Chapter 18: Problem 3

Give the algebraic explanation to the determination of equilibrium income and output in the Keynesian theory.

Short Answer

Expert verified

Answer: The equation for equilibrium income and output in the Keynesian theory is: \[Y = \frac{a + b(Y - T) + I + G + (X - M)}{1 - b}\] The equilibrium income depends on the parameters of the consumption function (autonomous consumption and marginal propensity to consume), investments, government spending, exports, imports, and taxes.

Step by step solution

Define Aggregate Demand and Aggregate Supply functions

In the Keynesian model, the AD and AS functions are used to determine the equilibrium income and output levels. Aggregate Demand (AD) is the total spending in an economy, while Aggregate Supply (AS) represents the total output produced in an economy. Aggregate Demand function: \[AD = C + I + G + (X - M)\] Where: \(C\) = Consumption \(I\) = Investment \(G\) = Government spending \(X\) = Exports \(M\) = Imports Aggregate Supply function: \[AS = Y\] Where: \(Y\) = National income (output)

Introduce the consumption function

The consumption function is represented in the Keynesian model as follows: \[C = a + bY_d\] Where: \(a\) = Autonomous consumption (consumption when \(Y_d = 0\)) \(b\) = Marginal propensity to consume (MPC) \(Y_d\) =Disposable income (\(Y - T\))

Define disposable income and introduce taxes

Disposable income (Y_d) is the difference between national income (Y) and taxes paid (T): \[ Y_d = Y - T \] Now, we incorporate taxes in the Aggregate Demand function: \[AD = C + I + G + (X-M) = (a+b(Y-T)) + I + G + (X-M)\]

Set Aggregate Demand equal to Aggregate Supply

In order to find the equilibrium income and output level, the Aggregate Demand (AD) function must be equal to the Aggregate Supply (AS) function. This means the total spending in the economy must match the total output available in the economy. \[ AD = AS \] \[(a+b(Y-T)) + I + G + (X-M) = Y\]

Solve for equilibrium income (Y)

We can solve for the equilibrium income (Y) by rearranging the equation from Step 4: \[Y = \frac{a + b(Y - T) + I + G + (X - M)}{1 - b}\] This equation represents the equilibrium income and output level in the Keynesian theory. When AD equals AS, the economy has reached an equilibrium, and there is neither excess demand nor excess supply. The equilibrium income, in this case, depends on the parameters of the consumption function, investments, government spending, exports, imports, and taxes.