Now we will investigate households' consumption and firms' investment decisions, as they are integral parts of the economic fluctuations and make up the vast majority of the demand for goods. These decisions introduce the crucial role of the financial markets, especially their feedback effects. Furthermore, most of the recent empirical work in macroeconomics has honed in on these fields.
8.1 Consumption under Certainty: The Permanent-Income Hypothesis
Assumptions
Consider an individual living for T periods whose lifetime utility is
\(U = \sum\limits_{t = 1}^T u(C_t)\) and \(u'(•) > 0, u''(•) < 0\)
where the instantaneous utility function relies solely on consumption in period t. The individual has a given initial wealth (A) and labor income (Y) and can save or borrow at an exogenous interest rate:
\(\sum\limits_{t = 1}^T u(C_t) \leq A_0 + U = \sum\limits_{t = 1}^T Y_t\)
Behavior
Since the marginal utility of consumption is always positive, the individual satisfies the budget constraint with equality. The Lagrangian is thus
\(\mathcal{L} = \sum\limits_{t = 1}^T u(C_t) + \lambda (A_0 + \sum\limits_{t = 1}^T Y_t - \sum\limits_{t = 1}^T C_t)\)
The first-order condition for consumption is
\(u'(C_t) = \lambda\)
The marginal utility of consumption is therefore constant. Furthermore, this means that consumption itself is constant. Substituting this fact into the budget constraint yields
\(C_t = \frac{1}{T} (A_0 + \sum\limits ^T _{T = 1} Y_t)\)
Thus the individual divides his or her lifetime resources equally among each period of life.
Implications
This analysis implied that consumption in a given period is determined not by income that period, but rather by income over his or her entire lifetime. Friedman used the term permanent income to refer to the right side of the equation and the difference between current and permanent income to be transitory income.
Our analysis also implies that although the time pattern of income is not important to consumption, it is crucial to saving. The individual's saving in period t is the difference between income and consumption:
\(S_t = Y_t - C_t\)
\(= (Y_t - \frac{1}{T} \sum\limits ^T _{T = 1} Y_t) - \frac{1}{T} A_0\)
Thus, saving is high when income is high relative to its average (when transitory income is high).
What is Saving? & Empirical Application
The basic idea of the permanent-income hypothesis is that saving is future consumption. This means that it is driven by preferences between future and present consumption. For example, take the common phrase "keeping up with the Joneses." If one consumes rather than saves now, he has more to make up in the future and may "fall even further behind."
The traditional Keynesian consumption function posits that consumption is determined by current disposable income. He further claimed that the higher the absolute level of income, the higher proportion of the savings. This is contrary to subsequent empirical studies, finding that across time the country's aggregate savings is proportional to the level of income, even broken down by subgroups. It looks like Friedman's hypothesis was closer to the mark:
\(Y = Y^P + Y^T\) but \(C = Y^P\).
Consider a regression of consumption on current income:
\(C_i = a + bY_i + e_i\)
In a univariate regression, the estimated coefficient on the right-hand-side variable is the ratio of the covariance of the right-hand-side and left-hand-side variables to the variance of the right-hand-side variable. In this case, this implies
\(\hat b = \frac{Cov (Y,C)}{Var (Y)}\)
\(= \frac{Cov (Y^P + Y^T, Y^P)}{Var (Y^P + Y^T)}\)
\(= \frac{Var (Y^P)}{Var(Y^P) + Var(Y^T) }\)
In addition, the estimated constant equals the mean of the left-hand-side variable minus the estimated slope coefficient times the mean of the right-hand-side variable. Thus,
\(\hat a = \bar C - \hat b \bar Y\)
\(= \bar Y^P - \hat b (\bar Y^P - \bar Y^T)\)
\(= (1 - \hat b) \bar Y^P\)
where the last line uses the assumption that the mean of transitory income is zero. Thus the permanent-income hypothesis predicts that the key determinant of the slope of an estimated consumption function ( \(\hat b\) ) is the relative variation in permanent and transitory income. This has illuminating ramifications when realizing how households save at different points during the business cycle and their lifecycle.
8.2 Consumption under Uncertainty: The Random-Walk Hypothesis
Individual Behavior
Now we will account for uncertainty, supposing that the individual maximizes:
\(E[U] = E[\sum\limits ^T _{T = 1} (C_t - \frac{a}{2} C_t^2)], a > 0\)
To describe the individual's behavior, we can use our usual Euler equation approach. Specifically, suppose the individual has chosen first-period consumption optimally given the information available and on and on. Now consider a reduction in \(C_1\) of \(dC\) from the value the individual has chosen and an additional increase in consumption at some future date. If the individual is optimizing, this change will not matter. Knowing that the individual will maximize due to the budget constraint, we can write
\(C_1 = \frac{1}{T} (A_0 + \sum\limits_{t = 1} ^T E_1 [Y_t])\)
That is, the individual consumes\(\frac{1}{T}\) of his or her expected lifetime resources.
Implications
Generally, the expectation of future consumption factors into current consumption. Since the changes in consumption are unpredictable, we can write:
\(C_t = E_{t - 1} [C_t] + e_t\)
\(C_t = C_{t - 1} + e_t\)
where e is a variable whose expectation is zero. This is Hall's famous result that the permanent-income hypothesis implies that consumption follows a random walk. The intuition is straightforward: if consumption is expected to change, the individual can do a better job at smoothing consumption. In adjusting marginal utilities to be equal presently and in the future, the individual can adjust his or her current consumption to the point where consumption is not expected to change.
Mathematically, we can write that
\(C_2 = C_1 + \frac{1}{T - 1} (\sum\limits_{t = 1} ^T E_2 [Y_t]) - \sum\limits_{t = 1} ^T E_1 [Y_t])\)
Note that the individual's behavior exhibits certainty equivalence: as the previous equation shows, the individual consumes the amount he or she would if his or her future incomes were to equal their means, thus uncertainty about future income has no effect on consumption. To see the intuition behind this certainty-equivalence consider the Euler equation relating consumption in periods 1 and 2:
\(u'(C_t) = E_1 [u'(C_2)]\)
when utility is quadratic, marginal utility is linear. Thus expected marginal utility of consumption is the same as marginal utility of expected consumption.
8.3 Empirical Application: Two Tests of the Random-Walk Hypothesis
Campbell and Mankiw's Test Using Aggregate Data
- The random-walk hypothesis implies that the change in consumption is unpredictable, thus there is no way to forecast the change in consumption from period to period
- C and M test this by using instrumental variables against an alternative- that some consumers spend their current income and others follow the random walk
- Their results suggest a large and significant departure from the random-walk model: consumption appears to increase by about fifty cents in response to an anticipated one-dollar increase in income
Shea's Test Using Household Data
- Using wage-earners covered by long-term union contracts, Shea looks at micro-level data that shows great predictability
- Furthermore, he tests for liquidity constraints-- the fact that some households cannot borrow and their current income is less than their permanent income-- by dividing the households by whether or not they have liquid assets. He finds that consumption trends are the same for both groups
8.4 The Interest Rate and Saving
Many economists have argued that more favorable tax treatment of interest income would increase saving, and thus increase growth. But if consumption is relatively unresponsive to the rate of return, such policies would have little effect.
The Interest Rate and Consumption Growth
We begin by extending our analysis to allow for a nonzero interest rate. Now, the individual's budget constraint is that the present value of lifetime consumption not exceed initial wealth plus the present value of lifetime labor income:
\(\sum\limits _{t = 1} ^T \frac{1}{(1 + r)^t} C_t \leq A_0 + \sum\limits _{t = 1} ^T \frac{1}{(1 + r)^t} Y_t\)
where r is the interest rate and all variables are discounted to period 0. When we allow for a nonzero interest rate, it's also helpful to allow for a nonzero discount rate. This simplified the analysis to assume that the instantaneous utility function takes the constant-relative-risk-aversion form used in Ch. 2:
\(u(C_t) = \frac{C_t ^{1 - \theta}}{1 - \theta}\)
where theta is the coefficient of relative risk aversion, which is the inverse of the elasticity of substitution between consumption at different dates. Thus the utility function becomes
\(U = \sum\limits _{t = 1} ^T \frac{1}{(1 + \rho)^t} \frac{C_t ^{1 - \theta}}{1 - \theta}\)
where rho is the discount rate. Now consider our usual experiment of a decrease in consumption in some period t accompanied by an increase in consumption in the next period by 1 + r times the amount of the decrease. Optimization requires that a marginal change of this type has no effect on lifetime utility. We can find this condition by:
\(\frac{C_{t + 1}}{C_t} = (\frac{1 + r}{1 + \rho})^{1 / \theta}\)
This analysis implied that once we allow for the possibility that the real interest rate and the discount rate are not equal, consumption need not be a random walk: consumption is rising over time if r exceeds rho and falling if r is less than rho. Thus if there are variations in the real interest rate, there are variations in the predictable component of consumption growth. Empirically, studies have found that consumption responds relatively little to changes in the real interest rate, thus theta is high.
The Interest Rate and Saving in the Two-Period Case
The complication is that the change in the interest rate has not only a substitution effect but also an income effect. That is, if the individual is a net saver, the increase in the interest rate allows him or her to attain a higher path of consumption than before.
This can be seen in analysis with an individual life of two periods (hearkening back to Diamond). Here, we'll use a standard indifference-curve-budget-constraint diagram. The slope of the budget constraint is \(- (1 + r)\), meaning that giving up 1 unit of first-period consumption allows the individual to increase second-period consumption by \((1 + r)\).
In Panel A, the individual is at point \((Y_1, Y_2)\), thus saving is zero. In this case, the increase in r has no income effect and the individual's initial consumption bundle continues to be on the budget constraint. First period consumption necessarily falls and so saving necessarily rises.
In Panel B, saving is positive. An increase in the interest rate has an income effect-- meaning the individual can now afford strictly more than his or her initial bundle. The income effect acts to decrease saving whereas the substitution effect acts to increase it. The overall change is ambiguous (the case shown gives savings as unchanging).
In Panel C, the individual is initially borrowing. In this case, both the substitution effect and income effect reduces first-period consumption and saving necessarily rises.
Since the stock of wealth in the economy is positive, individuals are on average savers rather than borrowers. Thus the overall income effect of a rise in the interest rate is positive. An increase in the interest rate thus has two competing effects on overall saving, a positive one through the substitution effect and a negative one through the income effect.
Complications
This discussion appears to imply that unless the elasticity of substitution between consumption in different periods is large, increases in the interest rate are unlikely to bring about substantial increases in saving. That conclusion is limited for two important reasons.
First, most changes do not just involve changes to the interest rate. For tax policy, it usually means interest income is taxed differently (perhaps leaving government revenue unchanged), and problem 8.7 shows that this change has only a substitution effect and thus shifts consumption toward the future.
Secondly, if individuals have long horizons, small changes in saving can accumulate over time into large changes in wealth. As a result, when horizons are finite but long, wealth holdings may be highly responsive to the interest rate in the long run even if the intertemporal elasticity of substitution is small. Classically, this is an instance of the power of retirement saving while young.
8.5 Consumption and Risky Assets
Now we extend our analysis to a world where individuals can invest in all sorts of assets, almost all of which have uncertain returns.
The Conditions for Individual Optimization
Consider an individual reducing consumption by a tiny amount and using that saving to buy an asset, i, with a an uncertain stream of payoffs (\(D_{t + 1} ^i, D_{t + 2} ^i\)). If the individual is optimizing, the marginal utility forgone from the reduced consumption in period t must equal the expected sum of the discounted marginal utilities of the future consumption provided by the asset's payoffs. Letting \(P_t ^i\) denote the price of the asset, we have:
\(u'(C_t) P_t ^i = E_t [ \sum\limits _{k = 1} ^\infty \frac{1}{( 1+ \rho)^k} u'(C_{t + k}) D_{t + k} ^i]\)
In words, this means that the change in utility times the price should be equal to the expected value of the sum of future utilities times the payoffs. Note that the payoff includes dividends as well as the selling price of the asset. If the individual holds the asset for one period, this becomes:
\(u'(C_t) = \frac{1}{1 + \rho} E_t [(1 + r_{t + 1} ^i) u'(C_{t + 1})]\)
This states that the discounted expected return is the future change in utility with added interest. Since the expectation of the product of two variables equals the product of their expectations plus their covariance, or,
\(E(X, Y) = E(X)E(Y) + cov (X, Y)\)
Thus we can rewrite the expression:
\(u'(C_t) = \frac{1}{1 + \rho} [E_t (1 + r_{t + 1} ^i) E_t (u'(C_{t + 1})) + Cov_t ( 1 + r_{t + 1} ^i , u' (C_{t + 1}))]\)
where the covariance term represents the covariance conditional on information available at time t. If we assume that utility is quadratic, then the marginal utility of consumption is linear (\(1 - aC\)). Substituting this for the covariance term gives simply:
\(u'(C_t) = \frac{1}{1 + \rho} [E_t (1 + r_{t + 1} ^i) E_t (u'(C_{t + 1})) + a Cov_t ( 1 + r_{t + 1} ^i , u' (C_{t + 1}))]\)
The above equation implies that in deciding whether to hold more of an asset, the individual is not concerned with how risky it is, since the variance of the asset's return does not appear in the equation. Intuitively, a marginal increase in holdings of an asset that is risky, but whose risk is not correlated with the overall risk the individual faces, does not increase the variance of the individual's consumption. Only the expected return is considered.
It also implies that the aspect of riskiness that matters to the decision of whether or not to hold more is the relation between the asset's payoff and consumption. This discussion implies that hedging risks is crucial to optimal portfolio choices. For example, individuals should not invest in their employer or that holdings should be skewed against domestic companies or sell domestic stocks short. However, we can readily observe that people exhibit a sort of home bias, where people are usually very heavily invested in their own countries.
The Consumption CAPM
This discussion takes assets' expected returns as given. But ironically, individuals' demands for assets determine these expected returns. If, for example, an asset's payoff is highly correlated with consumption (pays well when your consumption is high), its price must be driven down to the point where its expected return is high for individuals to hold it.
We can see this by solving the general first-order condition for the price of the asset:
\(P_t ^i = E_t [ \sum\limits _{k = 1} ^\infty \frac{1}{(1 = \rho)^k} \frac{u'(C_{t + k})}{u'(C_t)} D_{t + k} ^i]\)
the utility-consumption fraction (with the discount factor) shows how the consumer values future payoffs, thus how much he or she is willing to pay for various assets. This is referred to as the pricing kernel or stochastic discount factor.
Furthermore, if we solved for the expected returns, we could see that the higher the covariance of an asset's payoff with consumption, the higher its expected return must be. If we solved for the return on a risk-free asset, we would see that the expected-return premium that an asset must offer relative to the risk-free rate is proportional to the covariance of its return with consumption.
This model of the determination of expected asset returns is known as the consumption capital-asset pricing model, or consumption CAPM.
Empirical Application: The Equity-Premium Puzzle
One of the most important implications of this analysis of assets' expected returns concerns the case where the risky asset is a broad portfolio of stocks. It is easiest to return to the Euler equation and assume constant-relative-risk-aversion utility again rather than quadratic utility. Euler's equation becomes:
\(C_t ^{- \theta} = \frac{1}{1 + \rho} E_t [(1 + r_{t + 1} ^i) C_{t + 1} ^{- \theta}]\)
If we solve for the expected return, let a g term represent the growth rate of consumption over the periods, and write the second-order Taylor approximation, we will eventually find:
\(E[r^i] - E[r^j] = \theta Cov(r^i, g^c) - \theta Cov (r^j, g^c) = \theta Cov (r^i - r^j, g^c)\)
This states that the difference between the expected returns on two assets is the difference in their covariance with consumption. However, numerous empirical studies have shown this does not explain the returns on stocks and bonds. Consumption growth has become more stable over the years and less correlated with returns, and the average equity premium rate is now 7%. This large equity premium, particularly when coupled with the low risk-free rate, is thus difficult to reconcile with household optimization. Perhaps the theory of instant gratification holds in reality.
8.6 Beyond the Permanent-income Hypothesis
- One of the hypothesis's key predictions is that there should be no relation between the expected growth of an individual's income over his or her lifetime and the expected growth of his or her consumption: consumption growth is determined by real interest rate and the discount rate, not the time pattern of income. This has been proven incorrect, for example, individuals in countries where income growth is high typically have high rates of consumption growth over their lifetimes. Managers, C-level executives, etc. generally have earnings and consumption profiles that follow a similar pattern.
- Another idea is that individuals engage in precautionary saving due to the uncertain nature of the future. If we take into account the third derivative of utility from consumption, it seems that it is positive and thus a marginal reduction in current consumption increases expected utility.
- As for liquidity constraints, when they are binding the individual can obviously consume less than he or she otherwise would. Even if it is a potential, the presence of liquidity constraints seems to cause individuals to save as insurance against the effects of future falls in income.
- Both of these would seem to raise saving alone. However, when taken marginally, we notice that a large fall in income forces a corresponding fall in consumption, and thus a large rise in the marginal utility of consumption. Taken with obvious departures from perfect optimization due to information imperfection, it seems very likely that a high degree of impatience (weighting present consumption heavily) explains these departures from the permanent-income hypothesis.