2.1.8 Mathematical approach to the determination of equilibrium national income

The mathematical approach to the determination of equilibrium national income is based on the Keynesian cross model, which is a simple framework used to analyze the relationship between aggregate demand (total spending) and aggregate supply (total output) in an economy. The equilibrium national income occurs when aggregate demand equals aggregate supply, meaning that the economy is producing at a stable output level without any tendency for changes.

Let’s denote the following variables:

Y = National income or output C = Consumption function (represents consumption as a function of national income) I = Investment function (represents investment as a function of national income) G = Government spending X = Exports M = Imports

In a closed economy (without international trade), the aggregate demand (AD) is the sum of consumption (C), investment (I), and government spending (G):

AD = C + I + G

In equilibrium, the aggregate demand is equal to the aggregate supply (AS), which is the national income (Y):

AD = AS C + I + G = Y

Now, we need to specify the consumption and investment functions in terms of national income to solve for the equilibrium national income. Let’s assume that the consumption function (C) is a linear function of national income, and the investment function (I) is autonomous (not dependent on national income):

C = a + bY I = I0

Where: a = Autonomous consumption (consumption when income is zero) b = Marginal propensity to consume (MPC), which represents the change in consumption for every additional unit of income I0 = Autonomous investment (investment that does not depend on income)

To find the equilibrium national income (Y*), we set aggregate demand (AD) equal to aggregate supply (AS):

C + I + G = Y (a + bY) + I0 + G = Y

Next, we isolate Y on one side of the equation:

bY = Y – (a + I0 + G) bY – Y = – (a + I0 + G) Y(b – 1) = – (a + I0 + G) Y = – (a + I0 + G) / (1 – b)

This equation gives us the equilibrium national income (Y*) when aggregate demand equals aggregate supply. It represents the level of income where planned spending (C + I + G) is equal to total output (Y).