The consumption function
Consumption C(Y) depends positively on GDP in the cross model
Remember that in the classical model, consumption depends on the real interest rate. In the cross model it depends on GDP. Note that it is not possible to include r in the cross model as it is fixed. However, we need to justify the dependence of C on Y.
Consumption and GDP
At first, it might seem obvious that consumption will depend on Y. If GDP is doubled in real terms over a number of years, private consumption, government consumption and investment will also each roughly be doubled. If you draw a graph of GDP and consumption over time you see that consumption does grow by about the same rate as GDP.
However, from this reasoning, we cannot conclude that C depends on the Y because growth has been removed from our variables C and Y. We need to think of Y as a variable that varies over time around some average. Sometimes it is above the average and sometimes it is below the average but there is now upward trend. The same is true for C.
The crucial question then is whether consumption is above its average in periods when GDP is above its average and vise versa (technically, if the detrended variables are correlated over time). Keynes would have said yes, while classics would have said no.
Keynes’ motivation: In good times, when Y is high (above its trend), national income is high (above it trend). Consumers will take the opportunity to buy things they otherwise cannot afford. In bad times, on the other hand, consumers simply cannot buy things they would have bought if income was higher.
The classical motivation: Consumers want to smooth their consumption over time. In good times, consumers know that this is a temporary state. Instead of increasing consumption, they save and use their savings in bad times.
Fig. 11.1: Classical and Keynesian consumption function.
The rest of the world in the cross model
Imports Im(Y) depends positively on Y in the cross model
In the classical model, imports does not depend on Y. The discussion whether imports depends on Y or not is the same as for consumption. However, in the cross model, it is always assumed that when Y increases, consumption will increase by more than imports. This makes sense since C is usually larger than Y. For example, suppose that C is 1000 while Im is 100 and that Y increases by 10%. If C and Im increase by 5% each, C will increase by 50 while Im will increases by only 5.
Net exports NX = X - Im will depend negatively on the Y and rest of the world savings SR = Im -X depends positively on Y in the cross model. If we want to be explicit about these dependences we write:
The government in the cross model
Net taxes NT(Y) depends positively on real GDP in the cross model
In this model, when national income increases, the amount individuals pay in income taxes will increase. This is because income tax is specified as a percentage of total income. Other taxes may also increase when Y increases. However, government transfers to households will decrease. Therefore, net taxes NT will increase when Y increases.
Even though NT depends on Y, is still under the control of the government. NT may change even if Y does not change. This means that NT is part exogenous (as it may be controlled by the government) and part endogenous (as it will automatically change when Y changes). Therefore, we write NT(Y) but we must remember the exogenous nature of net taxes. Government savings, which is also part endogenous and part exogenous, depends positively on Y and we write:
Household savings SH(Y) and total savings S(Y) depend positively on Y
Household savings depends on Y because SH = Y - C - NT and C and NT both depend on Y. How it depends on Y cannot be conclusively be determined from this relationship as C and NT both depends positively on Y. We always assume that this dependence is positive and the following example illustrates why this assumption makes sense.
Suppose that NT = t- Y where tis a constant between 0 and 1.1is the proportion of income that we pay in taxes. Next, suppose that C = c- Yd where c is a constant between 0 and 1. c is proportion of disposable income that we use for consumption. If income Y increases by 1, NT increase by t, disposable income increases by 1 - t and C increases by c(1 - t). Thus, SH increases by 1 - c(1 - t) - t = (1 - c)(1 - t) > 0.
Since S = SH + SG + SR and all parts on the right hand side depends positively on Y, total saving S will depend on positive Y and we write S(Y) for total savings (net total supply of savings).
Aggregate demand in the cross model
Since C and Im depends positively on Y while G, I and X are exogenous, aggregate demand YD will depend positively on Y:
When Y increases, C and Im increases but since C increases more than Im, aggregate demand will increase when Y increases.
You may react to the the notation YD(Y). But if you think of Y as the national income (GDP = national income) then YD(Y) simply tells us that aggregate demand depends on income. Aggregate demand is the total quantity of finished goods and services that all sectors (consumers, firms, government and the rest of the world) together wish to buy under different conditions. The notation YD(Y) tells us that the only endogenous variable that affects aggregate demand is national income. The higher the income, the more we wish to buy. YD, C, Im, S, SH, SG, SR and NT all depend on Y while I, G and X are exogenous. We can illustrate this using the following diagrams.
Fig. 11.2: Aggregate demand and its components.
Each diagram has real GDP on the x-axis.
• The first diagram shows exports (X), imports (Im), net exports (NX) and rest of the world savings (SR). In this diagram, X = 1.3 and Im = 0.56 + 0.2 Y.
• The second diagram shows private consumption (C), investment (I ), government spending
(G), net exports (NX) and aggregate demand (YD = C + I + G + NX). Here, C = 0.22 + 0.4Y,
I = 0.5, G = 0.7.
• The third diagram shows private savings (SH), public savings (SG), the rest of the world savings (SR) and the total savings (S = SH + SG + SR). They are created from NT = 0.26/.
This diagram summarizes all variables in the cross model and how they depend on Y. Actually, these dependences will be the same in all of the Keynesian models.