A lemon-growing cartel consists of four orchards. Their total cost functions are \\[ \begin{array}{l} \mathrm{TC}_{1}=20+5 Q_{1}^{2} \\ \mathrm{TC}_{2}=25+3 Q_{2}^{2} \\ \mathrm{TC}_{3}=15+4 Q_{3}^{2} \\ \mathrm{TC}_{4}=20+6 Q_{4}^{2} \end{array} \\] \(\mathrm{TC}\) is in hundreds of dollars, and \(Q\) is in cartons per month picked and shipped. a. 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). b. 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? c. At this shipping level, which firm has the most incentive to cheat? Does any firm not have an incentive to cheat?

Understanding the total cost function is critical for any business or in this case, a lemon-growing cartel. The total cost function combines all the costs associated with production, including both fixed and variable costs. In our example, each of the four orchards has its own total cost function represented as \( TC_1, TC_2, TC_3, \) and \( TC_4 \) with the equations provided in the problem.

These equations allow us to calculate the total cost for any number of cartons, \( Q \), produced and shipped by each orchard. The total costs vary with the square of the quantity produced due to the quadratic term in the function, indicating that costs rise at an increasing rate as production increases. This is a common feature in production where increasing output leads to higher incremental costs.

The average cost (AC) is the total cost (TC) divided by the number of units produced (Q). It tells us the cost of producing each unit and is vital for setting the price to ensure profitability. In our case scenario, For firm 1, at an output level of 1 carton, \( AC = TC / Q \), which simplifies to the total costs divided by 1, yielding the average cost for that output level.

As production increases, it’s interesting to observe how the average cost changes. Often, AC decreases as production increases due to the spreading of fixed costs over more units, a concept known as economies of scale, up to a certain point before it may start increasing again.

The marginal cost (MC) is a concept that refers to the additional cost of producing one more unit. It is derived from the total cost function and in this example, it is the derivative of the TC equation with respect to quantity, \( Q \). For instance, for firm 1, the marginal cost at any given output level is calculated as \(10 \times Q\).

Understanding MC is crucial for decision-making. In a competitive market, firms ideally produce until MC equals the price. A cartel, such as the one in our scenario, may manipulate output and prices, but marginal cost still provides essential information about the cost of increasing production.

A cartel’s production allocation is about distributing the total output among individual firms in a way that minimizes costs for the entire group. The allocation is based on the principle of equating marginal costs across all firms, which essentially means that the last unit produced by each firm costs the same.

For our lemon cartel, this involves calculating the MC for each firm and comparing them to ensure that cumulative production meets the market demand, in our case, 10 cartons per month. The allocation optimally assigns production quantities that take advantage of each firm's cost efficiencies.

The incentive to cheat in a cartel arrangement arises when a firm can benefit by secretly producing more than the agreed quantity. For our lemon cartel, the firm with the lowest marginal cost at the point of allocated production has the most to gain by doing this since it can produce additional units at a lower cost than the agreed price.

On the flip side, if a firm's marginal cost exceeds the cartel's price, it has no incentive to cheat since producing more would result in a loss rather than profit. Analyzing marginal costs relative to the price can indicate which firm might be most tempted to step outside the cartel's agreement.