Assignment 3:

A IS-LM model:

We consider a economic system, where our GDP, which is given by:

Z = C(Y T) + G + I(r),

C(Y T) = C0 + C1(Y T),

I(r) = I0 I1r,

Where Z is planned expenses, our Y is our GDP, T our taxes, G is our government purchases, I is our investments, r is interest rates.

How do you deduce the IS Curve, i.e. deduce the Y as a function of r, G, and T?

Y = C(Y - T) + I(r) + G + NX(e)

>Y= a+b(YT)]+(cdr)+G+NX(e)

C(Y - T) = a + b (Y - T)

I(r) = c -dr

Explanation:

The goods market includes Net exports

Y= aggregate income

b= Marginal Propensity to Consume

d= Interest rate sensitivity of Investment

a= Autonomous Consumption

c= Autonomous Investment

e= exchange rate

The government spending multiplier dY/dG is:

dY/dG =1 /(1-b)

It is greater than 1 because 0 < b < 1 that makes the denominator less than the numerator.

The tax multiplier dY/dT is:

dY/dT = -b/(1-b)

The value of |dY/dT| < |dY/dG| because the numerator of the tax multiplier is smaller, given the same denominators.

Explanation:

Consider the following equation; to find the equation for the IS curve, deduce the equation to make Y as a function of r, G, and T.

Y = C(Y - T) + I(r) + G + NX(e)

Y = a + b(Y - T) + c - dr + G + NX(e) .... (1)

Here,

C(Y - T) =a + b(Y - T)

I(r) = c - dr

Solve equation (1) to find Y in terms of r, G, and T.

Y = a + bY - bT + c - dr + G + NX(e)

(1 - b)Y = a - bT + c - dr + G + NX(e) ... (2)

To find the multiplier dY/dG, differentiate equation (2) with respect to G.

dY/dG = 1/(1-b)

The value of the multiplier is 1/(1-b). Here, b is the marginal propensity to consume (MPC) which is initially represented as C1 in the question. It is given that 0 < C1 < 1, that is, 0 < b < 1. This implies that 0 < 1-b < 1. Therefore, the numerator in the multiplier is greater than the denominator which means that the value of the multiplier will be greater than 1.

Now, find the multiplier dY/dT by differentiating equation (2) with respect to T.

dY/dT = -b/(1-b)

The absolute value of tax multiplier, dY/dT is:

|dY/dT| = |-b/(1-b)| = b/(1-b)

Now, compare the value of the tax multiplier with the government spending multiplier dY/dG. The denominator of both the multipliers is the same. The numerator of tax multiplier is b which is less than 1. This implies that the numerator of dY/dT is less than the numerator of dY/dG.

b/(1-b) < 1/(1-b)

In other words, |dY/dT| < |dY/dG|

We are adding the following equations to our model:

M^d (Y, r) = M0 + M1Y M2r

Md = M P ,

Where M and P is exogenous variables and M0, M1, M2 > 0. and Md is money / liquidity savings in the economy.

M1 and M2 both are positive as they include various modes of money and transactions that occur in daily life.M1 includes cash, checkable deposits, and traveler's checks. On the other hand, the M2 includes savings and time deposits, certificates of deposits, and money market funds in addition to M1.

The Lm curve equation is the following:

Y=M1PMM1M0+M1M2r

The equation of money demand is:

Md (Y, r) = M0 + M1Y M2r

In the money market, at equilibrium money demand is equal to the money demand.

So, from the given equations, we can derive the Y as the function of r and M/P.

Md=PMM0+M1YM2r=PMM1Y=PMM0+M2rY=M11(PMM0+M2r)Y=M1PMM1M0+M1M2r

The above equation of Y as the function of r and M/P represents the LM curve equation.

The new question is now:

Do not take r as exogenous anymore. There is a negative thrust in the economy: show the expectations for the future to get people to save more and lower their expenditures. We are modeling that by removing C0

Find out how much the centralbank shall increase the money supply with to compensate for the fallen C0 ie. Find dM/dC0 under the condition that GDP is unchanged?