Question

Suppose that a CEO’s goal is to increase profitability and output from her company by bolstering its sales force and that it is known that profits as a function of output are $\mathbf{\pi }\mathbf{=}\mathbf{40}\mathbf{q}\mathbf{-}\mathbf{2}{\mathbf{q}}^{\mathbf{2}}$(in millions of U.S. dollars). Graph the company’s profit function. Compare and contrast output and profits using the following compensation schemes based on the assumption that sales managers view output and profits as “goods”: (a) the company compensates sales managers solely based on output: (b) the company compensates sales managers solely based on profits: (c) the company compensates sales managers based on a combination of output and profits.

Short answer

1. If sales managers are compensated exclusively on the basis of output, the managers are more likely to try harder to sell everything.
2. If the company will compensate the sales managers solely based on profits, the sales manager should sell where the profit will be at its maximum.
3. If the company will compensate the sales managers based on the combination of output and profits, the managers will still receive a minimum compensation regardless of how many units the sales managers will sell.

Step-by-step solution

Finding the Profit function and representing in the graph

A profit function can be mathematically expressed as:

$\mathrm{\pi }=\mathrm{R}-\mathrm{C}$

In this case, the profit is:

$\pi =40q-2q^2$

The condition may be if there is an increase in the output, the revenues will increase more than the costs.

But if there is a decrease in output, it would reduce costs by more than it would reduce revenue.

Suppose the sales managers view both output and profits as "goods", which means that both output and profit should rise.

The graph of the company's profit function is:

By the graph, the profit increased along with the output until it reached the profit of$\$200$ million with$10$ million units of quantity produced. It was shown that right after reaching$\mathrm{\$}200$ million profits, the profit started to decrease while the output increased.

Comparing and contrasting output and profits if sales managers are compensated exclusively on the basis of output

a)

If sales managers are compensated exclusively on the basis of output, the managers are more likely to try harder to sell everything.

As per the graph, the profit rises from $0$to $\$200$ million.
And, the output rises from $0$to$10$ million units.

We can say, if the managers are able to sell$\$10$ million worth of items, they will make a profit of$\$200$ million.

But if the managers were able to sell $20$million units, the profit will fall to$0$ .

Therefore, the managers should only sell $\$10$million units of goods, in order for them to get compensated.

Comparing and contrasting output and profits if sales managers are compensated solely on the basis of profit

b)

If the company will compensate the sales managers solely based on profits, the sales manager should sell where the profit will be at its maximum.

As per the graph, the maximum level of profit is$\$200$ million.

Therefore, the managers should sell $10$million of units of goods.

As long as the sales managers are able to sell units of goods, their compensation will be based on how much they have sold.

Comparing and contrasting output and profits if sales managers are compensated based on the combination of output and profits

c)

If the company will compensate the sales managers based on the combination of output and profits, the managers will still receive a minimum compensation regardless of how much unit the sales managers will sell.

The chances of additional compensation that will be coming from the firm's profit will still be based on how much the sales manager would sell.

As a result, the managers will try their best to sell more so that the number of units that they will sell will reach the maximum level of profit.