A Basic Macroeconomic Equation System

Here is a basic model to explain the behavior of aggregate output (Y) and aggregate consumption (C). Start with some definitions:

[1] Y = Gross Domestic Product, the total value of all goods and services produced annually in the economy.

[2] C = Consumer Expenditures, the total value of all goods and services bought by consumers annually

[3] I = Business Investment, the total value of all capital goods purchased by businesses annually

Next, we introduce the concept of circular flow. It means that one person's spending is another person's income. When I buy a meal at a restaurant, the money I spend ends up as income for restaurant owners, cooks, wholesale food distributers, farmers, and so on. What circular flow means in terms of our definitions is that Y equals total income as well as total expenditures.

Using circular flow, we can write

[4] Y = C + I

In this simple economy, all output is either sold to consumers or to businesses. (There is no government or foreign trade sector.) Consumer spending plus business investment equals total output. Also, because of circular flow, the sum of consumer spending and business investment is equal to income.

In the United States, consumer spending is about $6.7 trillion per year, and investment is about $1.8 trillion per year. If we were to treat this as the entire U.S. economy, then Y for the U.S. would be $8.5 trillion.

Next, we let consumer spending be a function of income. This consumption function can be written as

[5] C = C_{0} + *c*Y

This is an ordinary linear equation, of the kind that you met in middle school when you took algebra. *c* is the slope, and C_{0} (pronounced "C-nought") is the intercept. It seems simple, but when I was a graduate student and taught this at Harvard, about eight students came up to me after class and said, "I don't understand C-nought. I'm lost." I think you should catch on easier than those algebra-phobic Harvard freshmen, but don't be embarrassed to ask questions.

As a numerical example, let *c* equal 0.6, and let C_{0} equal $1.6 trillion. Then we have

C = $1.6 + (0.6)($8.5) = $6.7 trillion

I have cleverly chosen the parameters so that the consumption function fits our economy, in which consumer spending is $6.7 trillion and income is $8.5 trillion.

Equations [4] and [5] make up a system of two equations in two unknowns. The two unknowns are Y and C. Any time something else in the system changes, new values of Y and C are determined.

For example, we said that business investment is $1.8 trillion per year. What would happen if investment rose to $2 trillion per year?

Looking just at equation [4], we have

Y = C + I = $6.7 + $2.0 = $8.7 trillion

However, we cannot stop there, because the consumption function is not satisifed if we leave consumer spending at $6.7 trillion. With the new income of $8.7 trillion, equation [5] gives

C = $1.6 + (0.6)($8.7) = $6.82 trillion

However, we cannot stop there! We need to plug the new figure for C into equation [4], which will give us a new Y, and then a new C, and so on. Keynes termed this process the multiplier.

Does the multiplier ever stop? Yes. We can use ordinary algebra to solve [4] and [5] for C and Y, given values of I, C_{0}, and *c*.

In this example, plugging in the given numbers, we have

Y = C + $2.0 and C = $1.6 + 0.6Y

Using simple substitution, we have Y = $1.6 + 0.6Y + $2.0 = $3.6 + 0.6Y

Solving for Y, we have 0.4Y = $3.6, or Y = $9.0 trillion

Forget the numerical example, and go back to the general equations

Y = C + I and C = C_{0} + *c*Y

Solve these general equations for Y as a function of I, C_{0}, and *c*.

Terminology

A macroeconomic model consists of a set of equations with variables and coefficients. A variable is a measured economic statistic. In this model, the variables are Y (total output), C (consumer expenditures), and I (business investment).

A coefficient is an estimate of a component of a behavioral relationship. In this model, the coefficients are C_{0}, and *c*. A synonym for coefficient is "parameter." (In fact, using the term "parameter" in this context is correct. 99% of the time when you hear the word "parameters," it is being mis-used as if it were a synonym for perimeters. Every time I hear the word "parameter" used incorrectly, I cringe.)

Some coefficients are so significant that we give them special names. In our model, we call *c* the marginal propensity to consume. The marginal propensity to consume is important because it determines the size of the multiplier. The multiplier is

1/(1-*c*)

so that when the marginal propensity to consume is high, the multiplier is high, and conversely.

There are two types of macroeconomic equations. An identity is a relationship that is true by definition. For example, in this model Y = C + I is true by definition. Given the structure of our simple economy, we cannot violate this relationship. There is no controversy whatsoever about the validity of identities. Every economist agrees that identities are true.

The other type of equation is a behavioral equation. In our model, the consumption function

C = C_{0} + *c*Y

is a behavioral equation. It represents a hypothesis about how a key economic variable (in this case, consumer spending), is determined. Unlike identities, behavioral equations are not true with certainty. There is a margin of error in every behavioral equation. In fact, it gets worse. There is not a single behavioral equation that is beyond controversy. For every behavioral equation that I will present in this class, there is at least one very different alternative equation which some economists believe better represents reality.

In macroeconomic models, some variables are called exogenous, or pre-determined variables. This means that their values are taken as given. For example, in our simple model, business investment (I) is exogenous. We are simply told that investment is $1.8 trillion, or $2.0 trillion, or what have you.

Other variables have their values determined by the equations in the model. These are called endogenous variables. In our model, income or output (Y) and consumer spending (C) are the endogenous variables. We solve the equations for the values of the endogenous variables, taking as given the values of the exogenous variables and the coefficients.

Treating some variables as exogenous and others as endogenous is a matter of arbitrary choice. Macroeconomic models that are used for teaching purposes will use relatively few equations and relatively few endogenous variables, in order to try to minimize confusion. Macroeconomic models that are used to make forecasts and to simulate alternative economic policies can have hundreds of equations and hundreds of endogenous variables, because the goal is accuracy rather than simplicity.

For most of the rest of this section of the course, we will be adding variables and equations to the macroeconomic model. Each new equation adds an important element of realism and teaches an important concept. However, before we go on from the simple two-equation model, it is worth looking at one of its interesting properties.

The Paradox of Thrift

Suppose we start out with our basic two equations

Y = C + I and C = $1.6 trillion + 0.6Y

If investment is $1.8 trillion, then we know that solving for income gives $8.5 trillion, with $6.7 trillion in consumer spending. What is the aggregate level of saving?

All income has to be accounted for. Income that is not consumed is saved. If we subtract C from Y, we get saving. That means that S = I, so we know that saving will be $1.8 trillion.

What happens if people try to save more? In particular, what happens if the marginal propensity to consume falls to 0.5, because people try to save more of their income?

Our equations are now

Y = C + I = C + $1.8 trillion and C = $1.6 trillion + 0.5Y

Solve for the new values of Y and C. What is the new value of saving? Were consumers successful in increasing their saving? Did it help the economy to have consumers attempt to save more of their income? Explain what happened, and why.