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A marginal analysis compares the increased benefits of an activity to the higher expenses experienced by the same activity. Marginal analysis is used by businesses as a decision-making technique to assist them to optimise their prospective revenues. The focus on the cost or benefit of the next unit or individual, for example, the expense of producing one more widget or the profit made by adding one more worker, is referred to as marginal.

While selling a commodity, one must forecast the demand to avoid overproduction and thus, monetary losses.

Consider the following data for a cookie shop supply and demand :

Selling price per dozen = $ 0.69

Cost per dozen = $ 0.49

Salvage value = $ 0.29 per dozen

Calculate the profit/ loss per dozen cookies as shown below:

$\begin{array}{rcl}\mathrm{Profit}\left({\mathrm{M}}_{\mathrm{p}}\right)& =& \mathrm{Selling}\mathrm{price}-\mathrm{Cost}\mathrm{price}\\ & =& \$0.69-\$0.49\\ & =& \$0.20\end{array}$

$\begin{array}{rcl}\mathrm{Loss}\left({\mathrm{M}}_{\mathrm{L}}\right)& =& \mathrm{Cost}\mathrm{price}-\mathrm{Salvage}\mathrm{value}\\ & =& \$0.49-\$0.29\\ & =& \$0.29\end{array}$

Draw a table that shows the profits and loss for each possible quantity of cookies by inserting the formulas into a spreadsheet as shown below:

For better clarification see zoomed excel spreadsheet:-

Maximum expected profit = $416 and corresponding supply = 2200 dozens

Therefore optimal no. of cookies = 2200

For marginal analysis, use the following formula to calculate the probability of cookies not being sold:

$$\begin{gathered} \mathrm{P}≤\frac{{\mathrm{M}}_{\mathrm{L}}}{{\mathrm{M}}_{\mathrm{L}}+{\mathrm{M}}_{\mathrm{P}}} \\ \mathrm{Here}, \\ {\mathrm{M}}_{\mathrm{L}}=\mathrm{Marginal}\mathrm{Loss} \\ {\mathrm{M}}_{\mathrm{P}}=\mathrm{Marginal}\mathrm{Profit} \end{gathered}$$

Now the values of marginal profits and loss are obtained at the start of the question.

Obtain the value of probability as follow:

$\begin{array}{rcl}\mathrm{P}& ≤& \frac{{\mathrm{M}}_{\mathrm{L}}}{{\mathrm{M}}_{\mathrm{L}}+{\mathrm{M}}_{\mathrm{P}}}\\ & ≤& \frac{0.19}{0.19+0.16}\\ & ≤& 0.54\end{array}$

Thus, the probability of the cookies not being sold is 0.54

Obtained the table of cumulative probability as follows:

Now, since the probability of cookies not being sold is 0.54 which lies between 0.5 and 0.78 in the above table.

Thus the optimal number of cookies as per marginal analysis is 2,200