microeconomics, homework, elasticityOf course I don’t want you guys to solve this problem, but I hope someone can help me with this question of a microeconomics test. It’s not really a homework exercise, but I put that “homework” tag to be more honest.
I have to maximize a monopolist’s profit. His demand curve is $q=200-bp$ and his total cost function is $TC=c+5q$, where $c$ is a constant.
I find his maximization quantity by doing the following:
$p = \dfrac{200-q}{b} = \dfrac{200}{b} - \dfrac {q}{b}$
$ Marginal_revenue = MR = \dfrac{200}{b} - \dfrac {2q}{b}$
$ Marginal_cost = MC = 5 $
$ MR = MC $
$ \dfrac{200}{b} - \dfrac {2q}{b} = 5 $
$ 2q = 200 - 5b $
$ q^* = 100 - 2.5b $
$ p^* = \dfrac{100}{b}+2.5 $
$ \pi = p^* \cdot q^* = \dfrac{10000}{b} - 6,25b - c - 500$
Now the problem… I have to find how profit changes when the parameter $b$ changes. So I find the derivative:
$\dfrac{\partial \pi}{\partial b}=\dfrac{-10000}{b^2}+6.25$
And then I should explain this result with reference to demand elasticity. Now: how would you explain that derivative with reference to demand elasticity? With a greater b, we should have a more elastic demand and lower equilibrium profits. Does my derivative result confirm this? Thank you in advance!
Remember the definition of elasticity.
$\epsilon = \dfrac{\partial q}{\partial p} \dfrac{p}{q}$ where $\dfrac{\partial q}{\partial p} = b$ but $p$ and $q$ also depend on $b$ (as you’ve solved for above), so you have an explicit relationship between elasticity, prices, quantities, profits, etc.
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