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Let a demandfunction be defined as $D(p)=B-bp$, where $b,B>0$. A firm has some production cost, $c$, and can set the price $p$ under the constrain given by the Demand.

  • What is the optimal price?
  • Is the priceelasticity higher or lower than $1$ (absolute value)
  • How does price depend on $b$?

How would one do such a maximization problem?

My attempt

The elasticity would be given by

$\frac{p}{q}\cdot(-b)$ since $D'(p)=-b$

Then the problem is \begin{equation*} max_p(p-c)D'(p) \end{equation*}

which then follows that

\begin{equation*} D(p)+(p-c)D'(p)=0 \Leftrightarrow \end{equation*}

\begin{equation*} D(p)\left ( 1+(p-c)\frac{\epsilon}{p} \right )=0\Leftrightarrow \end{equation*}

and then the optimal price $p^*$ \begin{equation*} p^*=\frac{\epsilon}{1+\epsilon}c=\frac{\left | \epsilon \right |}{\left | \epsilon \right |-1}c=\frac{\left | \frac{-bp}{q} \right |}{\left | \frac{-bp}{q} \right |-1}c=\frac{c\left | b \right |\left | p \right |}{\left | b\right |\left |p \right |-\left | q \right |} \end{equation*}

and then we notice that

\begin{equation*} \left | \epsilon \right |>1 \end{equation*}

...............................................................................................

I feel like this is not correct nor the right approach. Help appreciated.

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Start with setting up profit equation:

$$\max_p \pi = (p-c)D(q)$$

Then substitute demand in:

$$ \pi = (p-c)(B-bp)$$

Take derivative of profit and equate it to 0:

$$\frac{d \pi}{dp}=B-2pb + cb=0$$

Now solve for optimum price:

$$p^*= \frac{B+cb}{2b}$$

There is also alternative way how to get to the same solution that uses elasticity of demand.

There is well known monopoly optimal pricing rule (see Peitz and Belleflamme Industrial Organization: Markets and Strategies) says that optimal monopoly prices are given by:

$$\frac{p-mc}{p}=-\frac{1}{El}$$

In your case marginal costs (mc) are $c$, and your elasticity of demand is actually:

$$EL= -\frac{bp}{q}= -\frac{bp}{B-bp}$$

In your attempt you forgot to substitute for Q in your elasticity.

Plugging these into optimal pricing rule gives us:

$$\frac{p-c}{p}=-\frac{1}{\left( -\frac{bp}{B-bp} \right) }$$

Which when solved for $p$ gives us again:

$$p^*= \frac{B+cb}{2b}$$

Both approaches give the same solution - the first one is quicker in this case in my opinion.

Now regarding what we say about price and parameter $b$, we can just rearrange the optimal price formula as:

$$p^*= \frac{B}{2b}+\frac{c}{2}$$

So clearly we can see that when $b$ parameter increases optimum price decreases.

We can also prove it more formally by inspecting derivative of $p^*$ with respect to $b$ which gives us:

$$ \frac{d p^*}{d b} = -\frac{B}{2b^2}$$

Since the derivative is negative we confirmed that the price varies negatively with $b$.

Next, per request in comment on what is elasticity at equilibrium price we can calculate that just by substituting the equilibrium price into elasticity which we already calculated above:

$$EL(p^*)= -\frac{b p^* }{B-bp^*}= -\frac{b \left( \frac{B+cb}{2b} \right)}{B-b \left( \frac{B+cb}{2b} \right)} \\= - \frac{B+bc}{B-bc}$$

The last expression will be bigger than 1 in absolute value because the numerator will be larger than the denominator ($B+cb>B-bc$) and absolute value will make the value positive.

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  • $\begingroup$ I notice what I did wrong. Thanks for the explanation. Regarding what happens if I lower or higher $b$ and the value of the elasticity; Isn't it impossible to really say anything about these without a given $b$? $\endgroup$ – user31331 Dec 1 '20 at 12:32
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    $\begingroup$ @bymathformath hi I added explanation for that at the end. Also if you think this answer solved your question consider accepting it $\endgroup$ – 1muflon1 Dec 1 '20 at 12:48
  • $\begingroup$ 1 last thing; WIll the absolute value of the price elasticity be bigger than 1 or smaller? I assume it is bigger but can not formally show it. $\endgroup$ – user31331 Dec 1 '20 at 14:14
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    $\begingroup$ @bymathformath that depends on what B, b and P are the question only mentions B,b>0. For B=1,b=0.5 and P=1 the price elasticity of demand will be exactly 1 if B=1, b=0.6 and P=1 it will be bigger than 1. If B=1, b=0.4, p=1 it will be smaller than 1. Does the question mention anything else about restrictions on coefficients? Or perhaps the question only asks for elasticity at optimal price? Without more info you can only say it could be either lower larger or equal to 1 in absolute value depending on coefficients (this could be valid answer). $\endgroup$ – 1muflon1 Dec 1 '20 at 14:19
  • $\begingroup$ Yes. This is ONLY at the optimal price with B and b bigger than 0. I should have specified that. How would that be in that case? $\endgroup$ – user31331 Dec 1 '20 at 14:29

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