Describe how equilibrium real GDP is established in the Keynesian model

The main purpose of the Keynesian model is to establish the relationship between the income and expenditure in an economy. The basic constituents are given as consumption expenditure, the investments of the consumer, the governmental expenditures and the net expenditure on the export-import of the economy.

Let us describe the different aspects of the development of equilibrium real GDP in the Keynesian model.

The Consumption Expenditure :

The main expression for the consumption is given as :

$C=\alpha +\beta Y$

Here in the above expression,

$\alpha$is defined as Autonomous Consumption

$\beta$is defined as the Marginal Propensity to Consume and

$Y$is defined as Disposable Income

The Investment Expenditure :

Investment Expenditure is defined as the expense that every firm in an economy bears for investing in the consumers and also on the goods in an economy.

The expression of investment is given as : $\mathrm{I}={\mathrm{I}}^{\text{'}}$

The Governmental Expenditure is defined as the expenditure that the government needs to bear for the public and other projects occurring in an economy.

The expression is given as :

$\mathrm{G}=G\text{'}$

The Net Export - Import is defined as the expenditure that happens because of the net export or import in an economy.

The expression is given as :

$NX=EX-IM.$

The final expression for the expression is given as :

$\mathrm{Y}=\mathrm{C}+\mathrm{I}+\mathrm{G}+\mathrm{NX}$

Now,

the final expression will be coming as : $AE=\alpha +\beta Y+{\mathrm{I}}^{\text{'}}+{\mathrm{G}}^{\text{'}}+\mathrm{NX}.$

Now as we know that from the expression the outcome or as we can say the income will be similar to the overall expenditure, we can write the equation as :

$Y=AE$

So,

$\mathrm{Y}=\alpha +\beta \mathrm{Y}+{\mathrm{I}}^{\text{'}}+G^{\text{'}}+\mathrm{NX}.$

The expression we obtain for the variable is given as :

$\mathrm{Y}=\frac{1}{(1-\beta )}[\alpha +\mathrm{I}+\mathrm{G}+\mathrm{NX}\text{.}$
Where, $\frac{1}{(1-\beta )}$Multiplier, it is the factor by which the return deriving from an expenditure exceeds the expenditure itself.

The graph can be drawn as :