We are interested in price of a commodity, traded at regular intervals. Why it is reasonable to take $a$, $c$, and $d > 0$ and $b < 0$? We are interested in the price of a commodity which is traded at regular intervals. 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, $Q_k = c + d p_{k - 1}$.


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

*Suppose we make the assumption that the supply is always equal to the demand. Find the difference equation satisfied by the sequence $\{p_k\}$.  
Okay after some research I feel as though I may have a start to answering the first question. If you know how supply and demand works you basically want supply high because its graph generates an increasing function where as the demand graph generates a decreasing function.
I'm still not positive about question one but think I have an answer for question two if anyone could confirm or deny it would be great, set $Q_k$ = $D_k$ then solve for $p_k$ I get $p_k$ = $\frac{c - a + dp_{k-1}}{b}$
 A: We can assume $b$ is negative, as demand decreases as price increases. Intuitively, we have an initial demand if something is free. If we are getting paid to get something, don't we demand more of it? If we are paying for it, wouldn't we demand less of it? Think of $a$ as an initial endowment.
Similarly, with $c, d > 0$, a higher price incentivizes the producer to supply more product, so $Q_{k}$ increases with $p_{k-1}$. Similarly, if price goes down, the producer won't make as much, so won't produce as much. The constant $c$ is how much the producer would make if $p_{k} = 0$. So if a business is starting up, they have to make the product to sell, right? So $c > 0$.
Does this make sense?
Edit: Your answer for $p_{k}$ is correct.
Let's talk about the story a bit more. So at equilibrium, we have $D_{k} = Q_{k}$. People are demanding exactly what is being supplied. Now suppose I increase the price. Then the supplier, based off the previous iteration's demand, is incentivized to produce more of the good. If I know $50$ people bought my good at \$2 per item, then at \$3 per item, I make an extra \$50. That's what the supplier sees.
Now those who demand the good react to the price at the current iteration. If it costs more, demand decreases, if nothing else than due to the budget constraint. You can only afford so much, correct? So now fewer than $50$ people demand the good, and the supplier has excess good and there is a difference between $D_{k}$ and $Q_{k}$.
Does that clarify some more?
