We are interested in the price of a commodity which is traded at regular intervals. We let Q_k denote the supply of the commodity, D_k the demand for the commodity, and p_k the price at the k^th time. The demand depends on the current price,

D_k= a + b p_k

and the supply depends on the previous price,

Q_k = c + d p_k-1

(a) Explain why it is reasonable to take a, c, and d, to be positive and b to be negative.

(b) Suppose we make the assumption that the supply is always equal to the demand. Find the difference equation satisfied by the sequence {p_k}.

(c) Suppose that the sequence of prices {p_k} converges to a limiting price p. What must p be?

(d) Find a condition on the coefficients so that you can prove that p_k -> p. Why is it reasonable that the conditions depend on d and b?

(e) Use specific choices of the coefficients and find numercial evidence that the prices can oscillate wildly if the condition is not satisfied.

(f) Iterate the equation for p_k and use the partial sum of the geometric series to prove that the sequence does not converge if the coefficients do not satisfy the condition in (d).