Question

Retailers will buy 45 wi-fi routers from a wholesaler if the price is $10 each but only 20 if the price is 60.
The wholesaler will supply 56 routers at $42 each and 70 at $50 each.
Assuming that the supply and demand function are linear, find the market equilibrium.

Answer 1

Given:

The supply and demand function is linear and is in equilibrium.

To Find :  

We have to find the equilibrium point at which both the supply and demand functions met.

Solution :

The Retailer will buy 45 routers if the price is $10 each and 20 if the price is $60 each.

$\[\begin{array}{l}\left( {{d_1},{p_1}} \right) = \left( {45,10} \right)\\\left( {{d_2},{p_2}} \right) = \left( {20,60} \right)\end{array}\]$

By using the equation of a line:

                                   $\[\begin{array}{l}d - {d_1} = \frac{{{d_2} - {d_1}}}{{{p_2} - {p_1}}}\left( {p - {p_1}} \right)\\\\d - 45 = \frac{{20 - 45}}{{60 - 10}}\left( {p - 10} \right)\\\\d - 45 = \frac{{ - 1}}{2}\left( {p - 10} \right)\\\\d - 45 = - \frac{p}{2} + 5\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,d = - \frac{p}{2} + 50\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {Eq.1} \right)\end{array}\]$

The wholesaler will supply 56 routers at $42 each and 70 routers at $50 each.

$\[\begin{array}{l}\left( {{s_1},{p_1}} \right) = \left( {56,42} \right)\\\left( {{s_2},{p_2}} \right) = \left( {70,50} \right)\end{array}\]$

By using the equation of a line:

                                   $\[\begin{array}{l}s - {s_1} = \frac{{{s_2} - {s_1}}}{{{p_2} - {p_1}}}\left( {p - {p_1}} \right)\\\\s - 56 = \frac{{70 - 56}}{{50 - 42}}\left( {p - 42} \right)\\\\s - 56 = \frac{7}{4}\left( {p - 42} \right)\\\\s - 56 = \frac{{7p}}{4} - \frac{{147}}{2}\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,s = \frac{{7p}}{4} - \frac{{35}}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {Eq.2} \right)\end{array}\]$

Equation $\[Eq.1\]$ and $\[Eq.2\]$:

                                 $\[\begin{array}{c}d = s\\\\ - \frac{p}{2} + 50 = \frac{{7p}}{4} - \frac{{35}}{2}\\\\50 + \frac{{35}}{2} = \frac{{7p}}{4} + \frac{p}{2}\\\\\frac{{100 + 35}}{2} = \frac{{7p + 2p}}{4}\\\\\frac{{135}}{2} = \frac{{9p}}{4}\\\\p = \frac{{270}}{9}\\\\\therefore \,p = 30\end{array}\]$

Substitute $\[p = 30\]$ in $\[Eq.1\]$:

                                            $\[\begin{array}{c}d = - \frac{p}{2} + 50\\\\ = - \frac{{30}}{2} + 50\\\\ = - 15 + 50\\\\\therefore d = 35\end{array}\]$

Hence, the market equilibrium point is (35,30).