FREC 240 Assignment #4
(1.)  A competitive firm sells output Q for $12/unit, so its Total Revenue TR = 12Q.  
    Its total cost TC includes fixed cost FC = $67.50, plus variable cost VC = 0.01Q3 (for positive values of Q).  
  1. Specify the firm's total profit as a function of output Q.
  2. Take the derivative of TR with respect to Q to obtain marginal revenue (MR)
  3. Take the derivative of TC (= FC + VC) to obtain the marginal cost function (MC)
  4. Take the derivative of the total profit function to obtain the marginal profit function (this should be the same as MR minus MC).
  5. Solve this marginal profit function for the value of Q at which the derivative of the profit function equals zero.
  6. Plug this value of Q into the profit function to determine the firm's maximum (or minimum) total profit.
  7. Take the second derivative of the profit function, and plug in this value of Q.  Is the result negative or positive at this value of Q?  (A negative second derivative means means that the slope of the profit function is going from positive to negative as Q increases, which means that profit is maximized.  A positive second derivative would imply the slope of the profit function is going from negative to positive, which implies a profit minimum.)
  8. Specify the firm's average total cost (ATC = TC/Q = FC/Q + VC/Q) as a function of output Q.
  9. Take the derivative of the ATC function.  
  10. Solve for the value of Q at which ATC is minimized.  
  11. Plug this value of Q into the ATC function to determine the firm's break-even price (equals minimum point on ATC schedule).
  12. Take the second derivative of the ATC function.  Is it negative or positive at this value of Q?
  13. Specify the firm's average variable cost (AVC = VC/Q) as a function of output Q.
  14. Is there a positive value of Q at which AVC is minimized?  (In economic terms, is there an output price below which the firm would shut down in the short run?)
(2.)  A competitive firm uses two inputs, K and L, to produce output Y.  In the short run, K is fixed but L is variable; in the long run, both K and L are variable.  Its production (or total product) function is Y = K0.4L0.4 (economists call this a Cobb-Douglas production function).
  1. Take a minute to check out the properties of this production function.  If you plug in values K=100 and L=100, what output do you get?  If you double L while holding K constant, do you get less than double the output (diminishing returns to L)?  If you double both K and L, do you get more or less than double the output (increasing or decreasing returns to scale)?
  2. Take the partial derivative of Y with respect to K to obtain the marginal product (MP) function of K. Take the partial derivative of Y with respect to L to obtain the MP function of L.
  3. If the price of Y = $25 so that total revenue TR = 25K0.4L0.4, what are the marginal value product (MVP) functions of K and  L?
  4. If the cost of K is P(K) = $2/unit, the cost of L is P(L) = $4 and and K is fixed at 500 units, specify the firm's short-run total profit (TR-TC) as a function of L only.  
  5. Take the partial derivative of this short-run profit function with respect to L.  This should be the same as MVP(L) minus P(L).
  6. Set this marginal profit function equal to zero, and solve for the short-run profit-maximizing level of L where K=500.
  7. Use this value of L to calculate the firm's short-run output level and profit.
  8. If P(K) = $2/unit and P(L) = $4/unit and both K and L are variable, specify the firm's long-run total profit (TR - TC) as a function of both K and L.  
  9. Take the partial derivative of profit with respect to K.  This should be the same as the MVP(K) minus P(K).
  10. Take the partial derivative of profit with respect to L.  This should be the same as MVP(L) minus P(L).
  11. Set both of these marginal profit functions equal to zero, and solve these two simultaneous equations for the profit-maximizing levels of K and L.  This is tricky, but note that if MVP(L) = 4 and MVP(K) = 2, then MVP(L)/MVP(K) = 4/2 = 2.  Economists refer to the ratio MVP(L)/MVP(K) as "marginal rate of technical substitution of L for K" and the ratio P(L)/P(K) as "marginal rate of transformation of L for K."  Sound familiar?
  12. Use these values of K and L to calculate the firm's long-run output level and profit.
  13. Suppose P(K) remains at $2/unit but P(L) increases to $6/unit.  Determine the new marginal profit functions with respect to K and L, set these equal to zero and solve them for the new profit-maximizing values of K and L.  Use these values of K and L to calculate the firm's new long-run output level and profit.