Question

A clothing brand plans to use a model to decide the print run (offer) of
a new exclusive garment to maximize your profits. By policies
Your supplier can only produce one package with one of the following
quantities 5,10,25,50 or 100 items with the following prices per pledge
respectively 15,9,7,5,6,5,5 Additionally, it is known that the curve of
manda of its exclusive products tends to be
D (x) = 100/x + 5
(a) Calculate the total cost of each garment package
(b) Determine the equilibrium price for each of the packages
(c) Calculate the total profits generated by each package when sold
completely at equilibrium price
(d) Calculate the net gains by subtracting the total cost from the total
of the package
(e) Determine which package is the most convenient

Answer 1

A clothing brand plans to use a model to decide the print run (offer) of
a new exclusive garment to maximize your profits.

Following is the chart for the quantity and price combinations respectively,

Units per package	Per unit cost
5	15
10	9
25	7
50	5
100	6

(Please note: Two prices provided does not correspond to any quantity i.e. 5, 5 at the end)

Now given is the inverse demand function with quantity in the RHS; inverse demand function is equal to average revenue, or equal to the price.

D (x) = 100/x + 5

Or we can also write, P = 100/x + 5

(a) Calculate the total cost of each garment package

Total production cost of each garment package is calculated multiplying the quantity in each package with the cost per unit; following chart shows the cost of each package,

Unit per package	Per unit cost	Total cost per package
5	15	75
10	9	90
25	7	175
50	5	250
100	6	600

The above chart shows that the cost per package increases with the increase in quantity, however, following the law of demand, cost per unit is decreased initially, from 15(for 5 unit) till the cost of 5 per unit (for 50 units). The last package with 100 units is however charged at the rate of 6 per unit that shoots the total cost of the last package high.

(b) Determine the equilibrium price for each of the packages

The equilibrium price is determined where the demand and supply equate. At the equilibrium, the marginal revenue earned should be equal to the marginal cost.

To calculate equilibrium price at demand= supply, we use the quantities per package in the inverse demand equation D (x) = 100/x+5; where x is the equilibrium quantity to be sold.

For instance for the first package, with 5 units, x is put equal to 5 to calculate the price using the given demand (inverse) function.

Implies, P ₁ = 100/ (5) +5 = 25, similarly, P ₂ = 100/ (9) +5 = 15 and so on. The chart below shows the equilibrium prices for each package.

Unit per package	Per unit Cost	Equilibrium price per unit
5	15	25
10	9	15
25	7	9
50	5	7
100	6	6

(c) Calculate the total profits generated by each package when sold completely at equilibrium price

The chart below shows the profit that is generated from each package if the cost remains as given and the equilibrium price is charged for each unit respectively. The Cost and revenue has been calculated earlier and the Profit is equal to (Total revenue- Total cost). For instance for the first package, Total revenue is 5 units*$25, that is equal to $125. And the total cost calculated earlier was 75 (for package 1) thus the difference, of 50 is the profit from the first package. Similarly calculating for package 2, the profit is 60 and so on. Interestingly, the profit increases gradually as the units of production increases and the price decreases to match demand, however, for the last package sold at a slightly higher price (6) with maximum units produced, see a drastic fall in profit to 0.

units per package	Per unit price	Total cost per package	Equilibrium price per unit	Total revenue earned	Profit= Total cost - Total revenue
5	15	75	25	125	50
10	9	90	15	150	60
25	7	175	9	225	50
50	5	250	7	350	100
100	6	600	6	600	0

The fall in profit is due to the marginal increase in the price from the last package, with an increased quantity of production.

(d) Calculate the net gains by subtracting the total cost from the total of the package

In case of including all the packages cost and revenue in calculation, would give us the probable net gain from the packages. Summing all the costs and all of the revenues from the 5 packages would give the net gain. Below chart gives the net gain calculated from all the packages.

Total cost per package	Total revenue earned
75	125
90	150
175	225
250	350
600	600
1190	1450
Net gain from all packages	260

Thus, the net gain from all the packages is $260

(e) Determine which package is the most convenient

It would be convenient for the garments supplier to order the package that would be available at the least cost per unit and also the package that would provide maximum profit from transaction.

We see that the package 4, with per unit cost of $5, which is the lowest, that can be sold at equilibrium price of $7, that is also low (and would not deter demand for the product due to price). Finally, the forth package with the equilibrium price and quantity as above fetches the maximum profit of $100. Hence with minimum cost, competitive price and maximum profit, the package that consists of 50 units, or the forth package from the above calculation would be the most convenient for the garments seller.