**Question 1Β **

Consider the demand and supply curves π! = πΌ β π½π πβ = π + πΎπ where πΌ, π½, π, and πΎ are suitable positive constants.

Find the equilibrium price P*, and the corresponding quantity Q*.

Suppose a tax of $π per unit is imposed on the producer. Find the new equilibrium price. How does it change?

Compute and compare the total revenues obtained by the producer before the tax is imposed (R*) and after (RΛ).

Use diagrams to illustrate the difference between the two market equilibria.

**Question 2**

Given π΄ = β 0 0 0 4 0 0 10 5 0 2, find π΄# , π΄$, πΌ + π΄ + π΄# and (πΌ β π΄)(πΌ + π΄ + π΄# ) where πΌ denotes the identity matrix of order

Use the results in (a) to find (πΌ β π΄)%&.

**Question 3**

Let π be the n Γ n matrix where all the elements are 1. a. Show that (πΌβ + ππ)(πΌβ + ππ) = πΌβ + (π + π + πππ)π for all real numbers a and b. b. Use the result in (a) to find the inverse of π΄ = β 4 3 3 3 4 3 3 3 4 2. 4. Let the IS equation be where 1 β b is the marginal propensity to save, g is the investment sensitivity to interest rates, and A is an aggregate of exogenous variables.

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