and consumption we obtain on Portuguese data a ˆl = 0.45. This result suggests that consumption appears to increases by about 45 cents in response to an anticipated 1-euro increase in disposable income. Francesco Franco Macroeconomics: Fluctuations and Growth 36/54
Extension to multiple assets (n) and risk. We are slightly changing the budget constraint, think about the differences. U = E 1  t=1 b t 1 u(c t ), n  S i t+1 = n Â(1 + r i t+1)s i t + Y t C t u 0 (C t ) = be t (1 + r)u 0 (C t+1 ) for i = 1 u 0 (C t ) = be t (1 + r i t+1 )u 0 (C t+1 ) for every i 6= 1 Francesco Franco Macroeconomics: Fluctuations and Growth 37/54
We can rewrite the second set of Euler conditions: u 0 (C t )=b E t (1 + r i t+1)e t u 0 (C t+1 )+Cov(1 + r i t+1, u 0 (C t+1 )) Assume for asset i : (1 + r) =E t (1 + rt+1 i ) (here we take as given the return on the asset i) andcov(1 + rt+1 i, u0 (C t+1 ) > 0!you buy more of i than the risk free asset which decreases the Cov until it reaches zero: you are hedging risk. Francesco Franco Macroeconomics: Fluctuations and Growth 38/54
The rates of return are endogenous: if agents demand a large amount of asset i for it is positively correlated with marginal utility of consumption its price will increase implying a decrease in its return: E t (1 + r i t+1) = 1 b u0 (C t ) Cov(1 + r i t+1, u0 (C t+1 )) E t u 0 (C t+1 ) Francesco Franco Macroeconomics: Fluctuations and Growth 39/54
use the risk free rate: E t (1 + r) = 1 b u0 (C t ) E t u 0 (C t+1 ) and you get: E t (r i t+1) r = Cov(1 + r i t+1, u0 (C t+1 )) E t u 0 (C t+1 ) which says that the expected-return premium an asset must offer relative to the risk-free rate is proportional to the covariance of its return with consumption. This a fundamental model in finance: the CAPM. Francesco Franco Macroeconomics: Fluctuations and Growth 40/54
Our derivation of the random walk hypothesis was based on the assumption of quadratic utility which in turn implied a linear marginal utility: 1 ac t aconstantsecondderivative a and a third derivative equal to zero. What if marginal utility is not linear and the third derivative is positive? Start from the Euler condition our two periods model (in the case of zero interest rate and preference rate): u 0 (C 1 )=E 1 u 0 (C 2 ). Francesco Franco Macroeconomics: Fluctuations and Growth 41/54
When marginal utility was linear we saw that E 1 [u 0 (C 2 )] = u 0 (E 1 [C 2 ]) which implied E 1 [C 2 ] = C 1. However if the marginal utility is not linear but convex (u 000 > 0) E 1 [u 0 (C 2 )] > u 0 (E 1 [C 2 ]). This is illustrated in the following graph where consumption in period two is stochastic: with probability 1 2 C 2 will be equal to C H and with probability 1 2 C 2 will be equal to C L. Francesco Franco Macroeconomics: Fluctuations and Growth 42/54
From the graph it is easy to see that in the case of convex marginal utility E 1 [u 0 (C 2 )] > u 0 (E 1 [C 2 ]). But then if E 1 C 2 = C 1 the marginal utility in period 2 is greater and requires a marginal reduction in C 1. Thus the combination of uncertainty and a positive third derivative of the utility function reduces current consumption and increases saving. this saving is known as precautionary saving. An increase in uncertainty increases precautionary savings (do graph). Francesco Franco Macroeconomics: Fluctuations and Growth 43/54
The second important extension of the basic model is to shut down the hypothesis that individuals can borrow and save freely at the same interest rate. Liquidity constraints raise saving so that the household has a buffer stock of assets to smooth shocks. Liquidity constraints can also explain why consumption follows income. Francesco Franco Macroeconomics: Fluctuations and Growth 44/54
Consider a three periods model with quadratic utility (to shut off the precautionary savings motif) no risky assets and r = r = 0. Assume you have a probability p L of having a liquidity constraint on period 2. max U = C 1 a 2 C 1 2 + E 1 C 2 a 2 C 2 2 + E 1 C 3 a 2 C 3 2 Francesco Franco Macroeconomics: Fluctuations and Growth 45/54
start by solving the problem as if you were in period 2 (you take A 2 as given) U 2 C 2 : C 2 = E 2 (C 3 ) C 2 = A 2 + Y 2 + E 2 Y 3 w/o liquidity constraints apple 2 A2 + Y 2 + E 2 Y 3 C 2 = min, A 2 + Y 2 with liquidity constraints 2 Francesco Franco Macroeconomics: Fluctuations and Growth 46/54
in period 1 : U 1 C 1 : C 1 = E 1 (C 2 ) C 1 = A 1 + Y 1 + E 1 Y 2 + E 1 Y 3 (1 p L ) apple 3 A1 + Y 1 + E 1 Y 2 + E 1 Y 3 + min, A 1 + Y 1 + E 1 Y 2 3 2 p L using the fact that A 2 = A 1 + Y 1 C 1. Francesco Franco Macroeconomics: Fluctuations and Growth 47/54
Therefore: C 1 apple A 1 + Y 1 + E 1 Y 2 + E 1 Y 3 3 Liquidity constraints can raise saving in two ways: first whenever a liquidity constraint is binding, it causes the individual to consume less than he or she otherwise would (period 2). Second the fact that the constraints might bind in the future reduces current consumption period 1. Francesco Franco Macroeconomics: Fluctuations and Growth 48/54
Two periods OLG (overlapping generations) model. No bequests and cannot borrow against children income (this a sort of borrowing constraint). Now two types of agents:old and young, the population size N t is changing in time. No uncertainty. There is a government and agents can borrow and lend to a foreign country. income tax consumption young yt y tt y ct y old yt+1 o tt+1 o ct+1 o size of generation t N t Francesco Franco Macroeconomics: Fluctuations and Growth 49/54
preferences u(c y t )+bu(c o t+1) budget constraint ( they can borrow or lend only from t to t + 1) b t+1 = y y t t y t c y t c o t+1 = y o t+1 +(1 + r)b t+1 c o t+1 ct y + co t+1 1 + r = y t y tt y + y t+1 o tt+1 o 1 + r Francesco Franco Macroeconomics: Fluctuations and Growth 50/54
Assume b(1 + r) =1, the we know: ct y = ct+1 o = 1 + r 2 + r b t+1 = and the aggregate: apple y y t t y t + y o t+1 t o t+1 1 + r 1 2 + r [y y t t y t + y o t+1 t o t+1] C t = N t c y t + N t 1 c o t. Francesco Franco Macroeconomics: Fluctuations and Growth 51/54
The government runs forever and has the following budget constraint: B G t+1 =(1 + r)b G t + G t N t t y t N t 1 t o t. You can also lend to a foreign country: B F t+1 =(1 + r)b F t + Y t C t G t. Francesco Franco Macroeconomics: Fluctuations and Growth 52/54
Exercise: temporary decrease in taxes in t financed with debt. Assume that N t = n. 1)non uniform reduction in tax (only for young) in t : Dt y t = 1, in t + 1 : Dt o t+1 = 1 + r. 2)uniform reduction in tax in t : Dt y t = Dt o t = 1, in t + 1 : Dt y s = Dt o s = r for s = t + 1, t + 2,... Francesco Franco Macroeconomics: Fluctuations and Growth 53/54
Readings I Olivier Blanchard, Alessia Aminghini and Francesco Giavazzi Macroeconomics, a European Perspective. Prentice Hall, 2010, chapter 16-17. Romer Advanced Macroeconomics Francesco Franco Macroeconomics: Fluctuations and Growth 54/54