Evaluating Economic Policy While Considering Worker’s Price Expectations: Another Application of Matrix Algebra

This posts develops a model of the macroeconomy in matrix algebra form.  The model consist of  markets for goods, money, and labor in addition to a general economic production function.  In accordance with the adaptive expectations hypothesis, asymmetrical information about the general price level in the economy will be available to producers and consumers.   More specifically, the model derived here assumes that workers supply their labor by considering the expected future price level, while producers decide how many workers to hire based on correct price expectations.


The Goods Market

The goods market will consist of demand and supply of output.  The demand for output will be denoted by consumption (C), investment (I), and government spending (G).  The supply of output is represented by the letter Y and will be determined by a production function.  The nature of this production function will be described in the production function section of the model.

Consumption is a function of output (aka income), lump sum taxes, and output based taxes.  Investments are a function of the real interest rate and government spending in completely exogenous, or determined outside of the model.

In order to understand the comparative statistics of the goods market, and the entire model for that matter, the total differential of every equation will be derived.  The total differential of the goods market is:

After taking the first differential and moving the endogenous and exogenous variables to the left and right hand side respectively.  The third equation represents the total differential of the good market equation.  The total differential of the production function and the other markets in this model will be derived in a similar way.

The Money Market

The money market equilibrium consist of the intersection between supply and demand for money.  The supply of money is completely exogenous to the model and will be considered in real terms be accounting for the general price level in the economy. The demand for money is a function of output (aka income) and real interest rates.

Taking the total differential and separating the endogenous from exogenous variables yields the equation that will be part of the matrix system.

The Production Function

The production of all goods in the economy will be a function of labor and capital.  Labor and capital are both homogeneous inputs and their derivatives with respect to the output function are both positive.  The total differential of the production function is

The Labor Market

The labor market consist of a supply and demand function of labor.  The supply of labor is given by the workers in the labor market and firms demand their services.  The interactions of both supply and demand yield the equilibrium quantity of labor and the market wage.

Workers supply their labor based on a comparison between the nominal wage and the expected price level.  It is assumed that producers have more information about market conditions so their labor demand decisions are based on the current price level. Workers expectations of future prices depend on current prices.  Their expectations and the total differential of their expectations is as follows…

This result will be an important substitute into the general labor market equilibrium and total differential

Using the result about the total differential about workers price expectations we arrive the following equation after combining like terms…


Once the model is in matrix notation it is fairly straight forward to use Cramer’s to solve for different the system.  The Jacobian of the endogenous matrix, that containing the output, real interest rate, price level, and labor market equilibrium coefficients is less than zero. This can be verified by taking the determinant of the first matrix above. The Jacobian will be denoted by |J| from this point forward.


Expansion in Government Expenditure and Output

An increase in government expenditure can increase the total output in the economy if labor demand does not equal labor supply and workers have incorrect price expectations.  If either one of these conditions aren’t meet than an expansion of government expenditure will not have an impact of total output.

Increase in the Money Supply and Interest Rates

The only ambiguous term in this equation is the 3rd term.  Assuming that the demand of labor equals the supply ( f = g) and that the ratio of the derivative of the expected price level and the actual price level is between zero and one we get that an increase in the money supply decrease the real interest rates. If price expectations are correct then monetary policy will be ineffective else increase in the money supply decrease interest rates.


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