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.

C0, C1, I0, I1 > 0 are all paramters and C1 < 1.

Our T and G is Exogenous variables. Also in this part of the assignment r is Exogenous

Now for the question:

Explain why our function: C(Y - T) looks the way it does and why 0 < C1 < 1 is a fair assumption?

C(Y-T) looks like the way it does because level of consumption depends on disposable income (Y-T), or the total personal income that they have after income taxes have been deducted.

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for every additional income, there will be proportionately smaller changes in marginal propensity to consume (c1). Hypothetically, we cannot increase our marginal propensity to consume (c1) more than what our income (Y) allows us.

Explanation:

C(Y-T) looks like the way it does because level of consumption depends on disposable income (Y-T), or the total personal income that they have after income taxes have been deducted.

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for every additional income, there will be proportionately smaller changes in marginal propensity to consume (c1). Hypothetically, we cannot increase our marginal propensity to consume (c1) more than what our income (Y) allows us.

So if, for instance, we assume that c1=0.7, when income(Y) increases by a dollar, then marginal propensity to consume (c1) increases by 70 cents. The additional change in consumption is less than the additional change in income.

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 new question is now:

How you deduce the mulitpliers dY/ dG and explain why its bigger than 1.

Why is, (The absolute value of) the multipilers with regard to T (taxes) less than G: : |dY /dT| < |dY /dG|?

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|

Thus, the absolute value of the tax multiplier is less than the government spending multiplier because the numerator of the tax multiplier is smaller, given the same denominators.

In theory, the government spending multiplier is greater than 1 and then the tax multiplier is less than the government spending multiplier because when the government increases its spending, the amount of money in the economy increases by the whole amount of government spending which then multiplies through increased consumption in the next round and subsequently increasing Y further, which makes dY/dG > 1. However, in the case of tax stimulus where the government reduces the tax rate so that people have more disposable income in hand, people save a portion of it and spend the rest. This implies that the initial stimulus will be of lesser amount than the tax redemption given by the government, which reduces the tax multiplier below the government spending multiplier.

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.

The new question is now:

Explain why M1 and M2 both are positive, and deduce the LM curve ie. deduce Y as a fuction off r and M/P?