Question

A lemon-growing cartel consists of four orchards. Their total cost functions are

TC₁ = 20 + 5Q₁²

TC₂ = 25 + 3Q₂²

TC₃ = 15 + 4Q₃²

TC₄ = 20 + 6Q₄²

TC is in hundreds of dollars, and Q is in cartons per month picked and shipped.

1. Tabulate total, average, and marginal costs for each firm for output levels between 1 and 5 cartons per month (i.e., for 1, 2, 3, 4, and 5 cartons).
2. If the cartel decided to ship 10 cartons per month and set a price of $25 per carton, how should output be allocated among the firms?
3. At this shipping level, which firm has the most incentive to cheat? Does any firm not have an incentive to cheat?

Short answer

a. The tabulate total, average, and marginal cost for each firm is given below:

b. Firms 1 and 4 will produce 2 units each, and firms 2 and 3 will produce 3 units each.

c. Firm 2 has the incentive to cheat. Firm 3 and 4 have no incentive to cheat, and firm 1 is indifferent.

Step-by-step solution

Explanation for part (a)

The total cost, average and marginal cost for firm1 is calculated below:

$$\begin{gathered} {\text{TC}}_{\text{1}}{\text{= 20 + 5Q}}_{\text{1}}^{\text{2}} \\ {\text{TC}}_{\text{Q = 0}}\text{= 20 + 5}{\left(\text{0}\right)}^{\text{2}} \\ \text{= 20} \\ {\text{AC}}_{Q=0}\text{=}\frac{\text{TC}}{\text{Q}} \\ \text{=}\frac{\text{20}}{\text{0}} \\ \text{= -} \\ {\text{MC}}_{Q=0}{\text{= TC}}_{\text{n}}{\text{- TC}}_{\text{n - 1}} \\ \text{= -} \\ {\text{TC}}_{\text{Q = 1}}\text{= 20 + 5}{\left(\text{1}\right)}^{\text{2}} \\ \text{= 20 + 5} \\ \text{= 25} \\ \text{AC =}\frac{\text{TC}}{\text{Q}} \\ \text{=}\frac{\text{25}}{\text{1}} \\ \text{= 25} \\ {\text{MC = TC}}_{\text{n}}{\text{- TC}}_{\text{n - 1}} \\ \text{= 25 - 20} \\ \text{= 5} \end{gathered}$$

Similarly, the total, average, and marginal costs are calculated from 0 to 5 units.

The table below shows the total cost, average cost, and marginal cost of all 4 firms.

Explanation for part (b)

The cartel will assign the production in such a manner that the lowest marginal cost is achieved:

Cartel Unit Assigned	Firm Assigned	MC
1	2	3
2	3	4
3	1	5
4	4	6
5	2	9
6	3	12
7	1	15
8	2	15
9	4	18
10	3	20

Thus, firms 1 and 4 will produce 2 units each, and firms 2 and 3 will produce 3 units each.

Explanation for part (c)

The firm whose marginal cost is lowest beyond its allocation has an incentive to cheat. Thus, firm 2 has the lowest marginal cost for producing the 4^thunit than the other; also, marginal cost is lower than the price; hence, firm 2 has an incentive to cheat. The other firm has marginal cost either greater than or equal to price. Firms 3 and 4 have marginal costs greater than the price; thus, there is no incentive to cheat. Firm 1 is indifferent as its marginal cost is less than the price.