Question

1. (26 marks total) Math Review: Recall the IS-LM model from your intermediate macro course. In particular, the goods-market equilibrium condition was Y C (Y-T) + 1 (r) + G, and the money-market equilibrium condition was m L(r, Y). Here, the exogenous variables are G (government spending). T (taxes), and m (real money supply). The endogenous variables ar Y (output, or income) and r (real interest rate). C() is the consumption function, which is increasing in disposable income Y -T, but less than one-for-one (i.e., 0 < C<1). I () is the investment function, which is decreasing in r (i.e., ' <0). L() is the demand function for real balances, which is decreasing in r, and increasing in Y (i.e., Lr < 0, Ly > 0). Recall that short-run equilibrium occurs when both the goods and money markets are in equilibrium. Let Y" and r* denote these short-run equilibrium quantitites of Y and r Using total differentiation, answer the following. In each case, if possible, determine the sign of the relevant derivatives, and if it's not possible, explain why (a) (4 marks) The IS curve is the combinations of Y and r that put the goods market into equilibrium. Find the slope of this curve (i.e., find dr/dY for the IS curve). (b) (4 marks) The L.M curve is the combinations of Y and r that put the money market into (e) (6 marks) Find the impact of an increase in G on the short-run equilibrium (i.e., d (d) (6 marks) Find the impact of an increase in T (i.e, dY*/dT and dr*/dT). equilibrium. Find the slope of this curve (i.e., find dr/dY for the LM curve). Y*/dG and dr/dG). (e) (6 marks) Find the impact of an increase in m (i.e., dY' /dm and dr*/dm).

please show all the steps and work to solve the problem, but also please visually emphasize your final answer.(e.g. by putting a box around).

Answer 1

(a) The IS Curve is given by the goods-market equilibrium equation. We know that it is given by

$Y = C(Y-T) &plus; I(r) &plus; G$

where C = Consumption as a function of disposable income Y-T

I = Investment as a function of Interest rate

G = Government Spending

Given that G, T(Taxes) are exogenous.

Let us assume the following Consumption and Investment functions:

$C = a &plus; b(Y - T)$

where b = Marginal Propensity to Consume, 0<b<1

$I = j -kr$

a, b, j, k >0

We now have the IS equation as:

$Y = a &plus; b(Y - T) &plus; j - kr &plus; G$

Rearranging to collect constants, endogenous and exogenous terms, we get:

$Y - bY = (a &plus; j) - kr &plus; (G - bT)$

$\Rightarrow Y (1 - b) = A - kr$

where A = (a + j +G - bT)

Taking total differential, we get

$(1 - b) dY = - k dr$

$\Rightarrow \frac {dr}{dY} = -\frac{(1 - b)} {k}$

The negative sign means that the IS Curve is negatively sloped, i.e. Y and r are negatively related.

(b) The LM Curve is given by the money-market equilibrium equation. We know that it is given by

m = L(Y, r)

We know that money income is positively related to Y and negatively related to r. So we have the following equation for the money market

$m = uY - vr$

where m (real money supply) is exogenous.

Rearranging and expressing in terms of r, we get:

$r = \frac {u}{v}Y - \frac{m}{v}$

Taking total differential, we get

$dr = \frac {u}{v} dY$

$\Rightarrow \frac {dr}{dY} = \frac {u}{v}$

The positive sign means that the LM Curve is positively sloped, i.e. Y and r are positively related.

(c) We have already derived the IS equation above

$Y(1 - b) = (a &plus; j) - bT) - kr &plus; G$

$\Rightarrow Y = \frac {(a &plus; j)}{(1-b)} - \frac{b}{(1-b)}T - \frac {k}{(1-b)}r &plus; \frac{1}{(1-b)}G$

Differentiating with respect to G, we get:

$\frac {dY*}{dG} = \frac {1}{(1-b)}$

(d)

$\Rightarrow Y = \frac {(a &plus; j)}{(1-b)} - \frac{b}{(1-b)}T - \frac {k}{(1-b)}r &plus; \frac{1}{(1-b)}G$

Differentiating with respect to T, we get:

$\frac {dY*}{dT} = -\frac {b}{(1-b)}$