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An industry consists of eleven identical firms with costs [tex]c(q) = 4q + q^2[/tex]. The market demand is [tex]Q = 100 - p[/tex].

1. What are the equilibrium price, output, and profits of each firm if all eleven participate in a cartel?

2. If a single firm cheated, what would its output and profits be, assuming the other firms maintain the cartel price?

Answer :

Let's break down the question step-by-step:


  1. Equilibrium Price, Output, and Profits in a Cartel:

    In a cartel, the firms work together like a monopoly to maximize joint profits by setting a common output level and price. For this, we need to calculate the total cost, total revenue, and then find the profit-maximizing output and price.


    • Total Cost (TC) for one firm:
      [tex]TC = 4q + q^2[/tex]
      where [tex]q[/tex] is the quantity produced by each firm.


    • Total Cost for the entire industry: Assuming each firm produces [tex]q[/tex] units, total quantity produced by all firms is:
      [tex]Q = 11q[/tex]


    • Market Demand:
      [tex]Q = 100 - p[/tex]
      Substituting [tex]Q = 11q[/tex] into the demand equation, we get:
      [tex]11q = 100 - p \implies p = 100 - 11q[/tex]


    • Total Revenue (TR):
      [tex]TR = p \times Q = (100 - 11q) \times 11q[/tex]
      Simplifying:
      [tex]TR = 1100q - 121q^2[/tex]


    • Profit ([tex]\Pi[/tex]):
      [tex]\Pi = TR - TC = 1100q - 121q^2 - 11(4q + q^2)[/tex]
      [tex]\Pi = 1100q - 121q^2 - 44q - 11q^2[/tex]
      [tex]\Pi = 1056q - 132q^2[/tex]



    To find the optimal output, take the derivative of the profit function and set it to zero:
    [tex]\frac{d\Pi}{dq} = 1056 - 264q = 0[/tex]
    [tex]q = \frac{1056}{264} = 4[/tex]

    Substituting [tex]q = 4[/tex] into [tex]p = 100 - 11q[/tex]:
    [tex]p = 100 - 11(4) = 56[/tex]


    • Equilibrium Price: $56

    • Equilibrium Output per firm: 4 units


    To find profit for each firm:
    [tex]\Pi = 56 \times 4 - (4 \times 4 + 4^2) = 224 - 16 - 16 = 192[/tex]


    • Equilibrium Profit: $192 per firm



  2. Output and Profits if One Firm Cheats:

    If one firm cheats and decides to produce more while others remain at the cartel level, assume the cheating firm produces [tex]q_c = q + x[/tex] where [tex]x[/tex] is the additional output.

    Under the cartel price:


    • Revenue for the cheating firm:
      [tex]R_c = 56 \times (4 + x)[/tex]


    • Cost for the cheating firm:
      [tex]C_c = 4(4 + x) + (4 + x)^2 = 16 + 4x + 16 + 8x + x^2 = 32 + 12x + x^2[/tex]


    • Profit for the cheating firm:
      [tex]\Pi_c = R_c - C_c = 56(4 + x) - (32 + 12x + x^2)[/tex]
      [tex]\Pi_c = 224 + 56x - 32 - 12x - x^2[/tex]
      [tex]\Pi_c = 192 + 44x - x^2[/tex]



    The cheating firm will set [tex]x[/tex] to maximize [tex]\Pi_c[/tex]. Taking the derivative with respect to [tex]x[/tex] and setting it to zero:
    [tex]\frac{d\Pi_c}{dx} = 44 - 2x = 0[/tex]
    [tex]x = 22[/tex]

    So, this firm will allot [tex]q_c = 26[/tex] units. Substituting back:
    [tex]\Pi_c = 192 + 44(22) - 22^2 = 192 + 968 - 484 = 676[/tex]


    • Output and Profit if one firm cheats:

      • Output: 26 units

      • Profit: $676






Therefore, under cheating conditions, the firm's profit is significantly higher compared to maintaining the cartel agreement.

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