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Keynesian Cross Model, Macroeconomics Models, Assignment Help, Homework Help
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The Keynesian Model
In this chapter we will look at the Keynesian cross model. This model is a simple version of what we call the "complete Keynesian model" or simply the Keynesian model. The Keynesian model has as its origin the writings of John Maynard Keynes in the 1930s, particularly the book "The general theory of Employment, Interest, and Money". Although this book was written as a criticism of the classical model, the similarities between the Keynesian model and the classical model are definitely greater than the differences. Lets point out the three most important differences directly:
• Say's Law does not apply in the Keynesian model.
• The quantity theory of money does not apply in the Keynesian model.
• The nominal wage level W is an exogenous variable in the Keynesian model.
Remember that W being exogenous means that it is predetermined outside the model. It does not necessarily mean that it is constant over time  even though this is a common assumption. However, the nominal wage must be known at any point in time in this model. To simplify our description of the Keynesian model, we will begin by assuming that W is constant.
The Keynesian model is slightly more complicated than the classic model, and it is developed in four stages by analyzing four separate models. Each model has, however, a value in itself. The models we will consider and the major characteristics of each are:
• Cross model: W, P and R are constant (and exogenous).
• ISLM model: W, P are constant and R is endogenous.
• ASAD model: W is constant, P and R are endogenous.
• The full Keynesian model: W is exogenous (but not constant), P and R are endogenous.
Once we have developed the full Keynesian model, we will combine it with the clasmodel which will lead to the neoclassical synthesis. The final chapter covers the MundellFleming model an extension of the neoclassical synthesis to an open economy where we also analyze the exchange rate.
Summary of the cross model
The following list summarizes the cross model and relates it to the classical model:
• Labor Market: The real wages W/P is exogenous in the cross model (W is exogenous in all the Keynesian models and P is exogenous in cross model). The determination of L is very different from the classical model, •
• Aggregate supply Ys is determined by the production function Y
_{S}
= f(L, K). Again, we always remove any trend in GDP and its components.
• Aggregate demand is not always equal to the aggregate supply. Say's Law does not apply in any of the Keynesian models. Therefore, we must describe how aggregate demand and GDP is determined in the cross model.
• The Quantity theory of money does not apply anymore. Fortunately, we don't need it since P is given in the cross model.
• Consumption was a function of the real interest rate in the classical model. In the cross model it is a function of Y.
• Investment was also a function of r in the classical model. In the Keynesian model it is exogenous.
• Government spending (G) is exogenous but the net tax NX is endogenous (in the classical model, they were both exogenous). Net tax is assumed to be a function of Y which means that government savings will be endogenous ( S
_{G}
(Y)= NT(Y)  G ).
• Exports (X) is exogenous, as it is in the classical model, but imports (1m) is endogenous.
Imports will also be a function of Y. Net imports and external savings will therefore also be endogenous variables (NX(Y) = X  Im(Y) and SR(Y) = Im(Y)  X).
• Household savings and total savings were functions of the real interest rate in the classical model. In the cross model they are functions of Y.
• The real interest rate is exogenous in cross model. This follows by the fact that the nominal interest rate is exogenous and prices are constant (u
^{e}
must be zero, and r = R).
We can divide our analysis of the cross model into three parts:
Aggregate demand. Aggregate demand is a major component of the cross model. The main purpose of this section is to arrive at the conclusion that aggregate demand depends on real GDP.
• Determination of GDP. GDP is determined very differently in the cross model compared to the classical model.
• Labor market. One of the main points of the Keynesian model is to allow for involuntary unemployment. In the classical model of the labor market, we are always in equilibrium and there is no involuntary unemployment.
Aggregate demand
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.
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:
NX(Y) = X  Im(Y)
SR(Y) = Im(Y)  X
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 controled 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:
S
_{G}
(Y) = NT(Y)  G
Savings
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 t is a constant between 0 and 1. t is 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 = S
_{H}
+ S
_{G}
+ S
_{R}
and all parts on the right hand side depends positively on Y, total saving Swill 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 demands Y
_{D}
will depend positively on Y:
Y
_{D}
(Y) = C(Y) + I+ G + X  Im(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 notation Y
_{D}
(y). But if you think of Y as the national income (GDP = national income) then Y
_{D}
(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 Y
_{D}
(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, S
_{H}
, S
_{G}
, S
_{R}
and NT all depend on Y while I, G and X are exogenous. We can illustrate this using the following diagrams.
Aggregate demand and its components
Each diagram has real GDP on the xaxis.
• 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 + 0.2Y.
• The second diagram shows private consumption (C), investment (1), government spending (G), net exports (NX) and aggregate demand (YD = C + 1 + G + NX). Here, C = 0.22 + 0.4 Y, I = 0.5, G= 0.7.
• The third diagram shows private savings (S
_{H}
), public savings (S
_{G}
), the rest of the world savings (S
_{R}
) and the total savings (S = S
_{H}
+ S
_{G}
+ S
_{R}
). They are created from NT = 0.26Y.
This diagram summarizes all varaiables in the cross model and how they depend on Y. Actually, these dependences will be the same in all of the Keynesian models.
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