# Technical Appendix

For those who are comfortable with simple algebra, the easiest way to develop the income-expenditure model is in terms of equations. The consumption function we have been using was:

(0) Consumption = \$5000 + (3/4)(expected income)

The equilibrium condition we began with was:

(8) expected income = actual income

Since we began by assuming that the only form of spending was consumption, and because spending is equal to actual income because one person's spending is another's income, we have this definition:

(9) actual income = consumption.

Using these three equations we obtain:

(10) expected income = \$5000 + (3/4)(expected income),

which yields, with a bit of algebra:

(11) (1-3/4)(expected income) = \$5000

or:

(12) expected income = 4x\$5000 = \$20000.

Adding investment only slightly alters the equations that represent the model. The equilibrium condition now includes investment as part of actual income:

(13) Expected Income=Actual Income = Consumption + Investment

The consumption function remains the same as above, but the model now needs an equation to describe investment behavior. The assumption used in the example was:

(14) Investment = \$2500

Substituting equations 0 and 14 into equation 13 gives:

(15) Expected Income = \$5000 + (3/4)(Expected Income) + \$2500

or:

(16) (1-3/4)(expected income) = (1/4)(expected income) = \$5000 + \$2500.

or:

(17) expected income = 4(\$7500) = \$30000.

Notice that the multiplier of four shows up in equation 11 and that equilibrium income equals the multiplier times the exogenous spending in the system (the \$5000 plus the \$2500). You can see in equation 10 that the mpc determines the size of the multiplier in this model. If, for example, the mpc in equation 9 was 2/3 rather than 3/4, the multiplier in equation 11 would be three rather than four. In very simple models (but not in more complex models) the multiplier will be 1/(1-mpc).

The paradox of thrift can be demonstrated in this simple model that excludes government using the equation:

(7) Sexp + T = I + G

as the equilibrium condition. To solve for equilibrium, we must find the savings function. Using the fact that expected savings is the difference between expected disposable income and consumption, and the consumption function in equation 0, we can find the savings function for our example as:

(18) S = Y - (\$5000 + (3/4)(expected income))

or:

(19) S = 1/4(expected income) - \$5000.

The one-fourth in this equation is the marginal propensity to save, the fraction of additional disposable income that people will save.

If people become more thrifty, they save more and consume less at each level of income. The 5000 in equations 0 and 18 becomes a smaller number. Suppose that it becomes 4000. Then substituting our revised savings function and the investment equation 14 into equation 7, we get:

(20) I+G = \$2500 = (1/4)(expected income) - \$4000 = S+T

or:

(20) \$6500 = 1/4(expected income)

or:

(21) \$26000 = expected income.

Comparing the new equilibrium income to the old one, we find that an extra \$1000 of thriftiness results in a \$4000 drop in income--the multiplier was at work again. Savings remain at \$2500.

Adding taxes complicates the consumption function. Taxes (abbreviated T) will create a gap between the total income of the nation and the disposable income that people can spend or save. Thus the consumption function is changed, with the revised consumption function assuming that consumption depends on expected after-tax income, or:

(22) C = a + (mpc)(expected income - taxes)

where the "a" term is the intercept. (It took the value of \$5000 in equation 0.) Two other equations must be added to the system to explain the behavior of government. The simplest assumption for both government spending and taxes is that they are determined outside the system as policy variables, or:

(23) G = g

(24) T = t

The solution to this revised set of equations is straightforward. First, recall that the equilibrium condition is that expected income equal the sum of consumption, investment, and government spending. Then suppose that government spending and taxes are both \$1000, that investment is fixed at \$2500 as before, and that the intercept and mpc in the consumption function remain \$5000 and 3/4, respectively. The substituting into the equilibrium condition, we get:

(25) expected income = \$5000 + (3/4)(expected income -1000)+2500+1000

or:

(26) expected income = 4(\$5000 + \$2500 +\$1000) - 3(\$1000) = \$31000.

You might have noted that the way we divide spending into parts is based on the way GDP can be split into its uses. The income-expenditure model tries to determine the size of each slice of GNP and then adds them together to discover the whole. We have not up to now included net exports, and for completeness they must be included. Exports act as an injection of spending into the system, while imports act as a leakage of spending from the system. Many textbook models eliminated net exports by assuming that the economy is "closed," which means that it has no imports or exports.

Finally, you might also notice the similarity of equation 7 to the budget constraint for the economy, which said that

(27) (I-S) + (G + Transfers - T) + Net Exports = 0

In the simple income-expenditure model, the only one of these variables that can adjust is savings, and the only way it can adjust is through a change in income. Thus if investment or government spending or net exports fall, the income-expenditure model asserts that income must also fall, because this is the only way it allows for savings to change. No modern economists believes a model this simplistic; they give some roles to changes in interest rates and prices. But those economists who believe that the income-expenditure model captures an important part of the story stress the adjustments in savings caused by changes in income.