Microeconomics

Question 1 

Consider the demand and supply curves ????! = ???? − ???????? ????” = ???? + ???????? where ????, ????, ????, and ???? are suitable positive constants.
Find the equilibrium price P*, and the corresponding quantity Q*.
Suppose a tax of $???? 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 (Rˆ).
Use diagrams to illustrate the difference between the two market equilibria.

Question 2

Given ???? = – 0 0 0 4 0 0 10 5 0 2, find ????# , ????$, ???? + ???? + ????# and (???? − ????)(???? + ???? + ????# ) where ???? denotes the identity matrix of order
Use the results in (a) to find (???? − ????)%&.

Question 3

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