18  Firm Supply

For more information on these topics, see Varian Chapter 22: Firm Supply.

18.1 Classwork 18: Firm Supply

Consider the cost function: \(TC = q^3 - 12 q^2 + 48 q + 100\).

  1. Fill in the blanks to find the essential cost curves:
  • Fixed Cost (FC) = 100 (the constant term)
  • Variable Cost (VC) = ___ (the parts of the cost function which vary with q)
  • Marginal Cost (MC) = ___ (derivative of TC)
  • Average Variable Cost (AVC) = ___ (VC/q)
  • Average Total Cost (ATC) = \(q^2 - 12 q + 48 + \frac{100}{q}\) (TC/q)

  1. Show that the minimum value for the AVC is 12. To find the minimum of this function, find where its derivative is equal to 0.

  2. Show that the minimum value for the ATC is around 27.3. Do this by plotting ATC and zooming in.

  3. In classwork 15, we learned that to profit maximize, the firm should set \(MR = MC\) and produce at that quantity. If the firm is price-taking (it’s small relative to the larger market and must supply at the market price in order to sell anything), then \(TR = PQ\) and \(MR = \frac{dTR}{dQ} = P\). But there is a caveat to the \(P = MC\) rule: if the market price gets too low, the firm won’t be able to cover its costs and would be better off producing 0 than producing where \(P = MC\). Let’s work through some specific prices to discover what that shutdown point could be. Hint: the quadratic formula helps you find the zeroes of a quadratic equation. It is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

    1. Explain why the firm has profit \(\pi = -100\) if it produces \(q = 0\).

    2. Take \(P = 30\), which is a price that’s higher than the minimum ATC. Show that if the firm produces until \(MC = P\), it will earn \(\pi \approx 19\). Is the firm better off producing where \(MC = P\), or better off producing 0?

    3. Take \(P = 27\), which is a price that’s lower than the minimum ATC. Is the firm better off producing where \(P = MC\), or better off producing 0?

    4. Take \(P = 12\), which is a price that’s equal to the minimum AVC. Is the firm better off producing where \(P = MC\), or better off producing 0?

    5. Take \(P = 10\), which is a price that’s lower than the minimum AVC. Is the firm better off producing where \(P = MC\), or better off producing 0?

    6. Given your answers to the previous question, what is the shutdown rule? Give an intuitive explanation. Do fixed costs matter in a firm’s supply decision?