Question

Wyandotte Chemical Company sells various chemicals to the automobile industry. Wyandotte currently sells 30,000 gallons of polyol per year at an average price of $60 per gallon. Fixed costs of manufacturing polyol are $360,000 per year and total variable costs equal $720,000. The operations research department has estimated that a 15 percent increase in output would not affect fixed costs but would reduce average variable costs by 50 cents per gallon. The marketing department has estimated the arc elasticity of demand for polyol to be -2.0. How much would Wyandotte have to reduce the price of polyol to achieve a 15 percent increase in the quantity sold? percent total revenues from $1,800,000 to $ . Total costs would be $ and total profits Such a price cut would would be $

Answer 1

a) Using the formula for the elasticity of demand,

$$$\begin{array}{l}{E_d} = \frac{{\% {\rm{ change in demand}}}}{{\% {\rm{ change in price}}}}\\ - 2 = \frac{{15\% }}{{\% P}}\\\% P = - 7.5\% \end{array}$$$

Here, the negative sign shows the decrement in percentage.

b) The percentage change in the price is:

$$$\begin{array}{l}\% P = \frac{{{P_2} - {P_1}}}{{\left( {\frac{{{P_1} + {P_2}}}{2}} \right)}}\\ - 0.075 = \frac{{{P_2} - 60}}{{\left( {\frac{{{P_2} + 60}}{2}} \right)}}\\ - 0.075{P_2} - 4.5 = 2{P_2} - 120\\{P_2} = \$ 55.663\end{array}$$$

The change in price is:

$$$\begin{array}{l}\Delta P = {P_1} - {P_2}\\\Delta P = \$ 60 - \$ 55.663\\\Delta P = \$ 4.337\end{array}$$$

The percentage change in the quantity is:

2 %Q= Q2-01 Q+Q2 0.15 = 02-30000 30000+02 4500+ 0.15Q2 = 2Q2 - 60000 Q2 = 34864.86 2

The initial revenue is:

Rinitial Pinitial X Qintial Rinitial = $60 x 30000 Rinitial = $1800000

The final revenue is:

$$$\begin{array}{l}{R_{{\rm{final}}}} = {P_{{\rm{final}}}} \times {Q_{{\rm{final}}}}\\{R_{{\rm{final}}}} = \$ 55.663 \times 34864.86\\{R_{{\rm{final}}}} = \$ 1940682.7\end{array}$$$

So, the price cut would increase total revenues from $1800000 to $1940682.7.

The average variable cost is:

$$$\begin{array}{l}AV{C_{{\rm{initial}}}} = \frac{{VC}}{Q}\\AV{C_{{\rm{initial}}}} = \frac{{\$ 720000}}{{30000}}\\AV{C_{{\rm{initial}}}} = \$ 24\end{array}$$$

A 15% increase in output reduces the average cost by 60 cents per gallon. So, the new average variable cost is:

$$$\begin{array}{l}AV{C_{{\rm{final}}}} = AV{C_{{\rm{initial}}}} - \$ 0.60\\AV{C_{{\rm{final}}}} = \$ 24 - \$ 0.60\\AV{C_{{\rm{final}}}} = \$ 23.4\end{array}$$$

So, the final total variable cost will be:

$$$\begin{array}{l}TV{C_{{\rm{final}}}} = AV{C_{{\rm{final}}}}{Q_{{\rm{final}}}}\\TV{C_{{\rm{final}}}} = \$ 23.4 \times 34864.86\\TV{C_{{\rm{final}}}} = \$ 815837.7\end{array}$$$

The initial total cost is:

$$$\begin{array}{l}T{C_{{\rm{initial}}}} = TV{C_{{\rm{initial}}}} + FC\\T{C_{{\rm{initial}}}} = \$ 720000 + \$ 360000\\T{C_{{\rm{initial}}}} = \$ 1080000\end{array}$$$

The final total cost is:

$$$\begin{array}{l}T{C_{{\rm{final}}}} = TV{C_{{\rm{final}}}} + FC\\T{C_{{\rm{final}}}} = \$ 815837.7 + \$ 360000\\T{C_{{\rm{final}}}} = \$ 1175837.7\end{array}$$$

So, the total cost will be $1175837.7.

The profit before the price cut is:

$$$\begin{array}{l}p{r_{{\rm{initial}}}} = {R_{{\rm{initial}}}} - T{C_{{\rm{initial}}}}\\p{r_{{\rm{initial}}}} = \$ 1800000 - \$ 1080000\\p{r_{{\rm{initial}}}} = \$ 720000\end{array}$$$

The profit after the price cut is:

$$$\begin{array}{l}p{r_{{\rm{final}}}} = {R_{{\rm{final}}}} - T{C_{{\rm{final}}}}\\p{r_{{\rm{final}}}} = \$ 1940682.7 - \$ 1175837.7\\p{r_{{\rm{final}}}} = \$ 764845\end{array}$$$

So, the total profit will be $764845.