THW's ISLM model

San Jose State University
Economics Department
Thayer Watkins

The ISLM Model

This is a presentation of a form of the ISLM model. The original ISLM model was developed by the American economist Alvin Hansen and the British economist John Hicks to explain the simultaneous determination of interest rates and national output. This model differs from the original in the inclusion of an equation representing a balance of payments equilibrium. It also differs in that investment depends upon output as well as the real interest rate.

The Model

Definitions of Symbols

Y = Gross Domesic Product GDI
C = Consumer Expenditures
I = Intended Investment Purchases
R = Real Interest Rate
G = Government Purchases
X = Exports
M = Imports
Y_D = Disposable Income
T = Taxes
Tr = Transfer Payments
M_S = Money Supply
NFFI = Net Foreign Financial Investment into the U.S.

The Equations

Y = C + I + G + X-M (Equilibrium condition)
C = a + bY_D (Consumption function)
Y_D = Y - T + Tr (Disposable Income determination)
T = T₀ + tY (Tax function)
I = I₀ - gR + hY (Investment function)
M_S = qY + L₀ - jR (Money market equilibrium)
X-M + NFFI = 0 (Balance of Payments equilibrium)
NFFI = NFFI₀ + sR (Net foreign financial investment function)

The Derivations

A Graphical Illustration of the Derivation of the IS Curve
The Scrollbar Slider at the top of display set the level of the real interest rate R, which determines the level of the interest-rate-sensitive demand and that determines the equilibrium level of Y. The IS curve is the levels of equilibrium Y plotted versus the level of the interest rate R.
Because of the problem of being able to view the creation of the IS curve at the bottom of the display while controlling the interest rate with the slider at the top of the display there is a duplicate display following the first one which has the interest rate scrollbar slider at the bottom of the display.

The Algebraic Derivation of the IS Curve

Y_D = Y - T + Tr = Y - T₀ - tY + Tr
Y_D = (1-t)Y + Tr - T₀
C = C = a + bY_D = a + b(Tr - T₀) + b(1-t)Y
X-M = -NFFI = -NFFI₀ - sR
Y = C + I + G + X-M
Y = a + b(Tr - T₀) + b(1-t)Y + I₀ - gR + hR - NFFI₀ - sR
Y = K(a + b(Tr - T₀ + I₀) - NFFI₀) - K(g+s)R
where K = 1/(1-b(1-t) - h)

A Graphical Illustration of the Derivation of the LM curve The
Scrollbar Slider at the top of display set the level of output Y, which determines the level of demand for money. The demand for money and the supply of money determines the equilibrium level of R. The LM curve is the levels of equilibrium R plotted versus the level of the output Y.

The Algebraic Derivation of the LM Curve

M_S = qY + L₀ - jR (Money market equilibrium)
jR = qY + L₀ - M_S
R = (q/j)Y + (L₀ - M_S)/j
A Graphical Illustration of the Changes in the Equilibrium Output and Interest Rate as Result of Policy Changes