Economics 813a 11/07/2018
1997 R.H. Rasche
V. Structure of Prototype Keynesian Model
The Keynesian Model as presented in Sargent consists of eight equations in eight endogenous variables (Y, N, P, C, q, r, I and Yd) and six independent exogenous variables (K, T, G, , w and M). K is considered an exogenous variable since the analysis is completely in terms of impact multipliers and K is assumed to be fixed at a point in time. Note that B is not an independent exogenous variable, since the value of B is totally determined given the government budget constraint and the open market operation constraint (dB = - dM). The equations of the model are:
1) w/P = FN(N,K)
2) Y = F(N,K)
3) I = I(q-1)
4) C = C(Yd, r-)
5) Y = C + I + G + K
6) q = q(K,N,r-,)
7) M/P = m(Y,r)
8) some definition of Yd.
The crucial difference between the prototype Classical model and the prototype Keynesian model is the specification of the labor market.
1) note that the production function and the labor demand specifications are the same in the two models.
2) in the prototype Keynesian model the labor supply function is assumed to be perfectly elastic at the prevailing nominal (money) wage rate. Alternatively we can state this as the nominal wage rate is an exogenous variable to the model.
Some kind of nominal wage rigidity is characteristic of all Keynesian, Post Keynesian, and New Keynesian models. The identifying feature of all of these models is that there is something inherent in the specification of the model that imposes some "stickiness" on nominal wages, so that real wages cannot adjust to maintain a constant level of employment in the face of changes in the exogenous variables on the demand side of the model. In Keynesian and Post-Keynesian models, the source of nominal wage “stickiness” is exogenous to the model. The thrust of New Keynesian analysis is to seek an economic rational (typically a microeconomic rationale) for the existence of such nominal wage “stickiness” and hence to define it as an endogenous part of the economic structure.
There are various definitions of disposable income that can be used:
1) we can use real perceived permanent disposable income (with or without M+B = 0) as used in the prototype Classical model. Then the only difference between the structure of the prototype Classical model and the prototype Keynesian model is the assumption about the labor supply curve and the endogenous or exogenous nature of the nominal wage rate.
2) we can use a current receipts definition of real disposable income. This is more in the spirit of Keynes, though not in the spirit of post-Keynesian analysis: Yd = Y - T - K.
Note that the marginal propensity to consume is a crucial feature of Keynesian models, so it doesn't make much sense to construct restricted versions of the model in which CY = 0 as we did in one case with the Classical model. A reasonable, and interesting, restricted case is to evaluate the model in the neighborhood of zero expected inflation ( = 0).
In order to examine the behavior of the reduced form of the Keynesian model in some small neighborhood of an initial solution, totally differentiate the above equations (including the current receipts definition of real disposable income):
1) dw/P - (w/P)(dP/P) = FNNdN + FNKdK
2) dY = FNdN + FKdK
3) dI = I'dq
4) dC = CYdYd + Crdr - Crd
5) dY = dC + dI + dG + dK
6) dq = qkdK + qNdN + qrdr - qrd
7) (1/P)dM - (M/P)(dP/P) = mrdr + mYdY
8) dYd = dY - dT - dK
Equations (1) and (2) typically combine to give an expression in two endogenous variables, dY and dP as a function of the exogenous variables dK and dw, which represents the aggregate supply curve of the economy: it indicates the changes in the price level that firms must receive in order to change the amount of output they are willing to produce for given levels of the capital stock and for given nominal wage rates.
Equations (2), (3)-(6) and (8) are usually combined into a single equation in two endogenous variables, dY and dr as functions of the exogenous variables, dK, d, dT and dG referred to as an IS curve. Note that (2) is required here only because of the relationship between q and N. With a more conventional investment demand equation: I = I(r-), I' < 0, then it would not be necesary to use (2) in the construction of the typical IS equation. Equation (7) is one equation in three endogenous variables, dY, dr, and dP/P as a function of the exogenous variable dM which is known as an LM curve. Note that this is the LM curve of the typical textbook model and is not the LM curve that Sargent constructs. Sargent eliminates from equation (7) dP/P using equations (1) and (2) to construct a relationship between dY and dr given the exogenous variables dM, dw and dK. In the conventional construction of an LM curve, which we use here, changes in the price level cause the LM curve to shift. In the LM curve constructed by Sargent, changes in P are subsumed into the slope of the LM curve.
When the model is reduced to three equations we obtain an aggregate supply curve of the form:
using the result that . It is important to note that the expression is just the reciprocal of the slope of the aggregate supply curve (plotted as lnP versus Y). The absolute value of the same term measures the horizontal shift of the aggregate supply curve in response to a one percent exogenous change in the nominal wage rate. Alternatively, the aggregate supply curve shifts up or down by one percent for a one percent change in w.
When equations (3)-(6) and the current receipts definition of real personal income are substituted into (5), the resulting expression is:
dY = CY[dY-dT-dK] + Crdr - Crd + I'[qKdK + qNdN + qrdr - qrd] + dG + dK.
But from (2): dN = (dY - FKdK)/FN so when all terms are collected the expression for the IS curve is:
IS: [1 - CY -I'qN/FN]dY + [-Cr -I'qr]dr = -CYdT + [-Cr -I'qr]d + dG +
[ - CY + I'qK - I'qNFK/FN]dK
Now put the aggregate supply equation (S), the IS equation and the LM equation together as a three equation system and compute the reduced form:
The determinant of the A0 matrix of this system is:
det = mY[Cr+I'qr]FN2/FNN + mr[1-CY-I'qN/FN]FN2/FNN - [M/P][Cr+I'qN]
= mr[FN2/FNN][{1-CY -I'qN/FN} - {Cr+I'qr}{(M/P)FNN/FN2 - my}/mr] > 0.
The reduced form of this system is:
or
The impact multiplier for real output with respect to a change in government purchases in this model is:
and
Recall that is the slope of the aggregate supply curve (plotted lnP versus Y), so where (P,Y|AS) is the elasticity of the aggregate supply curve. With these definitions:
The question is what are the economic mechanisms that generate this messy expression? First note that the term 1/{1-CY -I'qN/FN} measures the amount of the horizonal shift in the IS curve per unit change in government purchases. This differs from the simple 1/{1-CY} because in this model changes in real output and the accompanying changes in employment produce changes in q.
Now look at the LM curve (7):
mrdr = -[M/P]dP/P - mYdY + [1/P]dM
or:
dr = -[(M/P)dP/P + mYdY]/mr + (1/P)dM/mr.
As long as M is held constant, -[(M/P)dP/P + mYdY]/mr measures the amount that r must change to maintain equality between the demand for real balances and the supply of real balances when Y changes and the economy moves along the positively sloped aggregate supply curve. Since the slope of the aggregate supply curve is -FNN/[FN] 2. the amount that r must change to maintain portfolio equilibrium when real output changes by one unit is:
Therefore the second term in the denominator of dY/dG,
{Cr+I'qr}{(M/P)FNN/FN2 - my}/mr < 0
measures the amount that private aggregate demand is reduced by the changes in interest rates that are required to maintain portfolio equilibrium when real output increases by one unit with the corresponding price increase required to keep the economy on the positively sloped aggregate supply curve. The larger the absolute value of this term, the larger the denominator of the government expenditures multiplier, so the smaller the impact of changes in government expenditures on real output. This represents the source of "crowding out".
Now we can think of this in terms of a standard IS/LM diagram. When government expenditures change, the IS curve is shifted either to the right or the left. As output changes, the economy moves along a positively sloped aggregate supply curve, and the price level rises or falls as real output increases or decreases. Such changes in the price level cause the LM curve to shift up or down. The IS and LM equations can be represented as:
IS: AdY + Fdr = dG
LM: CdY + Ddr = E(dP/P)= EdY
where represents the slope of the aggregate supply curve. This equation
system can be solved for dY and dr to get:
or
dY =
In our prototype Keynesian model, the terms in the above expression are:
A = 1 - CY - I'qN/FN; F = -[Cr + I'qr]
C = mY; D = mr; E = -[M/P].
Other interesting multipliers for this prototype Keynesian model are:
the response of all the variables in the system to a simltaneous change in the nominal money stock and the nominal wage rate in the same proportions: i.e. an exogenous shock in which . Is Why?
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