a. Write down the expression for Jessica’s monthly profit. (Assume that there are exactly four weeks per month.) — James considering business venture selling custom embroidered

Jessica James is considering a business venture—selling custom embroidered baseball caps from a pushcart kiosk at College Mall. The caps will be available in 12 different colors and one-size fits all. The caps’ unique feature is that almost any name, phrase, or logo, can be stitched onto the cap while the customer waits. Thus, a customer can obtain a cap with his or her own name, monogram, special saying, or favorite logo in a wide variety of thread colors, sizes, and fonts. Based on preliminary market research and input from the franchising company, Jessica plans on selling each embroidered cap for $20.

Jessica plans to acquire the necessary technology (two industrial strength sewing machines hooked up to PCs with scanners and all of the necessary software) by obtaining a franchise from Made to Order Caps, Inc. Made to Order Caps, Inc., sells all of the necessary equipment and technology and provides the inventory of caps and other supplies.

In addition, Made to Order Caps, Inc., trains prospective franchisees such as Jessica in the basics of running the business and operating the machines.

Jessica will need to buy the caps from Made to Order Caps, Inc., for $4 per cap and pay a royalty to the franchising company of $2 per cap sold. In addition, Jessica believes that each cap will use about $2.50 worth of supplies (including thread, replacement of sewing needles, machine maintenance, etc.). Finally, Made to Order Caps, Inc., requires that each franchisee invest $250 per month on leaflets and brochures to advertise the product.

Jessica discovers that College Mall would supply her with the pushcart and all of the necessary equipment for proper display. The license from the mall also allows Jessica to use several electrical outlets and two telephone outlets. College Mall will pay for Jessica’s electricity consumption, but she will have to obtain two business telephone lines at a cost of $50 per line per month. College Mall is willing to license the space and equipment to Jessica on a monthly basis for $1,970 per month.

To generate maximum sales, Jessica wants to keep the pushcart open for business from 10 a.m. to 10 p.m. Monday through Friday, and from 10 a.m. to 6 p.m. on Saturday and Sunday. Jessica is willing to put in 50 hours at the kiosk, and she can obtain additional part-time help at $10 per hour (only one individual is required to operate the business). Jessica’s only other significant expense is setting up to accept credit cards.

Jessica anticipates that 75% of her sales will be credit card sales, and the credit card company charges Jessica a fee of 2% of the selling price.

Required:

a. Write down the expression for Jessica’s monthly profit. (Assume that there are exactly four weeks per month.)

b. Calculate Jessica’s monthly breakeven point in baseball caps. What does this translate to in revenue?

c. How much profit would Jessica earn in a month if she sold 1,000 caps? How many caps would Jessica have to sell to earn a target profit of $4,032 per month?

d. Jessica has been toying around with how the quantity of caps she can sell is likely to vary inversely with the selling price. Jessica is keenly aware that the lower the price per cap, the more caps she can sell and vice versa. She is determined to figure this relationship out and find the best price. After conducting extensive market surveys, Jessica believes that the local monthly demand for embroidered baseball caps is as follows:

Price per Cap Demand

$20 ……………………. 300

$25 ……………………. 250

$28 …………………….. 220

$30 ……………………. 200

$32 ……………………. 180

$34 …………………….. 160

How does this piece of information alter Jessica’s profit calculation? What price per cap should Jessica charge to maximize profit? How much profit does she earn at this price?

e. Jessica is almost done—she realizes that she forgot to include taxes as part of her decision model. Jessica believes that combined local, state, and federal taxes will be 25% of her profit before taxes. Does this piece of information affect Jessica’s decision in part (d)? Does this change the profit-maximizing price? Does it change Jessica’s profit?

f. What do you think of Jessica’s business venture?

SOLUTION

The profit is:

Profit before taxes= [(Price – unit variable cost) Quantity] – Fixed cost.

Based on the information provided, we have the following:

Profit = ($20.00 price – $4 cost of cap – $2 royalty per cap – $2.50 supplies per cap – [.02 .75 20] expected credit card fees per cap) × Quantity – ($250 for leaflets and brochures + $100 for phone lines + $1,970 for space and equipment + $1,040* for additional help)

* = (76 hours the shop is open per week – 50 hours worked by Jessica each week) $10 per hour for temporary help × 4 weeks per month.

This reduces to:

Profit = [$11.2 Quantity] – $3,360.

The breakeven point can be obtained by setting profit in the model from part [a] above equal to $0. Thus, we have:

$0 = ($11.20 Breakeven volume) – $3,360.

Solving, we obtain Breakeven volume = 300 caps.

At a planned price of $20 per cap, this translates to 300 $20 = $6,000 in revenue per month.

Thus, in order to breakeven on a monthly basis, Jessica needs to sell approximately 10 caps per day.

If Jessica sold 1,000 caps per month, her expected profit would be:

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Profit = ($11.20 1,000) – $3,360 = $7,840 (not bad!)

If Jessica wished to earn $4,032 per month, she would need to sell:

$4,032 = ($11.2 Q) – $3,360, or Q = 660 caps, roughly 22 caps per day.

First, notice that Jessica’s research informs her that at a price of $20, demand is expected to be 300 caps per month – as we calculated in part [b], this is Jessica’s breakeven point. Things don’t look good, but perhaps Jessica can do better by exploiting her market research and newfound knowledge of the demand schedule. Let us expand the table to compute Jessica’s expected profit at alternate prices.

Note that expected profit in the table is based on the profit equation we developed in part [a]. Specifically, we plug price and quantity into the following equation to arrive at expected profit. Importantly, as the selling price and quantity change, so too do the credit card company charges, which equal 2% of the selling price.

Profit = {[Price – 4 – 2 – 2.50 – (.02 .75 Price)] × Quantity} – $3,360.

OR, Profit = [(.985Price – 8.50) × Quantity] – $3,360.

Price per cap	Demand	Expected Profit
$20	300	$0.00
$25	250	$671.25
$28	220	$837.60
$30	200	$850.00
$32	180	$783.60
$34	160	$638.40

Jessica maximizes her profit if she sets her price at $30 per cap as her expected profit is highest ($850.00) at this price. Relative to setting the price at $20 per cap, the $30 price results in a lower volume (200 caps versus 300 caps). However, the higher contribution margin per cap ($21.05 rather than $11.20) offsets this.

Note (can be skipped without loss of continuity) :

In this case, the relation between demand and price is

Quantity = 500 – (10 × Price).

We can modify the expression for profit to incorporate this new piece of information. Thus, we have:

Profit= {[500 – (10 Price)] (.985 Price – 8.5)} – $3,360.

This simplifies to:

Profit = 492.50(Price) – 9.85(Price)2 – 4,250 + 85P – 3,360.

Or, the relation between Jessica’s profit and price is:

Profit = 577.50(Price) – 9.85(Price)2 – 7,610.

We can solve for the best P using calculus or Excel’s solver function. The exact answer is

P = $29.31 and expected profit is $854.63

Taxes affect profit in a relatively straightforward fashion. We can take the profit model developed in part [d] and add the tax variable – doing so yields:

Profit after taxes = Profit before taxes – taxes paid.

Here, because taxes are proportional to income, taxes paid = (tax rate pre-tax profit). Adding this column yields:

Price per cap	Demand	Expected Profit	After-tax profit
$20	300	$0.00	$0.00
$25	250	$671.25	$503.44
$28	220	$837.60	$628.02
$30	200	$850.00	$637.50
$32	180	$783.60	$587.70
$34	160	$638.40	$478.80

The optimal price continues to be $30 per cap.

Notice that the optimal price does not change – this occurs because (in this example), taxes are a linear function of pre-tax profit. Thus, Jessica still wishes to maximize pre-tax profit which, in turn, will also maximize after-tax profit. This aspect of the problem allows discussing the relation between pre-tax income and tax rates, including extending the discussion to non-linear relations between pre-tax and after-tax income.

Taxes will, of course, reduce the amount of Jessica’s profit. In our example, Jessica will now earn an after-tax monthly profit of:

$850 .75 = $637.50.

Note (Can skip without loss of continuity): In the calculus approach, incorporating taxes yields

Profit after taxes = (577.50(Price) – 9.85(Price)2 – 7,610) (1 – tax rate).

Since t = .25, we have:

Profit after taxes = (577.50(Price) – 9.85(Price)2 – 7,610) (.75).

Profit after taxes = 433.125(Price) – 7.3875(Price)2 – 5707.50.

Again, we can solve for the best P using calculus or Excel’s solver function. The exact answer is Price = $29.31 (which is exactly what we arrived at earlier).

While Jessica’s venture has intuitive appeal and our original calculations seemed to indicate that the venture might be a “go” (after all, selling 22 hats a day does not seem like a lot) the numbers simply do not add up. Given the totality of the costs involved and Jessica’s market research, it appears that (at best) Jessica will earn a very modest profit and will be unlikely to maintain a reasonable lifestyle with this business.

We can see, though, how modest amounts add up – for example, if Jessica were to “manage” 10 kiosks in various malls around the state, she could earn a reasonable sum of money – this is precisely what franchisers seek to do.