Question

Suppose that the inverse market demand for silicone replacement tips for Sony earbud headphones is p ​= pN ​- 0.1Q, where p is the price per pair of replacement​ tips, pN is the price of a new pair of​ headphones, and Q is the number of tips per week. Suppose that the inverse supply function of the replacement tips is p ​= 2​ + 0.012 Q. a. The effect of a change in the price of a new pair of headphones on the equilibrium price of replacement tips​ ( ​dp/dpN​) at the equilibrium is given by nothing ​(Round your answer to three places​.)

Answer 1

Complete Question

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Answer:

a

   The effect of a change in the price of a new pair of headphones on the equilibrium price of replacement tips​ ( ​dp/dpN​) is

                   $\frac{\delta p}{\delta p_N} =1$

b

 The value of Q and p at equilibruim is

          $Q_e = 250$    and    $p_d =$ 5

The consumer surplus is  $C= 3125$

The producer surplus  is   $P = 375$

Explanation:

      From the question we are told that

           The inverse market demand is  $p_d = p_N -0.1Q$

                The inverse supply function is     $p_s = 2+ 0.012Q$

a

The effect of change in the price  is mathematically given as

                  $\frac{\delta p}{\delta p_N}$

Now differntiating the inverse market demand function with respect to $p_N$

We get that  

                   $\frac{\delta p}{\delta p_N} =1$

b

   We are told that $p_N =$$30

        Therefore the inverse market demand becomes

                             $p_d = 30 -0.1Q$

At  equilibrium

                  $p_d = p_s$

So we have

               $30 -0.1Q_e = 2+ 0.012Q_e$

Where $Q_e$ is the quantity at equilibrium

                    $28 = 0.112Q_e$

                     $Q_e = \frac{28}{0.112}$

                    $Q_e = 250$

Substituting the value of  Q into the equation for the inverse market demand function

                $p_d = 30 - 0.1 (250 )$

                    $p_d =$ 5

Looking at the equation for $p_d \ and \ p_s$ we see that

     For  Q =  0

             $p_d = 30$

             $p_s =2$

 And  for Q =  250

                 $p_d = 5$

                 $p_s = 5$

Hence the consumer surplus is mathematically evaluated as

           $C = \frac{1}{2} * Q_e * (30 -5)$

Substituting value

        $C = \frac{1}{2} * 250 (30-5)$

           $C= 3125$

And

  The  producer surplus is mathematically evaluated as

                    $P = \frac{1}{2} *250 * (5-2)$

                    $P = 375$