Joe Cobb

Mathematical Macroeconomics

The System of Six Equations with Seven Unknowns
Macroeconomic Equilibrium would occur if all of these equations were satisfied

THE SEVEN UNKNOWNS
Y = income
C = consumption
I = investment
r = the interest rate
Md = the demand for money
Ms = the supply of money
P = the price level

The Six Equations

NOTE: Dividing nominal values by the price level ( 1/P ) gives the real economic values: the average level of all prices increases when the value of the dollar diminishes.

EQUATION (1)
I/P = ( Y/P – C/P ) and
S/P = ( Y/P – C/P ) [ S = savings ]
Real income is either consumed or saved. Total investment and total savings are the same thing. Anything not consumed is either intentionally or unintentionally invested.

EQUATION (2)
I/P = f( r )
Real investment (and saving) is a function of the interest rate. Investors look at rates of return.

EQUATION (3)
C/P = f( Y/P , r )
Real consumption is a function of real disposable income and the interest rate. You cannot consume what you don’t have, and you choose to save (or borrow) part of it based on yield.

EQUATION (4)
Md = P * f( Y/P , r )
The demand for money is determined by the price level multiplied by a function of real income and the interest rate. Size of the “dollar” and your other wealth influence how many dollars you want to have readily at hand, as well as your investment alternatives that give a higher yield than holding cash.

EQUATION (5)
Ms = f( r )
The supply of money is determined by the interest rate. Fractional reserve banks create money by originating loans. The money multiplier is endogenously determined by loan demand plus banks’ prudential reserves. Reserves are fixed amounts of “outside” (reserve) money.

EQUATION (6)
Md = Ms
The demand for money equals the supply of money. Otherwise you have inflation or deflation, which is a disequilibrium that produces instability.