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Unformatted text preview: e, . CSR = CSR (Q,) Figure #3: Family of SR Cost Curves
$ Y = f(X) / g(X) 2 7 CSR (1) CSR (2) CSR (3) CSR (4) CLR Q 17 In this example, the long run cost curve is the envelope (a curve that is tangent to each member curve in the family of curves) of the family of short run cost curves. We can calculate the envelope of a family of curves using the following steps... 1. Get the slope with respect to the parameter , 2. Set this slope equal to zero and solve for the optimal parameter *, 3. Substitute the optimal parameter * into the equation of the curve. Let's do an example, consider the following family of short run cost curves: CSR (Q,) = - 2Q2 + (15 )Q + 2 where Q denotes the output level and denotes the plant size. STEP 1: Get the slope with respect to the parameter . CSR (Q,) / = 2 Q STEP 2: Set this slope equal to zero and solve for the optimal parameter *. CSR (Q,) / = 0 2 Q = 0 or * = Q / 2 STEP 3: Substitute the optimal parameter * into the equation of the curve. CLR (Q) = - 2Q2 + (15 *)Q + *2 = - 2Q2 + (15 Q / 2)Q + (Q / 2)2 = 8/4Q2 2/4Q2 + 1/4Q2 + 15Q = 9/4Q2 + 15Q This is the equation of the envelope of this family of short run cost curves. Notice that there is no fixed cost component in the equation in the long run cost curve. This is consistent with the notion that in the long run all costs are variable. Review: MRS and MRTS MRS or Marginal Rate of Substitution is the rate at which one good can be substituted for another good along the same indifference curve. MRS = -Y (along the same IC or holding utility constant) X 18 This is equivalent to saying that MRS is equal to the (negative) slope of the indifference curve. We should also remember from ECON 201 that the MRS is the ratio of the two marginal utilities. MRS = MUX / MUY MRTS or Marginal Rate of Technical Substitution is the rate at which one factor can be substituted for another factor along the same isoquant. MRTS = -L (along the same IQ or holding output constant) K This is equivalent to saying that MRTS is equal to the (negative) s...
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- Spring '10