Blog 3. Fahad Alhusaini

Marginal analysis

In this assignment on marginal analysis we perform calculations on a hypothetical company, named XYZ Perfumes, Inc. The company is based out of Kansas City and is a manufacturer of perfumes. For the purpose of this report, we would assume that the company manufactures only one kind of perfumes and thus, price for each bottle of perfume is constant.

The company has worldwide clients and their perfume brand is very popular. XYZ Perfumes is a new firm, and only commenced operations about 4 years ago. However, despite that the company has managed to become profitable in such a short span of time.

The company owns its factory (a small industrial unit in Kansas City), so there are fixed costs involved. For every batch of production, the fixed costs are of the tune of $1200.

Hence, since we represent fixed costs by ‘a’, we can say that a = 1200

The variable costs represent the marginal cost associated with producing one bottle of perfume, after already having paid for the fixed costs. So, this represents the variable ‘b’, which equals the price of producing one additional bottle of perfume. In our situation, this price is $10 per bottle. Thus, we can write that b = 10

Let us further assume that XYZ Perfumes is able to sell each perfume bottle for a retail price of $50 per bottle, or in other words, p = 50

Cost of production = c = fixed cost + variable cost

Fixed cost is independent of quantity, while variable cost depends on number of bottles of perfumes (we denote this number or quantity by ‘q’).

c = a+bq (cost function)

Upon substituting the values of a and b, we get c = 1200+10q

In order to calculate the revenue, we simply multiply the selling price (p) by quantity (q)

Which means, R = pq = 50q

Subtracting costs from revenue, we get profit (P) as follows:

P=50q-1200-10q = 40q-1200

Now that we know the profit function, we can easily calculate the break-even point by equating P to zero (since at break-even, profit is zero).

Hence, P = 0 = 40q-1200

Hence, q = 30 bottles need to be sold to break-even

The following chart shows this break-even point, where the green line, representing profit crosses the x-axis (profit is zero at this point).

Figure 1: Graph showing cost, revenue and profit as functions of quantity

In the above graph, we can see that the revenue curve (red) is steeper than the cost curve (blue). Since the slope of these curves indicates the magnitude of marginal revenue and marginal cost, respectively, we can conclude that marginal revenue is greater than marginal cost. Therefore as production increases, so does profitability.

Assuming that the daily production of perfume bottles is 50, we can perform the cost and profit analysis by referring to Figure 1. For 50 bottles produced, the daily profit is $800, based on a revenue of $2500 and costs of $1700. The marginal cost of producing the 50^th bottle is $10. In order to calculate the average cost for the day:

Avg cost = total cost / no. of bottles = 1700/50 = $34

In order to graph the marginal and average costs on the same chart, we use Figure 1 (blue line is nothing but the marginal cost) and plot average cost alongside, as shown in Figure 2 below:

Figure 2: Graph showing marginal (c) and average (a) costs as functions of quantity

It must be noted that average cost is not defined for zero units, so it begins with infinity. For the purpose of this graph, we chose a high value of 5000 for zero quantity, in order to be able to draw the graph.

As seen in Figure 2 above marginal cost has a constant positive slope, while average cost as a non-constant decreasing slope.

Questions:

1)    Is the marginal revenue less than or greater than the marginal cost at q = n? Explain.

Marginal revenue is greater than marginal cost, simply because the slope of revenue curve is greater than that of cost curve, as stated above.

2)    Is the number of units sold daily (q =n) after or before the break-even point? What does this mean?

The number of units sold is greater than break-even, which means XYZ Perfumes gets into positive profit region towards the end of each day. The moment this company manufactures 30 bottles (assuming it sells 30 bottles as well), the company enters profitability. The 20 more bottles made each day represent net profit.

3)    If production is increased by one extra quantity per day (i.e. if q = n + 1)) will the company continue to make money? Explain. (be sure to reference the formulas R(q + 1) – R(q) and C(q + 1) – C(q) in your explanation)

If production is increased to 51 bottles each day, profitability will increase. The profitable bottles in this case will be 21, instead of 20.

4)    At q = n, does an increase of production increase or decrease the average cost for the company?

At q=50, an increase in production will decrease average cost.

5)    Explain whether increasing or decreasing average costs would be better for the company.

Decreasing average costs is better because company retains more profits in that case.

Analysis:

XYZ Perfumes is doing very well and will continue to do so, provided that the current recession does not affect the sales negatively for too long. The company sells more bottles than break-even quantity each day, which keeps the balance sheet strong.

The company will do very well in future, which can be proven using following scenario:

Production of bottles each day: 50 bottles

Per day profit (calculated above): $800

Annual profit (assuming 250 production days): 250*800  = $200K