Determination of Equilibrium Level of Income, The Case of a three Sector Model: In the new post, we will provide an explanation of how the equilibrium level of national income is determined in the two-sector model of an economy in which the households and business firms are the sectors. The role of the Government sector in the determination of equilibrium level of income was ignored.
The Government sector in both its spending and taxing activities profoundly affects the character and magnitude of the general level of economic activity. This is in the sense that; when it spends, it affects the incomes of all; when it taxes, it affects the spending ability of virtually all.
This being the case, it becomes pertinent to add the government sector (a third sector) to our two sector model in order to have the three sector model that would enable us to examine the effect of government expenditure (G) and taxation (T) on the level of income. As a starting point, the sectors in the economy will now consist of households, the business firms, and government. By this, we present aggregate demand as:
Y=C+I+G —- (4.12)
Where, all values are measured in constant Naira (that is, money value deflated by the general price level P). Writing the linear consumption function previously written as:
C=a+byd ——- (4.4)
By the above linear consumption function, the aggregate consumption level is determined by the level of disposable income (Yd). The linear consumption function shows that tax out of income is taken into consideration. Otherwise, aggregate consumption level would have been stated to depend on the level of income. C in equation (2) is as earlier defined and Yd is disposable income.
Since we are to consider tax in our analysis, we assume that consumption is a function of . disposable income. In which case, all income is not paid out to individuals, and with tax, personal income (net) equals’ personal disposable income. Also, we retain our previous assumption concerning investment, that, investment is autonomously determined which entails that it is constant at all levels of income. In which case we write that:
I=Io ——-(4.5)
Determination of Equilibrium Level of Income – the Case of a Four Sector Model
Previously, we explained and illustrated how equilibrium level of income can be determined in a one sector model, two sector model and three sector model. In all, we treated the economy as if it were a closed one having no interaction with other economies. The economy is assumed not to take part in international trade. By this, the economy has no exports or imports.
An economy that does not take part in international trade is not in existence in the real world. We only assumed in our previous analysis that such an economy exist given that it is a useful simplification when examining how total expenditures on consumption and investment and government expenditures and taxation interact to determine the levels of national income.
Moreover, by the assumption under reference, we have taken into consideration the typical device used in economics of starting off with a highly simplified model and gradually introducing additional complications. If we relax the assumption of a closed economy and consider as a matter of fact that all economies trade, it then becomes necessary to have an extension of the model of income determination to include a fourth sector – the foreign trade sector also.
The foreign trade sector is made up of a country’s exports of domestic goods and services as well as its imports of foreign goods and services. Given that a county exports domestic goods and services and imports foreign-made goods and services, it follows that the net export balance will be the sum of gross exports minus gross imports.
If we consider that X is gross exports (value of total exports of a country in any given time period) and M the gross imports (value of total imports in any given period), then the net export is given by X – M.
The gross export is simply goods and services sold to foreign buyers while the gross imports are the goods and services purchased from other economies by domestic residents. The gross export is assumed to be autonomous or exogenous (X.).
This is in the light of the fact that the determinant of a country’s exports (incomes of the other economies, relative prices of commodities, the differences in prices of goods in one economy versus those of other economies, all types of commercial policies – tariffs, quotas, exchange restrictions, and so on; the availability of reserves or exchange the currency acceptable in international markets; tastes; monopolistically produced goods – only one seller of item; political factors and trading blocs; custom . and tradition; national resources, and so on are not determined by or in the domestic markets; they are exogenous – external to the market under consideration.
However, considering the fact that some of the variables mentioned are slow to change and in the short run can be considered stable or constant, exports are usually taken as autonomous independent of the domestic level of income. Apart from the fact that exports are considered autonomous or exogenous, they are also considered as injection into the circular flow of income.
This has to do with the fact that goods sold abroad increase domestic incomes. The gross imports on the other hand are the goods and services purchased from other economies by domestic residents. It may be solely exogenous in the sense that if national income were zero, a country would still be importing some goods, for it would have to live by dissolving and by using any accumulated foreign exchange it might have for imports. On the other hand, gross imports may consist of the exogenous component and the component that depends on the level of income of domestic residents. It would appear that the level of income of domestic residents is the only determinant of imports. This is not really the case.
There are other determinants like relative prices of similar, related products – prices at home versus those abroad; tastes, foreign exchanges, commercial policies, political and ideological concerns, cultural and traditional elements among others.
It is because these other determinants of imports as mentioned are assumed to remain unchanged in the short run together with the idea that many of them can be subsumed under the level of income variable that this variable is taken as the only determinant of imports in the equilibrium national income analysis. By bringing together the exogenous component and the induced component of imports results in import function of the form:
M = Mo + mY
Where, M. is autonomous (exogenous) imports; m is the marginal propensity to import (the proportion of a change in income that affects the volume of imports). The autonomous (exogenous) and induced components of imports notwithstanding, imports are a leakage from the circular flow of income. This is borne out of the fact that incomes earned from production of imported goods accrue to foreigners, and the sales receipts do not return to domestic producers but flow abroad.
Thus, in order to extend the model of income determination from closed economy to open economy, it is now necessary to add the influence of exports and imports as net exports.
Recall that, the equilibrium condition for national income determination in a closed (three sector) economy following the aggregate supply approach is given by:
Y=C+I+G
Where, C, I, and G are components of aggregate demand; each component is defined as:
C = Personal consumption expenditure
I = Private investment expenditure
G = Government expenditure.
Adding the influence of exports and imports as net exports into the model:
Y=C+I+Gwe have:
Y=C+I+G+ (X-M)
Where C, I and G are as defined earlier. X represents gross export and M represents gross import. It should be noted that, the influence of foreign trade in determining aggregate demand is brought out by the term (X – M) indicating net export or net foreign expenditures.
The model: Y=C+I+G+ (X-M), is the open economy (four sector) model of equilibrium national income determination through the aggregate demand and aggregate supply approach. Retaining the previous idea of a linear consumption function of the form:
C = a + bYd together with the assumptions of an autonomously determined investment given by 1 = lo; exogenously determined government expenditure and net taxation given by: G = Go and T = To respectively; exogenous or autonomous gross export and gross import given by X = X, and M = M, respectively; the equilibrium level of national income for an open economy (four sector) can be determined through aggregate demand/aggregate supply approach by the algebraic method involving the substitution of the relevant terms and collection of terms as follows:
Y=C+I+G+ (X-M)
Y = a + bYa + Io + Go + Xo – Mo
Since Ya = Y-T and T = To we have:
Y = a + b(Y-To) + Io + Go + Xo – Mo
Y = a + bY – bTo + Io + Go + Xo – Mo
Y-bY=a-bTo + Io + Go + Xo – Mo
Y (1 – b) = a – bTo + Io + Go + Xo – Mo
y = a−bT0 + 1₂ +G₁ + Xo – Mo_1 (a-bT + Io + Go + Xo – Mo)
The last expression above gives the equilibrium level of national income of an open economy (four sector economy) under the idea of a linear consumption function and the assumption that investment, government expenditure, imports and exports are all autonomous.
The aggregate demand and aggregate supply approach was adopted in arriving at this equilibrium level of national income expression. Alternatively, the same equilibrium level of national income can be arrived at through the leakages equaling injections approach.
The leakages in the open economy (four sector economy) case are saving (S), taxation (T), and imports (M) while the injections are investment (1), government expenditure (G), and exports (X). With these leakages and injections the condition for equilibrium is given by:
S+T+M=I+G+X
Since S=Yd-C and taking into consideration that C = a + bYd; I = lo;
G = Go; X = Xo; M = Mo; T = To;
the equilibrium condition equation above can be written as:
Ya-C+To+ Mo= To + Go + Xo
Ya- (a + bYd) +To+ Mo = Io + Go + Xo
Ya-a-bYd+To+ Mo = lo + Go + Xo
But Yd = Y-T and T = To; therefore:
Y-To-a-b(Y-To) + To + Mo = lo + Go + Xo
Y-To-a-bY+bTo+To+ Mo = Io + Go + Xo
Solving for Y we have:
Y-bY= a-bTo+la+ Go + Xo – Mo
Y(1-b) = a-bTo + lo + Go + Xo – Ma
ya-bT+1+ Go + X-Me
1-b (a-bT+lo+Go+ X-M₂)
The above expression is the same equilibrium level of income arrived at through the aggregate demand and aggregate supply approach.
Notice that in the above determination of equilibrium level of income for an open four sector economy the exogenous component of imports was considered without the induced component. Thus, in what follows, we would consider the exogenous component together with the induced component depicted by the import function previously given as:
M = Mo + mY
Retaining the structural equations presented previously as:
Y=C+I+G+ (X-M)
C = a + bYa
1 = lo
G = Go
X = Xo
T = To
Yd = Y-T; and replacing M = Mo with
M = Mo + mY
Then the equilibrium level of national income for an open four sector economy in which the imports is not only exogenous but also have an induced component can be determined. In this case, we use the aggregate demand/aggregate supply approach. Following the aggregate demand/aggregate supply approach, we make the relevant substitution into:
Y=C+I+G+ (X-M) to have:
Y = a + bYa + lo + Go + Xo – (Mo + mY)
Y = a + b(Y-To) + lo + Go + Xo-Mo-my)
Y = a + bY – bTo + lo + Go + Xo-Mo- my)
Y-bY + mY = a- bTo+ Io + Go + Xo – Mo
Y = a-bTo + Io + Go + Xo – Mo