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National income, often symbolized as \(Y\), is a critical concept in macroeconomics. It represents the total value of all goods and services produced over a specific period within a country. In the four-sector model, national income is expressed as the sum of consumption (\(C\)), investment (\(I\)), government spending (\(G\)), and net exports (exports (\(X\)) minus imports (\(M\))). Here's the equation:

• \(Y = C + I + G + X - M\)

This formula sums up different types of expenditures to represent the economic activity within a country. Each component represents a key area of the economy:

• **Consumption (\(C\)):** Spending by households on goods and services.
• **Investment (\(I\)):** Spending by businesses on capital goods.
• **Government Spending (\(G\)):** Expenditures by the government.
• **Net Exports (\(X - M\)):** The value of exports minus the value of imports.

Understanding national income helps economists and policymakers gauge the health of an economy and make decisions about fiscal and monetary policies.

The autonomous export multiplier indicates the impact of a change in autonomous exports (\(X^*\)) on the national income (\(Y\)). It shows how sensitive national income is to changes in exports. To find this, we calculate the partial derivative of \(Y\) with respect to \(X^*\):

• \( \frac{\partial Y}{\partial X^*} = \frac{1}{1 - a + m} \)

Where \(a\) is the marginal propensity to consume and \(m\) is the marginal propensity to import.
A higher value of the autonomous export multiplier means that changes in exports have a larger impact on national income. For instance, if the autonomous export multiplier is large, then even a small increase in exports will significantly boost the national income.
It's crucial to note that this multiplier is inversely related to the expression \(1 - a + m\). A rise in the marginal propensity to consume or a decrease in the marginal propensity to import will increase the value of the multiplier.

The marginal propensity to import (\(m\)) refers to the proportion of additional income that a country spends on imports. The marginal propensity to import multiplier shows how changes in \(m\) impact national income (\(Y\)). Formally, it's given by the partial derivative of \(Y\) with respect to \(m\):

• \( \frac{\partial Y}{\partial m} = \frac{-(b + I^{*} + G^{*} + X^{*} - M^{*})}{(1 - a + m)^2} \)

This expression highlights that as the marginal propensity to import increases, national income decreases, given that more income is used for imports rather than domestic spending.
In practical terms, higher import propensity means that a larger portion of additional income is being spent on foreign goods, reducing the amount of money circulating within the domestic economy, which leads to a decrease in national income.
It's essential for policymakers to understand this relationship. By managing the marginal propensity to import through tariffs or import restrictions, they can influence the national income and stabilize the economy.

Equilibrium national income is the level at which the total amount of goods produced equals the total amount of goods demanded. In the four-sector model, this equilibrium is found by solving the equation:

• \( Y = \frac{b + I^{*} + G^{*} + X^{*} - M^{*}}{1 - a + m} \)

Where each variable represents a component of national income.
This equilibrium ensures that all markets within the economy are in balance, with no unplanned changes in inventories. For instance, considering the given values in the exercise (\(a = 0.8\), \(b = 120\), \(I^{*} = 100\), \(G^{*} = 300\), \(X^{*} = 150\), \(m = 0.1\), and \(M^{*} = 40\)), substituting these values in gives:

• \( Y = \frac{120 + 100 + 300 + 150 - 40}{1 - 0.8 + 0.1} = \frac{630}{0.3} = 2100 \)

Thus, the equilibrium national income is 2100. This calculation shows that the various components (consumption, investment, government spending, exports, and imports) collectively balance out, resulting in a stable economic state.
Understanding equilibrium national income is vital for economic forecasting and policy formulation. It helps in determining how changes in fiscal policy, like changes in government spending or taxation, would impact the overall economy.