A Mathematical View of the Keynesian Model

John Maynard Keynes did not present his theory mathematically, but his classic work was quickly translated into a mathematical form which, following the lead of Paul Samuelson, began to appear in macroeconomics textbooks. The mathematics of the model are presented below in three forms - tabular, algebraic and graphical. The implicit assumptions in the analyses are that there will be no change in interest rates in the capital market and wages in the labor market caused by changes in the output market and that the changes in the output market will manifest themselves primarily as quantity adjustments.

The Tabular Approach

The "mathematics" of the simple Keynesian model is presented in the following table. We begin by specifying the level of demand for the (I)nvestment, (NX)et Exports, and (G)overnment spending which we will assume remain unchanged - at least they are independent of the level of income. In this example we will assume that I = 180, Net Exports = 70, and Government = 150. We also need to fill in the consumption column which can be done only after we specify a relationship between (C)onsumption spending and (Y)income - a key piece of the Keynesian model. The Aggregate Demand (AD) column is simply the sum of C + I + G + NX.

To understand the table we can look at the situation if the level of income equals 3,000. At that level of income AD equals 4,800 so we are clearly not in an equilibrium situation. There is excess demand which the firms will recognize as inventories decline. The only way to satisfy the excess demand is to make sales from inventories. Similarly, if income equals 18,000, then AD equals 16,800 and in this situation inventories will be building up.

The questions we need to concern ourselves with is: Where will the economy go if there is no government intervention? What will be the equilibrium level of income? At what income level will AD = Income?

To understand the situation, consider what you would do if you were running a T-shirt shop and you saw your inventories being depleted. You would likely get on the phone / Internet and order additional stock. If many T-shirt shops found themselves in your situation, they would also place orders for more product and the level of production would rise. If we were at an income / output level of 3,000, then you would expect to see income / output rise. The situation would be quite different if inventories were building up - if you were unable to sell all that you were ordering. In this situation you would likely make the call to reduce your orders which would eventually become an order for lower production. If we were at an income level of 18,000, then you would expect to see income fall.

Following this logic through, we would find that there is only one level of income (12,000) at which AD = Income. This will be the equilibrium level of income which the economy will gravitate towards.

Before leaving our simple example there is one additional aspect of the table which we need to examine. If you compare the consumption and income columns you see that they tend to move in the same direction, which is what we would expect. As income rises we find consumption spending also rising. In fact, each time income rises by 1,000, consumption rises by 800. Stated differently, the change in consumption equals 80 percent of the change in income so we would say that the Marginal Propensity to Consume = .8.

 Income (Y) Consumption (C) Investment (I) Net Exports (NX) Government (G) Aggregate Demand (AD) Excess Demand 1000 2800 180 70 150 3200 2200 2000 3600 180 70 150 4000 2000 3000 4400 180 70 150 4800 1800 4000 5200 180 70 150 5600 1600 5000 6000 180 70 150 6400 1400 6000 6800 180 70 150 7200 1200 7000 7600 180 70 150 8000 1000 8000 8400 180 70 150 8800 800 9000 9200 180 70 150 9600 600 10000 10000 180 70 150 10400 400 11000 10800 180 70 150 11200 200 12000 11600 180 70 150 12000 0 13000 12400 180 70 150 12800 -200 14000 13200 180 70 150 13600 -400 15000 14000 180 70 150 14400 -600 16000 14800 180 70 150 15200 -800 17000 15600 180 70 150 16000 -1000 18000 16400 180 70 150 16800 -1200

To get to the real substance of the Keynesian contribution let's redo the analysis under the assumption that government spending has been increased by 200 [the Government column is now 350]. If you look down the table you will find that the new equilibrium level of income is 13,000. The increase in government spending of 200 brought about an increase in income of 1,000. In this case the multiplier is 1,000/2000 = 5.

 Income Consumption Investment Net Exports Government Aggregate Demand Excess Demand 1000 2800 180 70 350 3400 2400 2000 3600 180 70 350 4200 2200 3000 4400 180 70 350 5000 2000 4000 5200 180 70 350 5800 1800 5000 6000 180 70 350 6600 1600 6000 6800 180 70 350 7400 1400 7000 7600 180 70 350 8200 1200 8000 8400 180 70 350 9000 1000 9000 9200 180 70 350 9800 800 10000 10000 180 70 350 10600 600 11000 10800 180 70 350 11400 400 12000 11600 180 70 350 12200 200 13000 12400 180 70 350 13000 0 14000 13200 180 70 350 13800 -200 15000 14000 180 70 350 14600 -400 16000 14800 180 70 350 15400 -600 17000 15600 180 70 350 16200 -800 18000 16400 180 70 350 17000 -1000

The value of the multiplier could also be derived directly from the formula which we will derive in the algebra section. The formula for the multiplier is:

DY/DG = 1/(1-MPC)

You should think of this as one equation with three unknowns and you could rewrite the equation to help you solve three policy questions.

How much do I need to increase spending to achieve a certain target level of income? You might want to use this policy if you were trying to move the economy out of a recession.

DG = DY *(1-MPC)

How much will income change if spending changes by this amount? This formulation would be useful if you wanted to get a handle on the impact on the economy of some external spending shock.

DY = [1/(1-MPC)]*DG

What is the multiplier? If you have data on two output and spending levels, you could use these data to calculate the multiplier.

Multiplier = (DY/ DG) = 1/(1-MPC)

The Graphical Approach

The situation can be presented with the aid of the simple Keynesian - cross diagram. Although it is not clear in the diagram, the slope of the Consumption and Aggregate Demand curves are equal to the marginal propensity to consume = .8. Furthermore, the intersection of the AD and the 45 degree line occurs at an income of 12,000. The intersection occurs at this level since it is the graphical and the intersection of the C+I+G+X-M line and the 45 degree line is the equilibrium level of income.

To see the impact of an increase in aggregate demand, you can compare the two diagrams which differ only in that the second one describes the situation after a 400 increase in net exports. The equilibrium level of income rises to 14,000 which is what we would expect given our earlier work. In the tabular section, income increased by 1,000 for an increase in G of 200, while here the increase in income is 2,000 when the increase in spending is 400. In both cases the multiplier is 5 = 1/(1-MPC).

Initial Situation

Situation after 400 Increase in Demand

The Algebraic Approach

The algebra of the simple Keynesian model is presented in the set of equations below. The structural model is:

• C = 2000 + .8*Y
• I = 180
• G = 150
• (X - M) = 170
• AD = C + I + G + X - M

Where:

• C = consumption spending
• I = Investment spending
• G = Government spending
• (X-M) = Net export spending
• Y = Aggregate income

Following standard procedures, if we substitute all of the information into the last equation then we can solve for the equilibrium level of income as follows.

• Y = C + I + G + X - M
• Y = 2000 + .8*Y + I + G + X - M
• Y*(1-.8) = I + G + X - M
• .2*Y = I + G + X - M
• Y = [1/(1-.8)]*[I + G + X - M]

Completing the substitution we get:

• Y = 2000 +.8*Y +180 + 150 + 70
• Y-.8Y = 2400
• .2Y = 2400
• Y = [1/.2]*2400
• Y = 12,000

No surprise here - the equilibrium level of income is 12,000

To derive the multiplier all you need to do is redo the problem for a different level of spending (G, I, X, or M) and derive the new equilibrium.  If you allow the government spending to increase by 200 so that it is 350 rather than 150, you end up with the equation Y =(1/.2)*2600 = 13,000.  This is precisely what we found using the tabular approach.

The multiplier in this problem equals 1000/200 = 5 which is what we get if we plugged a marginal propensity to consume of .8 into the formula: DY/DG = (1/(1-.8)) = 1/.2 = 5.