Answer to Question #215906 in Macroeconomics for Collins

Question

The following parameters describe the structure of a hypothetical economy:
Autonomous consumption=240
Autonomous investment=1000
Autonomous taxes=100
Autonomous government expenditure=400
Real money supply (M/P)=600
Tax rate=0.25
Marginal propensity to consume=0.8
Interest elasticity of investment=50
Interest elasticity of demand for money=62.5
Income elasticity of demand for money=0.25
a) Find λ1, λ2, β1, β2. Determine and explain the relative effectiveness of fiscal and
monetary policies.
b) Use your answer in part a) above to determine equilibrium income and interest rate.
c) State the values of the fiscal and monetary policy multipliers if the economy is in a
liquidity trap. Explain.
d) If government expenditure is increased by 150 units, show how equilibrium interest
rate and equilibrium income will change. Can you determine the extent to which
investment is crowded out as a result? Explain.

Answer 1

IS curve represents the equilibrium in the goods market. That is, at each point on the IS curve, $AD=Y$

$AD = C + I + G$

Here, C denotes consumption given by: $C = C_0 + c(Y - T - tY) = C_0 + c((1-t)Y - T)$

Here, C₀ is the autonomous consumption, c is the MPC,  T is the autonomous tax, and t is the tax rate. 

I denotes the investment given by $I = I_0 - bi$

Here, I₀ denotes autonomous investment, b is the interest elasticity of investment, and i is the interest rate.

I denotes the investment given by $I = I_0 - bi$

Here, I₀ denotes autonomous investment, b is the interest elasticity of investment, and i is the interest rate.

Putting these together

$AD = C_0 + c((1-t)Y - T) + I_0 - bi + G$

We can now derive the IS curve by solving AD = Y:

$Y= C_0 + c((1-t)Y - T) + I_0 - bi + G$

the IS curve. $Y =\frac{(C_0+I_0+G−cT) − bi}{(1 − c(1−t))}$

Now we move on to the LM curve. The LM curve denotes equilibrium in the money market.

The money supply equation is given by$L=\bar{(\frac{m}{p})}$ where $\bar{(\frac{m}{p})}$ denotes real money supply

The money demand equation is given by $\frac{M}{P} = kY- hi$ , where $\frac{M}{P}$  denotes money demand, k denotes income elasticity of money demand and h denotes interest elasticity of money demand. 

We can now obtain the LM curve by solving  $\bar{(\frac{m}{p})}=\frac{m}{p}$

$\bar{(\frac{m}{p})}=KY-hi\\i=\frac{i}{h}[KY-\bar{(\frac{m}{p})}]$

Now that we have the IS and LM curves, we can solve for the general equilibrium. We use the value of i from the LM curve and substitute in the IS curve. Let

$a_G=\frac{1}{(1 − c(1−t))}, A_0=(C_0+I_0+G−cT)$

So, IS:  $Y=a_G(A_0−bi)$

Now, using i from LM gives us:

$Y = a_G[A_0−\frac{b}{h}(kY − MP)]$

Solving for Y we get the equilibrium

$Y = \frac{ha_G}{h+kba_G}A_0 + \frac{ ba_G}{h+kba_G}(\frac {M}{P})$

a) Effectiveness of fiscal policy refers to the effect of change in government expenditure (G) on the equilibrium income (Y). It is given by: $\frac{ha_G}{h+kba_G}$ the coefficient of $\frac{M}{P}.$  

b)

We can now calculate the equilibrium interest rate and income:

$A_0 = 240−0.8×100+1000+400=1560\\Y = \frac{ha_G}{h+kba_G}A_0 + \frac{ ba_G}{h+kba_G}(\frac {M}{P})\\=1.67×(1560)+1.33×600=3403.2$

Using this we calculate$i = \frac{1}{62.5}(0.25×3403.2−600)=4.0128$

c)

, we can calculate these:

$\frac{ha_G}{h+kba_G}=1.67\\\frac{ha_G}{h+kba_G}=1.33$

In a liquidity trap, monetary possible is ineffective while the fiscal policy is most effective. That is, in a liquidity trap,

$\frac{ha_G}{h+kba_G}=a_G=2.5\\\frac{ha_G}{h+kba_G}=0$

d) Now suppose G is increased by 150. Then, equilibrium income increases by, using the fiscal policy multiplier,  

$150×1.67 =250.5\\So\space Y = 3403.2+250.5=3653.7\\i = \frac{1}{62.5}(0.25×3653.7−600)=5.0148$

 interest increases by $5.0148-4.0128=1.002.$ So, using the interest elasticity of investment, we can calculate the crowding out of investment by: $-50×1.002=-50.1$ . Hence investment gets crowded out by -50.1