# 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 income-expenditure 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 income-expenditure model now has these equations:

(2) C = a + b(Ye-T)

(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 2-5 into 6 gives:

(7) Ye = a + b(Ye-t) + d + fYe + hi + g.

Rearranging terms and solving for Ye gives:

(8) Ye = (1/(1-b-f))(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.