Technical Appendix
(This is an optional section that takes the reader well
beyond what is traditional in introductory economics.)
The simple form of ISLM model can be presented with basic
algebra. The only change in the incomeexpenditure model
equations is in the investment equation, which becomes:
(1) I = d + fYe + hi.
In this equation d is the intercept, f tells by how much
investment will rise when expected income rises by $1.00, h
tells by how much investment changes when interest rates
change, and i is the interest rate. h should be a negative
number. Higher interest rates yield lower investment.
Our modified incomeexpenditure model now has these
equations:
(2) C = a + b(YeT)
(3) G = g
(4) I = d + fYe + hi
(5) T = t
In equilibrium expected income equals actual income, and
actual income is always equal to actual expenditures,
or:
(6) Ye = C + I + G
Substituting equations 25 into 6 gives:
(7) Ye = a + b(Yet) + d + fYe + hi + g.
Rearranging terms and solving for Ye gives:
(8) Ye = (1/(1bf))(a  bt + d + g + hi).
Because equation 8 has two unknowns, the interest rate
and expected income, it is the equation for a line. Assuming
that the multiplier is positive and h is negative, the line
will have a negative slope. The line is, of course, the IS
curve. However the model is now incomplete since it does not
have a unique solution for income. There are many
equilibrium incomes, one for each level of interest
rate.
Adding in the monetary sector completes the model. The
demand for money equation is
(9) Md = kYe + wi.
The assumption that money stock is determined by policy
gives:
(10) Ms = m.
The equilibrium condition that money supply equals money
demand is:
(11) Ms = Md.
Solving equations 9, 10, and 11 for Ye gives:
(12) m = kYe + wi
or, rearranging terms,
(13) Ye = m/k  (w/k)i.
This equation with two unknowns is also the equation for
a line, the LM curve. Since w is negative, the line will
have a positive slope. The steepness will depend on how
large w is relative to k. If w, which is the responsiveness
of money demand to interest rates, is large, then the slope
of the LM curve will be flatter. (Note that interest rate is
on the vertical axis.) A small change in interest rates (the
rise) will yield a large change in Ye (the run). If k is
large relative to w, the slope will be steeper. In the
extreme case of w/k equaling zero, the LM curve becomes a
vertical line.
Note that one could solve equations 1 to 11 directly for
equilibrium income. While easier algebraically, this method
does not yield the beautiful curves that look just like
supply and demand curves that economists enjoy so much.
Copyright
Robert Schenk
