We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends on the previous price, $Q_k = c + d p_{k - 1}$. (To get {$p_k$} assume $D_k$ = $Q_k$)
- Suppose that the sequence of prices{$p_k$} converges to a limiting price $\bar p$. What must $\bar p$ be?
- Find a condition on the coefficients so that you can prove that $p_k$ $\rightarrow$ $\bar p$. Why is it reasonable that the conditions depend on d and b?
I have calculated the sequence to be {$p_k$} = $\frac{c - a + dp_{k-1}}{b}$ , from here I was thinking that for number 1. $\bar p$ must be the average price that the supply and demand converges to, as for number 2. im lost as to what to do, possibly a Cauchy-$\epsilon$ argument?
EDIT- also the second part of 2. I believe it is reasonable that the conditions depend on b and d because they are the constants that scales the price at a k^th time?
If $p_k = \frac{c-a+dp_{k-1}}{b}$ and $\lim_{k \to \infty} p_k = L$ exists, then $L = \frac{c-a+dL}{b}$ or $bL = c-a+dL$ or $L(b-d) = c-a$ or $L = \frac{c-a}{b-d}$.
Just elementary algebra.