First Order Linear Differential Equation problem

The demand and supply functions of shoes are as follows:

Qd = a bP

Qs = c + dP

where a = 50, b = 1, c = 20, and d = 2. There is a possibility that the initial price P(0) does

not equal to long-term equilibrium price P* which is attainable in the long term-due to process

of time. In other words, all variables can change overtime or be functions of time (t). Based on

the information, answer the following questions

a. Find the market long-term equilibrium price P*!

b. Given sufficient time for the adjustment process to work, what is the price time path P(t)

by assuming that the rate of price change with respect to time (dP/dt), at any moment is

directly proportional to excess demand (Qd-Qs)? (hint: use nonhomogeneous first

order linear differential equation to find the complete solution P(t) that consists of

complementary function (Pc) and particular integral (Pp)!

c. Based on your solution in part b, will the price path over time P(t) converge to long-term

equilibrium price P* as t given current price is P(0) = 8? Use the qualitative

diagram to support your answer!

d. Give economic interpretation of complementary function (Pc) and particular integral (Pp)

in attaining the long-term equilibrium price P*!