IS-LM Model

integrates real (Iinvestment Savings: IS) with money (Liquidity Money: LM).

IS: (Investment - Savings:REAL)


Variables:


Equations

Y = C + I + G       (1)
C = a + b(Y - T)       (2)
I = I0 - ci       (3)

 

Note: Equation 3 is the investment function which says that the higher the rate of interest the more attractive are risk free government securities versus risky private investments. Thus the amount of private investment decreases as the interest rate increases. To invest in a private project, the rate of return must equal the market interest rate plus a premium for the higher risk.

LM: (Liquidity - Money: Money)

Variables:


Equations:

Ms = Md       (4)
Md/p = eY-fi       (5)

 

Note: Equation (4) is the Keynesian money market equilibrium equation. The first term on the right of (5) is the transactions demand for money and the second term is the speculative demand for money. Individuals hold a portfolio of assets such as money, stocks, bonds, real estate, etc. The individual holds money in the portfolio to take advantage of opportunities which may present themselves. As the interest rate rises the opportunity cost of holding money increases and he or she shifts from money to other assets.

Solution to the IS-LM model

How to solve the IS model is in the notes. The IS curve is:
Y = k(A - ci)       (6)
where k = 1/(1 - b) and A = a + I0 + G - bT Important terms
The LM curve is obtained by substituting (4) into (5) to obtain:
Ms/p = eY-fi       (7)

To obtain the solution to the entire model rewrite (6) and (7) as
Y + kci = kA       (6¢)
eY - fi = Ms/p       (7¢)

Note: (6') and (7') are 2 equations in Y and i that can be solved by High School algebra. [Pain in the ass] Note: (6'==IS) contains fiscal policy G and T and (7'==LM) contains monetary policy Ms. See 1st graph below. Remember IS slopes down to right while LM slopes up to right.

Solution to IS-LM Model

Y = K[A + (c/f)Ms/p]       (8)

Equilibrium Graph

Example: Given G = 200; T = 150; Ms = 100; p = 1.0; a = 100; b = 2/3; I0 = 600; c = 2500; e = 0.25; and f = 1250 which implies K = 6/5 or 1.2 Note: K = k/[1 + (ekc/f)]; A = a + I0 + G - bT

What is the equilibrium Y? Use (8), definition for A and value given for K
Y = 1.2(100 + 600 + 200 - 100 + (2500/1250)100) = 1200

 

and we can derive the following D equations:

 

Fiscal Policy

DY = KDG and DY = -bKDT    (9)

Note: For the first fiscal model for exam 1, we had D equations for Da and DI. For this model there are similar equations for Da and DI0. If you are an A student, if is in your interests to figure out what these equations are? Do the involve K or k, that is the question.

Fiscal Policy Graph

Let us assume you are hungry for a big A, then you should conisder how Da and DI0 affect the solution. Do they affect the IS or the LM curve and how? Suppose to reduce unemployment the government desire to raise Y by 48 what is the required DG?
Use DY = KDG hence 48 = 1.2DG or DG = 40

Crowding Out: Impact on Private Investment

DI = -(ce/f)DY       (10)

In the graph above increasing G or decreasing T raises the interest rate and this affect the level of private investment.
DI = -cDi. Equation (10) above is obtained by manipulation to obtain DI as a function of DY. If G is increased by the amount in b above, how much I is crowded out (Assuming p remains constant)? We use formula DI = -(ce/f)DY ABOVE that was derived in the notes. DI = -(2500(.25)/1250)48 = -24

 

 

Monetary Policy Graph

Suppose p starts increasing due to either cost push (OPEC) or demand pull (Viet Nahm) war. The LM curve starts moving to the left and the interest rate starts rising. The head of the Fed knows that expanding the money supply under such conditions is like throwing gasoline on a fire, so why does he( someday she) do so. Once interest rates go above 20% there is a risk that many many businesses will go bankrupt and cause a major recession, if not a depression. Therefore expanding the money supply is the lessor of two evils.

 

Compensating Monetary Policy

D(Ms/p) = f
Kc
 (k -K) DG       (11)

Ideal Fiscal with Accomodating Monetary Policy




 

IV. Now suppose the real money supply Ms/p is increased to compensate the crowding out. What is DMs/p? We use D(Ms/p) = [f/ Kc](k -K) DG to obtain DMs/p = 30

Another problem for you to work out


IS:


LM:


Variables:


The solution to the IS model is:


The solution to the IS-LM model is:


Change equations:


Given

  1. Why is K < k? Explain either mathematically or economically? Is denominator greater than 1? Economically: Think crowding out!
  2. What is the equilibrium Y? Which equation do you use?
  3. To reduce unemployment the government desires to raise Y by 108. What is the required DG? What is an alternative fiscal instrument which could be used to achieve the same objective? What about DT? Remember with DT the change equation is slightly different than for DG?
  4. How much investment is crowded out? Which change equation do you use?
  5. Now suppose the real money supply Ms/p is increased to compensate the crowding out. What is DMs/p? Again, which change equation do you use?
  6. Suppose Y before increasing G is less than the full employment Y. If increasing G increases Y beyond the full employment Y what will happen? Will p remain stable?
  7. Return to part III above. What is the required DG if at the same time DI0 is -10. How does this affect the rest of the problem?

 

 

 

 

File translated from TEX by TTH, version 2.25.
On 31 Oct 1999, 11:27.

Revised: Tuesday, 6 Nov 01