A Canonical Form for Multipliers For

Ec 813a 12/21/2018

 1998 R. H. Rasche

VI. A “Canonical form” for Reduced Form Multipliers of Macroeconomic Models

Impact multipliers can always be computed through matrix inversion of a linear or linearized macroeconomic model. Unfortunately after the relevant matrix inversion and multiplication, the result may not provide any intuition about how the model works, nor of the signs of the impact multipliers. Equally frustrating, matrix operations in computer programs like Mathematica can result in long expressions that are nearly impossible to sign or interpret. However, for a broad class of macro models it is possible to generate the multipliers in a “canonical form”, from which it is straightforward to determine the sufficient conditions to sign the various impact multipliers.

Consider a generic macroeconomic model that has either exogenous expected inflation or backward looking inflation expectations.

Systematically reduce the dimension of the model to three equations in three endogenous variables: dY, dP/P, and d(r-), where Y = real output, P = the price level,

r- = the real rate of interest. This can be done by substitution for other endogenous variables to create

1) an aggregate supply curve: collapse the labor demand labor supply curve and the aggregate production function to eliminate employment and the real wage rate as endogenous variables,

2) an “IS” curve: substitute equations for components of aggregated demand into the output market equilibrium equation, using whatever additional equations are necessary to reduce this to an equation in the three endogenous variables,

3) a “LM” curve: substitute the demand for real balances and the money supply equation into the asset market equilibrium condition and use the definition of the nominal interest rate as the real interest rate plus the expected rate of inflation to reduce this to an equation in the three endogenous variables.

Check that the aggregate supply curve involves only dY and dP/P (i.e. the nominal and/or real interest rate does not enter into this equation). If this restriction is not satisfied STOP! The following algorithm will not work for models that do not satisfy this restriction.

This class of model can be written in matrix form as:

where G is real government purchases and X1, ..., Xn-1, are n-1 other exogenous variables, and Bj, j = 1, ..., n are 3x1 vectors. Let the first equation in this representation be the aggregate supply curve, the second equation the “IS” curve and the third equation the “LM” curve. Note that A12 = 0 for models with a perfectly inelastic instantaneous aggregate supply curve. A22 = 0 for models in which the price level (or inflation rate) does not enter the “IS” curve. The determinant of the A matrix is

det = (A11A22A33 + A12A23A31 - A11A32A23 - A21A12A33) = A11[A22A33 - A32A23] + A12[A23A31 -A21A33]

The adjoint matrix is:

Assume that . Then the impact multiplier .

Case I: A12,A33  0 (The instantaneous aggregate supply is not perfectly inelastic and the instantaneous demand for real balances is interest elastic). Divide the determinant and the adjoint matrix by -A12A33.

(-A12A33)-1det = =

The sign of Z must be > 0 for the model to have sensible economic properties. This can be seen by considering the impact multiplier for Y with respect to G:

If the aggregate supply curve is normalized to A11 > 0, then A12 < 0 if the instantaneous aggregate supply curve is positively sloped. Thus the sign of the determinant must be the same as the sign of A33 for a meaningful macroeconomic model.

The scaled adjoint matrix is:

(-A12A33)-1adjoint ==

where Ck are 1x3 vectors. The impact multipliers are:

, j = 1, ..., n-1

and so to sign the impact multipliers all that is required is to sign CkBj+1, k = 1, .. 3; j = 1, ... n-1

Case II: A12 = 0; A33 0. (A perfectly inelastic aggregate supply curve; interest elastic demand for real balances). In such a model, a change in G must produce a change in P of the same sign as the economy moves along the vertical aggregate supply curve.

where W > 0.

But = the determinant of the 2x2 submatrix in the lower right hand corner of A. Since dY can always be assigned a positive coefficient in the aggregate supply curve, the sign of the determinant of A = A11[A33W] must have the same sign as A33 for a meaningful macroeconomic model with a vertical aggregate supply curve.

Further, since W > 0, the determinant of must have the same sign as A33 which is the determinant of the 1x1 submatrix in the lower right hand corner of A. Therefore this determinant must have the same sign as the determinant of A. From these properties it follows that the A matrix must be positive definite for this class of models to have an equilibrium.

Divide the adjoint matrix by A11A33:

where are 1x3 vectors. The impact multipliers are:

, j = 1, ..., n-1

and so to sign the impact multipliers all that is required is to sign Bj+1, k = 1, .. 3; j = 1, ... n-1

Case III: A12 = A33 = 0.0 (A perfectly inelastic aggregate supply curve and zero interest elasticity of the demand for real balances). In such a model, a change in G must produce a change in r- in the same direction assuming that private aggregate demand is negatively related to the real interest rate (A21, A23 > 0). Divide the determinant of A by –A11A32:

Divide the adjoint matrix by –A11A32:

where are 1x3 vectors. The impact multipliers are:

, j = 1, ..., n-1

and so to sign the impact multipliers all that is required is to sign Bj+1, k = 1, .. 3; j = 1, ... n-1