The IS/LM Model

List of parameters:

a: autonomous consumption

b: MPC out of disposable income

c: autonomous investment

d: interest responsiveness of investment

e: income responsiveness of investment

f: income responsiveness of money demand

g: interest responsiveness of money demand

List of variables:

C: Personal Consumption Expenditures

G: Government Consumption Expenditures and Gross Investment

I: Gross Private Domestic Demand

L: Demand for Real Balances

M: Nominal Money Supply

P: Price Level

t: Marginal tax Rate

TE: Total Expenditures

TR: Transfer Payments

Y: Gross Domestic Product

A. The Goods Market

equilibrium condition: Y = TE

This gives us:

which is solved for Y:

The above equation is the equation for the IS curve: The various combinations of i and Y for which the goods market is in equilibrium.

The IS equation has 2 unknowns: Y and i. To solve it, we need another equation in the same 2 unknowns. That equation comes from the money market.

B. The Money Market

equilibrium condition: L = M/P

This gives us:

This is the equation for the LM curve: the various combinations of i and Y for which the money market is in equilibrium.

The LM equation also has 2 unknowns: i and Y. Thus, we can use the IS and LM equations together to solve for the equilibrium level of output (Y) and the equilibrium interest rate (i) in the economy.

C. Internal Equilibrium: IS/LM

equilibrium condition: simultaneous equilibrium in both the goods and the money markets. This means, Y=TE and L=M/P.

Simultaneously solving the IS and LM equations together involves a lot of algebra, but is straightforward. Various methods can be used to arrive at the same result; I prefer to substitute the LM equation into the IS equation and solve for Y, then take that value and put it into the LM equation to solve for i.

This is an equation with only one unknown (Y). Once the values of the parameters are placed in the above equation, the resulting value of Y (equilibrium Y) can then be placed into the equation of the LM curve to give the equilibrium value of i.

D. Multipliers

The derivation of multipliers is exactly the same as for the previous model(s): totally differentiate the equation(s), discard the unchanged exogenous variables, and solve for the changes in Y and i. Note, that because there are now 2 endogenous variables (Y and i.), there will be two "change" equations that will need to be solved simultaneously. Before attempting to derive the desired multiplier, it is often helpful to graph the problem. It is also very important to keep track of the market being affected by the exogenous change (the goods vs. the money market).

As an example, assume there’s an increase in G. Since G (Government consumption expenditures and gross investment) affects the goods market, we know that the increase in G causes an increase in TE (i.e., an increase in spending), which causes an increase in Y (i.e., an increase in output). The IS curve shifts (parallel) right. As Y increases, there’s an increase in money demand (money is a normal good) so L increases. As L increases, the interest rate is bid up to achieve equilibrium in the money market. This higher interest rate leads to a decrease in investment, partially offsetting the original increase in Y that was due to the change in G. The final equilibrium point is at C. Here's the graph for a change in G.

The math, to figure the changes in Y and i from this change in G, is tedious but fairly straightforward. We again start with the 2 equations that were used just prior to the derivation of the equilibrium conditions:

This equation has 2 unknowns: the change in Y and the change in i. It cannot be solved by itself. Fortunately, we also have the equilibrium condition in the money market:

This is again an equation in 2 unknowns, which can be combined with the previously derived equation to solve for the 2 unknowns. Again, my preference is to put the money market equation into the goods market equation, but there are many alternatives ways of accomplishing the same task. Solving for gives us: