I have this:
$Qd = 21 − 6P$
$Qs =−6+30(P − 0.9)$
I'm asked to solve P and Q at equilibrium.
If I'm not mistaken, $P=1,5$ (because $21 - 6P = -6 + 30P - 27$)
And Q is: $21 - 6*1.5 = 12$
I hope it's right.
Now the tricky part is to determine the quadratic formula that expresses the equilibrium price and quantity as a function of τ (where τ is 0.9 here above) . Subsequently, I should express the revenue as a function of τ
I'm given 4 possible answers for the revenues and I have to pick one:
(a) $18τ−6.667τ^2$
(b) $16.5τ−5τ^2$
(c) $15τ−3.333τ^2$
(d) $13.8τ−2τ^2$
They all give the same right prediction (10.8) with $τ=0.9$ of course, which doesn't help.
But I don't know how to go from two linear equations to a quadratic one.
What I tried is this:
if $-6 + 30(P-τ) = 12$
Then $-6 + 30(P-τ) = 13.33333*0.9 = 13.33333τ$
Then, I just multiplied both sides by τ (for whatever reason, in order to have a $τ^2$ to toy with):
$-6τ + 30τ(P-τ) = 13.33333τ^2$
And (considering that $P=1.5$), I get:
$39τ - 16,666666τ^2$
But none of the formulas proposed here above are matching it (and my answer can't be reduced into any of these).
I'm probably couldn't be more wrong but that's what I get when messing around with those equations.
I looked on the internet but either I find linear or quadratic examples. None that goes from one to another. And to me it even make sense that nothing is to be found. I don't know why there should be any reason to go from one to another. It's like apple and oranges to me. Obviously there is something that I just don't get.
Any help please?
Thanks!