To recap, we are trying to develop a formula that gives optimal aggregate consumption as a function of variables that are observable currently. Since output is divided between consumption, investment, government spending and net exports, the behavior of consumption will help us predict the behavior of investment. In the extreme case where net exports are zero (i.e., it is a closed economy) and government spending is zero (or all government spending is embedded in the consumption or investment variables), we have

Y_t = C_t + I_t ,

or

I_t = Y_t – C_t.

Also, since Y = F(K,N;A), where K is capital, N is labor input and A is technology, a change in Y may reflect a change in N, the employment level. Therefore, the consumption function might help us determine the labor demand function. Of course, we should not get ahead of ourselves because the problem so far assumes that labor input is constant, so we should not try to figure out yet what causes it to change. If the economy is not closed (net exports can be positive or negative), the current account is

CA_t = B_t+1 – B_t = Y_t + rB_t – C_t – G_t – I_t,

so knowledge of C will gives us some information about the current account.

We began the analysis by assuming that there is a single representative individual that maximizes utility subject to a budget constraint. This individual is to maximize the expected value subject to

a) B₀ is a given.

b) B_t+1 - B_t = Y_t + r B_t - C_t and

c) The limit of B_N/(1+r)^N as N goes to infinity is zero.

An application of the Kuhn-Tucker theorem gives us the Euler equation

The budget constraints give, after some work, B_t(1+r) = E_t [] –

E_t []. The job now is to transform the Euler equation and infinite horizon budget line into serviceable equation, where serviceable means that current data points can be matched with other current data points.

The first step is to invoke the Hall assumptions, (1) linear quadratic utility, (2) constant real interest rate, and (3) b(1 + r) = 1. The linear-quadratic utility function assumption, when combined with a linear budget constraint, gives what is known as a "certainty equivalent," an expression which refers to the notion that one formula with lots of random variables in it (i.e., future values of income and consumption) can be replaced with another formula in which all of the random variables are replaced with their expected values. The Hall assumptions give

C_t = (r/(1+r))((1+r)B_t + E_t [] .

Now we can make a few assumptions about the Y_t process.

Case 1: Y_t = Y + e_t, where Y is average income, and E_t[e_t+j] = 0, for j larger than one.

This income process has E_t [Y_t] = Y_t, because Y_t is known at t. On the other hand, E_t[Y_t+1]= E[Y + e_t+1] = E[Y] + E_t[e_t+1] = Y. With a little work, we obtain the result that

C_t = r B_t + (1/(1+r))Y + (r/(1+r))Y_t.

To read this equation, let r = 0.02 (the real interest rate is 2%, a reasonable number). The equation says that current consumption is investment income (rB_t), plus 0.98 times average income (Y) plus 0.02 times current income. Current income matters very little. Average lifetime future income matters significantly. This is the essence of Milton Friedman’s permanent income hypothesis. It is intended as a criticism of the Keynesian consumption function because that function assumes a marginal propensity to consume (MPC) of approximately 0.75, not 0.02 as we have here.

The consumption function derived here cannot be tested directly because we do not have a good measure for permanent income. In other words, if we run the regression,

C_t = a + b Y_t + e_t,

and that b is closer to 0.75 than to 0.02 the equation might still be valid. If we had a good measure for permanent income, and called it YP, then we could test the model by running the regression

C_t = a + b₁ YP_t+ b₂ Y_t + e_t.

We would expect b₁ to be near 1 and b₂ to be near zero. The previous equation would have b nearer to 1 because Y_t would be "proxying" for YP_t.

The model of case 1 can also be tested by checking whether the original income sequence Y_t follows the assumed process. A simple version for this regression is

Y_t = a + b Y_t-1 + e_t.

If b is anything but zero, we have to reject the assumption that

Y_t = Y + e_t.

Case 2. This is a more complicated income process. It assumes that income shocks endure for more than one period. The process is

Y_t – Y = r(Y_t-1 - Y) + e_t.

The parameter r is known as a persistence parameter. It indicates how long output will be away from its mean of Y. When r = 0, we have case 1, zero persistence. The greater is the persistence, the closer r is to one, the more consumption should respond to current income. The reason, sensibly enough, is that when r is close to one, a positive innovation (increase) in this period’s income implies that later periods will have a similar increase. Thus, if a person’s income rises by 100% for one period, lifetime income might rise by 2%, so consumption should rise by 2%. However, if a person’s income rises by 100% for 50 periods, the corresponding lifetime increase might be 98%, and hence consumption should rise by 98%. The following algebra formalizes this intuition.

We need to reduce the expression for the present value of expected future income:

We need to find E_t [Y_t+j] in terms of Y_t. The steps are recursive. First, take the assumption Y_t – Y = r(Y_t-1 - Y) + e_t, and advance it one period, and take E_t[] of both sides to get:

Y_t+1 – Y = r(Y_t - Y) + e_t+1.

E_t [Y_t+1 – Y] = E_t [r(Y_t - Y) + e_t+1]

Advance the original equation 2 periods and take E_t[] of both sides to get:

Y_t+2 – Y = r(Y_t+1 - Y) + e_t+2.

E_t [Y_t+2 – Y] = E_t[r(Y_t+1 - Y) + e_t+2]

= E_t[r(Y_t+1 - Y)] + E_t e_t+2]

= E_t[r(Y_t+1 - Y)] + 0

= r( r(Y_t - Y))

= r²(Y_t - Y)

The fifth equality uses (1) from above. These substitutions can be continued, giving

E_t [Y_t+j – Y] = r^j(Y_t - Y).

Therefore,

E_t [Y_t+j ] = Y + r^j(Y_t - Y),

and

E_t =

= Y +

= ((1 + r)/r)Y + (Y_t - Y),

= ((1 + r)/r)Y + (Y_t - Y),

It follows that consumption is

C_t = r B_t + r/(1+r)( ((1 + r)/r)Y + (Y_t - Y),

or

C_t = r B_t + Y + (Y_t - Y),)

This is equation (35), p. 83 of Obstfeld and Rogoff, 1995.

The last equation is the consumption function for case 2 process. If r is equal to zero, we have case 1 again. If r is near one then current income has a large effect on consumption. If r is equal to one, each one dollar increase in current income will raise consumption by one dollar.

When r is near one, the consumption function looks very similar to the Keynesian consumption function (where the coefficient on (Y_t - Y) is approximately 0.75. This fact does not, however, support the Keynesian idea that a temporary tax cut or government spending increase will raise consumption, because these changes do not change expected future income, and this sum is the basis for current consumption according to this entire section (and most of Obstfeld and Rogoff).