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1.3 The Income-Expenditure Model

A. What is a model?
A model is a toy.  It is a simple simulation of some aspect of the real world.  We use models because the real world is too complex to understand in all of its detail.  We use models because we hope that a simplification will help us understand something important about the world.  An architect's model, made of cardboard, is a lot simpler than a real building, but it is enough like the real thing that it helps the architect understand some important things about how the building will look and work, and identify problems in the design.

In social science, we often build models by using mathematical equations.  Learning to use a model requires getting used to some math, and it's tempting to feel that once you've figured out the math you're done learning the model.  You're not!  That's like saying an architect is finished using a model when all the bits of cardboard are stuck together.  You're only done learning a model when you can relate it back to the real world that the model is supposed to portray, just as the architect uses what the model shows to modify the design of the real building.

B. Macroeconomic equilibrium
We're making a model of the entire national economy, or macroeconomy.  We base it on the following idea: if total output exceeds what people want to buy, there will be stuff left over -- which means that inventories of unsold goods will rise.  On the other hand if people want to buy more than is produced during some period, what will happen is that previously-accumulated inventories of goods will fall.

Think about a shoe store.  It has a certain amount of inventory that it wants to keep on hand at all times so as to be able to normal customer demand.  If customers buy shoes at a slower rate than it gets new shipments from the manufacturers, inventory will rise.  If on the other hand customers buy shoes at a more rapid pace than it gets shipments from the factory, inventory falls.  Think about the entire economy this way and you understand how we think output adjusts to demand.

Equilibrium just means a position of stability.  If inventories are rising, we are not in equilibrium because too many goods are being made, and sooner or later factories are going to have to cut production because the current level of output is not sustainable.  If inventories are falling, we are not in equilibrium because too few goods are being made, and sooner or later factories are going to have to increase production.  In other words macro equilibrium is when total output is equal to total desired expenditures.  We believe that the national economy will move toward such a macro equilibrium on its own.

When we set up our macro framework, we said that

In other words the three components of demand equal output.  So does this mean we're always in macro equilibrium?  No.  The key here is I, or capital investment.  "I" includes accumulation of inventory, whether firms want it or not.  In other words, in order to make our national income accounting framework hang together, we just assigned any unsold output to I -- it's like saying to a firm: if you made it and didn't sell it, you bought it.  In other words for the purpose of making our accounting framework add up, inventory change is a sort of shock absorber that makes demand equal to output.

So let's distinguish between "planned investment," or Ip, and total investment, which is Ip plus unintended inventory changes.  So while

is always true, is only true when we are in macro equilibrium.

In other words, since we said

we have macro equilibrium only when which is what we said above.
 

C. Aggregate Supply and Aggregate Demand
Let's repeat our macro equilibrium condition

And read it his way: we have macro equilbirum when people want to buy all the goods that firms produce.  The income-expenditure model therefore zeroes in on the problem that firms face in a modern capitalist economy of figuring out how much to make and offer for sale in any given period.  Because production and transport of goods takes a lot of time, a firm may have to predict consumer demand for its output a year or more in advance.

In the income-expenditure model, total output responds to the demand for it.  In other word, aggregate supply is driven by aggregate demand. ( Not all models work like this.)  That means that to figure out what the equilibrium level of output is, we have to figure out how much demand there is.  That means that we have to know what determines the levels of C, Ip, and G.

In this particular model, the answers for the last two are easy.  We will assume they are fixed and unchanging.  We will assume that businesses make plans about how much capital equipment they want to acquire, and do not change those plans.  We will assume that G is set through some political process.  We will similarly assume that T is fixed.  In fancier language, G, Ip, and T are "exogenous" to this model.

That leaves C, which is at the center of this model.  What determines how much consumers spend?  Well, their income, Y.  But we've just said that Y is partly determined by C, since C is an element of demand.  So C affects Y and Y affects C.  How exactly?  Read on.

D. The Consumption Function
How much do consumers wish to spend? We focus on the relationship between income Y (remember this is also the same thing as aggregate output) and consumption C.  The actual consumption households undertake depends on their disposable income, because they don't have any choice about paying taxes. So consumption and savings will be functions of disposable income, or (Y-T).

Since whatever is not consumed must be saved, as soon as we specify a consumption function we have necessarily specified a savings function. "Function" just means that one thing depends on another thing or things.

To keep things simple, we are going to specify consumption as a linear (straight line) function:

in which "a" represents some basic level of consumption people will undertake regardless of income (assume they dip into savings if their income is zero) and "b" represents the amount of each additional dollar earned people will spend on goods and services. (In the language of analytic geometry, "a" is the "intercept" and "b" is the "slope" of the line.)

This "b" has a special name: the Marginal Propensity to Consume (MPC). In economic terms, it tells the additional amount of aggregate consumption that the members of the economy will desire to undertake, for each additional dollar of income they receive.

The MPC is always positive (since when people earn more, they will consume more).

The MPC is also less than 1. That is we assume that some part of each extra dollar earned is saved. That gets us to the next point, We know from our savings identity that in all circumstances

So, once we know our consumption function, we can always derive the relationship between Y and S. We can also easily figure out the Marginal Propensity to Save. Since every extra dollar earned is either saved or consumed, E.g. if my MPC is .75, I spend seventy-five cents of each extra dollar earned on goods and services, so I must be saving the remaining quarter. Hence my MPS is .25.

E. Aggregate Expenditure and Equilibrium -- a numerical example
We now have C, Ip, and G.  That means we have all the information we need about the planned level of total (aggregate) expenditure in the economy.  Here is a numerical example, with a graph.

To begin with, let's specify the consumption function as:

and set G, Ip, and T We can fill in the T in the consumption function to get which is the same as Here's a graph of C as a function of Y.

We can also graph G and Ip as functions of Y.

which makes for a boring graph because they are constants and won't change if Y changes -- hence they appear as horizontal lines.  That's because, remember, they are exogneous to the model rather than being determined within the model. Finally we can put them all together:

including aggregate expenditure, which is just C + Ip + G, or the three components added up.  Finally, we put the level of output on the graph, as a dotted line.  Since output always equals income, it's just a simple line from the origin.

Here is how to solve for Y.  Start with our equilibrium condition

and substitute in, from the material above, what we know about each of the components of aggregate expenditure Don't go anywhere until you see where this comes from.  Next we multiply that .75 through the "(Y-400)" parentheses: simplify subtract .75 Y from both sides and then divide both sides by .25 (or, if you prefer, multiply by 4) to get Now if you look back up at the last graph, you can see that this was already solved on the graph: the level of Y at which output (Y) is exactly the same as the demand for it (AE, or C + Ip +G) is indeed 3000.

We can also check that we have the reight level of Y by seeing if

First, how much are C and S?  To figure out C, take the consumption function and substitute in the equilibrium level of Y that we just found and then we substitute this value for C, plus Ip and G which we know already, into so our answer checks out. What a relief!  (If you have to do this on an exam, always check that your numbers actually work.) Finally, let's work out how high saving is.  If then Notice that G > T -- we are running a fiscal deficit.  Fiscal deficits, or surpluses, are perfectly compatible with macro equilibrium.

Now let's show all this on one of our flow diagrams.

so you can see that everything adds up.  Notice the fiscal deficit, and the fact that government has to borrow from the flow of savings, via the financial sector. There is no reason here to seek a balanced budget in which G = T.  (Don't confuse macro equilibrium with a balanced budget.)

If you like algebra, here's a generalized solution for the income-expenditure model.
 

F. Changes in the numerical example
Starting with the model from the previous section

Let's try changing the level of government spending.  See if you can solve for equilibrium levels of Y, Yd, C, and S for each of these different levels of government spending.  Click to see answers and pictures. What, in general, happens to the equilibrium level of output as G changes?
 


©1999 Colin Danby