C. Swanson

Review of some elements in Obstfeld and Rogoff, Foundations of International Macroeconomics.

1. Current Account.
2. Consumption functions, derived from utility maximization

Equals: Total income from all sources (output plus bond income) less spending.

Equals: Increase in foreign bonds held.

Equality holds because income that is not allocated to current expenditures (on consumption, government spending, or investment) is used to purchase foreign assets. Negative values for the current account work the same way, but in reverse.

A) Three equations for the current account are:

1) CA_t = B_t+1 - B_t

2) CA_t = Y_t + r_tB_t -(C_t + I_t + G_t)

3) CA_t = NX_t + r_tB_t

B) The current account has several interpretations:

1) It indicates changes in national wealth and asset holdings.

2) It indicates how current expenditures relate to earnings.

3) It indicates how well the country is faring internationally.

C) In some cases, a negative current account can lead to a currency run.

A. Consumption is the opposite of saving. Production is used either for current consumption or for investment. In a closed economy, production goes to investment or consumption. In an open economy, domestic production goes to either domestic consumption, domestic investment or foreign consumption or investment (export). Exporting is very similar to investing in that it involves reducing current consumption so that future consumption can be increased.

Closed economy: Y_t = C_t + I_t + G_t

Open economy: Y_t + r_tB_t = C_t + I_t + G_t + (B_t+1 - B_t)

B. (Keynesian Consumption Function) The consumption function was elevated to great prominence by John Maynard Keynes in 1936 through his "General Theory" book. The Keynesian consumption function looks like: C_t = A + b Y_t

Here, b is the marginal propensity to consume, usually thought to be around 0.75, and A is autonomous spending. The idea is that for every $100 of income, about 75% is spent on current goods and the remainder is saved. Autonomous spending is positive, because the individual must spend something, even if earnings are zero.

The Keynesian consumption function is the basis of the Keynesian multiplier which is the heart and soul of the Keynesian model. The multiplier is derived from the assumption that output is demand driven: Y = C + I + G + NX, where all variables should have a superscript D indicating demand. Since consumption is C_t = A + b Y_t, we have

Y_t = A + b Y_t + I_t + G_t + NX_t

This becomes

(1- b)Y_t = A + I_t + G_t + NX_t

or

Y_t = (1/(1 - b))(A + I_t + G_t + NX_t)

= m (A + I_t + G_t + Nx_t)

where the number m,

m = 1/(1 - b)

is called the multiplier.

If b = 0.75, then m = 4. This means a $100 increase in G will, according to the theory, raise Y by $400.

C. (Consumption as a constrained optimum) The Keynesian consumption function, C_t = A + b Y_t, is built from logical first principles, but these principles are quite vague: a person ought to spend a fraction, but not all of current income on consumption. Is this sometimes true, always approximately true, approximately true in special circumstances, or what? We can derive the exact functional form for some consumption functions under special circumstances and see what they look like.

1) Maximize utility subject to a budget constraint.

2) Obtain first order conditions.

3) Make wild and crazy simplifications about the utility function.

4) Substitute the results of (2) and (3) into the budget constraint.

5) Make some assumptions about an exogenous process (usually Y_t)

6) Examine the resulting consumption function, compare it to data.

More explicitly,

1) max 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.

Condition (a) says that there is some initial known level of assets; it might be zero. Condition (b) says that asset accumulation occurs in proportion to income, less spending.

Condition (c) is one version of a No Ponzi scheme condition. It rules out debt bubbles (more on that later). Technically, it says debt cannot grow too fast. On the first go around, getting the first order conditions, only condition (b) enters the fray. In later steps, (a) and (c) become important.

2) The Kuhn-Tucker theorem gets us to the first order conditions (note the plural), also known as the Euler Equation (pronounced: "Oiler," as in Houston Oilers):

This equation is written with the time subscripts on the real interest rate. If the real interest rate is assumed constant, these subscripts can be removed. We now leave the Kuhn-Tucker theorem and all statements about maximization. So long as we use the Euler Equation, we are adhering to the maximization conditions.

3) Now we make some wild assumptions about the preferences and conditions. There are three classes of assumptions that we will make, and our choice will depend on where we want to go.

Specifications

a) Hall’s assumptions (linear-quadratic utility, constant r):

(1) U(C) = C - (a₀/2)C² a₀ > 0. (See Ob.&Rog. p. 81.)

(2) r_t is a constant.

(3) 1 = b(1 + r)

b) Constant relative risk aversion (CRRA):

(1) U(C) = (C^{1 - g} - 1)/(1 - g), U(C), g > 0.

(2) r_t not a constant.

c) Logarithmic utility

(1) U(C) = Log(C).

(2) r_t not a constant.

d) Exponential (constant absolute risk aversion).

(1) U(C) = , a > 0.

Item (c) on this list, Logarithmic utility, is a special case of item (b), constant relative risk aversion, although this fact may not be completely obvious. Also, the complicated form of the CRRA is complicated only when U(C) is written. The derivative has a nice clean form. Since the derivative gets used over and over and the original U(C) only once, the notation is set up to make the derivative clean at the expensive of a complicated form for U(C). It could easily have been set up so that U(C) was clean and the derivative complicated.

Continuing step (3): Assume Hall’s specification. The marginal utility of consumption is:

The Euler equation is:

.

Substitute the linear-quadratic marginal utility and note that r is constant, so it factors out:

.

The second Hall assumption is about the real interest rate and the discount factor:

.

The expectation of a constant is that constant:

.

Subtract and factor,

.

Cancel to get:

.

The final equation is the form of the Euler equation under Hall’s assumptions. This equation is algebraically very simple. All the stuff that preceded it is intended to make this final result look nice and clean. The simpleness is needed because the budget constraints complicate the picture again.

4) Now we need to substitute the Euler equation into the budget equation. We first transform the budget constraints into a single present value formula. The form is:

This present value formula has made no use of the Euler equation, so it does not depend on the utility function specification or Hall’s assumptions. This formula says that the present value of consumption is equal to the present value of income (Y_t) plus wealth from bond income as of period 0. To obtain it, we use Forward and Lag operators. We use these because they are a useful tool later. The budget constraint is

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

or

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

or

(1/(1 + r))( C_t - Y_t) = B_t - (1/(1 + r))B_t+1 = B_t - (1/(1 + r))F(B_t).

The last line uses the "Forward operator," F(B_t) = B_t+1.

Then factor,

(1/(1 + r))( C_t - Y_t) = (1 - F/(1 + r)) B_t.

Divide both sides by (1 - F/(1 + r)). What allows us to treat F as though it were an ordinary number rather than an operator? The answer is abstract algebra theory, but we don’t sweat that here. The result is

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

or

(1 - F/(1 + r))^-1 ( C_t - Y_t) = B_t(1 + r).

Now we go back to the geometric series and apply it to the operator F, again not worrying too much about the fact that F is an operator, not a number. When a is between 0 and 1, (1 - a)^-1 = 1 + a + a² + a³ +...=

Let a = F/(1 + r), and pretend a is a number, which it is not. Then,

(1 - F/(1 + r))^-1 = .

For any series X_t, F^j(X_t) = X_t+j, and

(1 - F/(1 + r))^-1 X_t = X_t = .

Replace X_t with C and then Y to obtain

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

which is the result we wanted.

To recap a little, the budget conditions and nothing else give:

B_t(1 + r) = - ,

The maximization problem and Hall specification (linear-quadratic) gives

C_t = E_t [C_t+1],

and

C_t+1 = E_t+1[C_t+2], etc.

Now we put these together. To do so we need yet another technical device, this one too having lots of use later. It is called the law of iterated expectations. In words, it says that what you expect to expect is what you now expect for some distant event. For example, say that an investor named Helen is forming expectations in January about a stock price in December. Helen knows that her beliefs will be revised in November. What Helen will believe in November about the December stock price is something that Helen will form expectations about in January. The law of iterated expectations says that if Helen’s January expectation (mean forecast) of December’s price is $85, then her January expectation of November’s forecast of December’s price is also $85. Mathematically, it is written as

E_t[C_t+j+k] = E_t[E_t+j[C_t+j+k]]

In the Hall model, this means that E_t[C_t+2] = E_t[E_t+1[C_t+2]] = E_t[C_t+1] = C_t.

This used the fact that C_t+1 = E_t+1[C_t+2], and C_t = E_t [C_t+1].

We can therefore write the expected value of the present value of consumption as

The present value and budget conditions are then:

E_t B_t(1 + r) = E_t - E_t ,

B_t(1 + r) = C_t - E_t ,

or

C_t = r B_t + E_t ,

This last equation is the consumption function. It is equation (32) on p. 81 of Obstfeld and Rogoff, with investment and government spending set to zero. It gives current consumption as a function of expected future income. In a sense, it is a permanent income theory, because the expected present value of future income is the determinant of current consumption. The final form for C_t depends on the assumptions made about the Y_t process. This is done next.