Question

Please answer this question in its entirely. PLEASE EXPLAIN CAREFULLY AND SHOW ALL THE STEPS. I need to see to understand. If you are not gonna follow my guidelines please abstain from answering. SOLVE all the subsections a b c dThank you!

1. [20 points] Consider first the goods market model with constant investment that we saw in Chapter 3. Consumption is given by C = co+c(Y- T) and I, G, and T are given. a. [5 points] Solve for equilibrium output. What is the value of the multiplier? Now let investment depend on both sales and the interest rate: I = bo + b,Y - b2i b. [5 points] Solve for equilibrium output. At a given interest rate, is the effect of a change in autonomous spending bigger than what it was in part (a)? Why? (Assume c1 + b<1 .) Next, write the LM relation as: M/P diY - d2i c. [5 points] Solve for equilibrium output. (Hint: Eliminate the interest rate from the IS and LM relations.) Derive the multiplier (the effect of a change of one unit in autonomous spending on output) d. [5 points] Is the multiplier you obtained in part (c) smaller or larger than the multiplier you derived in part (a)? Explain how your answer depends on the parameters in the behavioral equations for consumption, investment, and money demand.

Answer 1

a. Given:

CCo+ C1Y

Since investment is constant, government expenditure and taxes are given. The IS relation is given by:

Y=Co+ C1 (Y -++ G

Solving for the equilibrium output:

Co-C1 T+1+G Y = 1 C

To obtain the multiplier, we need to find the change in Y for a unit change in any autonomous variable, say G. To do so, we completely differentiate the above equation, keeping the taxes and autonomous consumption constant.

dG dY 1 C dG 1 C

b. Given, new investment function:

= b + bY

Using this equation for investment and the consumption function from part a., the IS relation is given as follows:

YCo+C1(Y bbYb2l G

Solving for the equilibrium output:

Co-C1T+bo bz1+ G Y= 1-C-b

"The effect of a change in autonomous spending" is just the multiplier. Using the same process as the previous part to obtain the multiplier. Keeping taxes,interest rate, autonomous consumption, and autonomous investment constant to derive the multiplier:

$\frac{dY}{dG}=\frac{1}{1-c_1-b_1}$

Since the denominator in this case is smaller than the denominator in the original multiplier, this implies that this multiplier is stronger than the original multiplier. Therefore, the effect of a change in autonomous spending is bigger in this case than in the previous case.

c. Given, the LM relation:

di d Y P

And the IS relation:

YCo+C1(Y bbYb2l G

Using the LM relation to substitute i in the IS relation:

Y Co+ C(Y T) + b+ brY(Y) d G

Solving for equilibrium output:

$Y=\frac{c_0-c_1T+b_0+\frac{b_2M}{d_2P}+G}{1-c_1-b_1+\frac{b_2d_1}{d_2}}$

The multiplier is calculated by completely differentiating the above equation:

$dY=\frac{dG}{1-c_1-b_1+\frac{b_2d_1}{d_2}} \rightarrow \frac{dY}{dG}=\frac{1}{1-c_1-b_1+\frac{b_2d_1}{d_2}}$

d. The multiplier obtained above can be compared with the multiplier obtained in part a. as follows:

The multiplier in part a is stronger than the new multiplier if:

bi0

The multiplier in part a is weaker than the new multiplier if:

$\frac{b_2d_1}{d_2}-b_1<0$

The multiplier in part a is the same as the new multiplier if:

- b1 0