<p>The monopolist is allowed to set the price. However, because his product has a perfect substitute, he can sell 0 products at prices $p&gt;2$. He also has no incentive seting the price $p&lt;2$, because the consumers will buy any number of products he wishes to sell at $p = 2$. This means that $p(q) = 2$. Its graph in the $(q,p)$ coordinate system would be a horizontal line at $p = 2$. Because of this, $p'(q) = 0$. Correcting for this you should be able to easily finish the exercise. </p>

CC BY-SA 3.0
2017-08-19T08:53:06.180Z
2017-08-19T08:53:06.180Z
Giskard
<p>I'm reviewing the following question from my microeconomics course notes. I don't understand how to do part (b).</p>

<blockquote>
  <p>A firm faces a perfectly elastic demand curve because the product it
  sells has a perfect substitute. The (inverse) demand function is
  $p=2$. Its production function is $f(K,L)=K^{1/4}L^{1/4}$.</p>
  
  <p>(a) Assuming input prices of $w=r=1$, show that the firm's cost
  function is $C(q)=2q^2$.</p>
  
  <p>(b) Assuming the firm operates as a monopolist, what price will it
  charge, what quantity will it produce, and what are its profits?</p>
  
  <p>(c) What is the deadweight loss from monopoly?</p>
  
  <p>(d) Would the answers to parts (b) and (c) change if the firm could perfectly price discriminate?</p>
</blockquote>

<p>Part (a) is easy; I'm stuck on part (b).</p>

<p>If the firm faces a "perfectly elastic demand curve" $p=2$, then the firm <em>isn't</em> in a position to operate as a monopolist, right? A monopolist ought to be able to <em>set</em> a price, not have to accept the market price $p=2$.</p>

<p>If I assume the monopolist sets a price $p$ for quantity supplied $q$, then I can define the profit function</p>

<p>$$\pi(q)=p(q)\cdot q-c(q)=p(q)\cdot q-2q^2$$</p>

<p>Differentiating gives the first-order condition</p>

<p>$$q\cdot p'(q)+p(q)=4q$$</p>

<p>But I can't proceed without knowing the monopolist's demand curve (which, again, I thought <em>couldn't</em> be perfectly inelastic).</p>

<p>If I <em>assume</em> $p(q)=2$, then I can deduce that $q=1/2$. But this can't be right, because then I can't calculate a deadweight loss.</p>

CC BY-SA 3.0
2017-08-20T13:29:19.270Z
If the price is effectively fixed, then I would agree that (c) and (d) are rather empty questions: price is equal to marginal revenue as no customer is willing to pay more than $2$, but enough customers are willing to pay any amount less than or equal to $2$ for more than the firm can produce
Henry
<microeconomics><monopoly>
How to determine the price a monopolist will set when the demand curve the firm faces isn't known?
josh milligan
# How to determine the price a monopolist will set when the demand curve the firm faces isn't known?

(Asked by: username_0 on 2017-08-18)

I'm reviewing the following question from my microeconomics course notes. I don't understand how to do part (b).

> A firm faces a perfectly elastic demand curve because the product it
> sells has a perfect substitute. The (inverse) demand function is
> $p=2$. Its production function is $f(K,L)=K^{1/4}L^{1/4}$.
>
> (a) Assuming input prices of $w=r=1$, show that the firm's cost
> function is $C(q)=2q^2$.
>
> (b) Assuming the firm operates as a monopolist, what price will it
> charge, what quantity will it produce, and what are its profits?
>
> (c) What is the deadweight loss from monopoly?
>
> (d) Would the answers to parts (b) and (c) change if the firm could perfectly price discriminate?

Part (a) is easy; I'm stuck on part (b).

If the firm faces a "perfectly elastic demand curve" $p=2$, then the firm *isn't* in a position to operate as a monopolist, right? A monopolist ought to be able to *set* a price, not have to accept the market price $p=2$.

If I assume the monopolist sets a price $p$ for quantity supplied $q$, then I can define the profit function

$$\pi(q)=p(q)\cdot q-c(q)=p(q)\cdot q-2q^2$$

Differentiating gives the first-order condition

$$q\cdot p'(q)+p(q)=4q$$

But I can't proceed without knowing the monopolist's demand curve (which, again, I thought *couldn't* be perfectly inelastic).

If I *assume* $p(q)=2$, then I can deduce that $q=1/2$. But this can't be right, because then I can't calculate a deadweight loss.

**Tags:** microeconomics, monopoly

> **Comments:**
> **username_1**: If the price is effectively fixed, then I would agree that (c) and (d) are rather empty questions: price is equal to marginal revenue as no customer is willing to pay more than $2$, but enough customers are willing to pay any amount less than or equal to $2$ for more than the firm can produce  

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## 1 answers

### Answer by username_2 on 2017-08-19. Score: 3

The monopolist is allowed to set the price. However, because his product has a perfect substitute, he can sell 0 products at prices $p>2$. He also has no incentive seting the price $p<2$, because the consumers will buy any number of products he wishes to sell at $p = 2$. This means that $p(q) = 2$. Its graph in the $(q,p)$ coordinate system would be a horizontal line at $p = 2$. Because of this, $p'(q) = 0$. Correcting for this you should be able to easily finish the exercise.


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