Question

4. Consider a hypothetical economy.
a.The consumption function is given by
𝑪 = 𝟐𝟎𝟎 + 𝟎. 𝟕𝟓(𝒀 − 𝑻 ).
The investment function is
𝑰 = 𝟐𝟎𝟎 − 𝟐𝟓𝒓
Government purchases and taxes are both 𝟏𝟎𝟎.
For this economy, graph the IS curve for 𝒓 ranging from 0 to 8.
b. The money demand function of the economy is
(𝑴/𝑷 )
𝒅 = 𝒀 − 𝟏𝟎𝟎𝒓.
The money supply 𝑴 is 𝟏, 𝟎𝟎𝟎 and the price level 𝑷 is 𝟐.
For this economy, graph the LM curve for 𝑟 ranging from 0to 𝟖.
c. Find the equilibrium interest rate 𝑟 and the equilibrium level of income
d.
d. Suppose that government purchases are raised from 100 𝑡𝑜 150. How much does the IS
curve.

Answer 1

c. To find the equilibrium interest rate and level of income, we need to find the intersection of the IS and LM curves. The IS curve is given by the equation 𝐼+𝐺+𝑇=𝐶+𝑌, where 𝑌 is income, 𝐺 is government purchases, and 𝑇 is taxes. Substituting the given functions, we get:

𝐼+100=200+0.75(𝑌−100)
𝐼+100=200+0.75𝑌−75
𝐼=275+0.75𝑌

The LM curve is given by the equation 𝑀/𝑃=𝐿(𝑌,𝑟), which can be rearranged to:

𝐿(𝑌,𝑟)=1000−200𝑟

Setting 𝐼=𝐿, we can solve for the equilibrium interest rate:

275+0.75𝑌=1000−200𝑟
0.75𝑌=725−200𝑟
𝑌=966.67−266.67𝑟

Substituting this back into the IS equation, we get:

𝐼=275+0.75(966.67−266.67𝑟)
𝐼=275+725−200𝑟
𝐼=1000−200𝑟

Setting 𝐼=𝐺+𝐺=150, we can solve for the equilibrium interest rate:

1000−200𝑟=150
−200𝑟=−850
𝑟=4.25

Substituting this back into the equilibrium income equation, we get:

𝑌=966.67−266.67(4.25)
𝑌=861.67

Therefore, the equilibrium interest rate is 4.25% and the equilibrium level of income is $861.67.

d. If government purchases are raised from 100 to 150, the IS curve will shift upwards, as government spending is a component of aggregate demand. This will lead to an increase in the equilibrium level of income and the interest rate.