Unformatted text preview: Cost Curves Definitions The long run (total) cost function, LRTC(y), is defined by: LRTC(y) = Min x1,x2 w1 x1 + w2 x2 s.t. f(x1, x2) = y The short run (total) cost function, SRTC(y), is defined when one of the factors is fixed. Suppose x2 is fixed at level z. Then SRTC(y) is defined by: SRTC(y) = Min x1, w1 x1 + w2 z s.t. f(x1, z) = y Note: For any y, SRTC(y) ≥ LRTC(y), and they are only equal if z happens to be the cost‐minimizing input level. When we want to talk generically about a cost function (whether short run or long run), we call it the cost function, c(y), or equivalently the total cost function, TC(y). Average cost: AC(y) = TC(y) / y Relationship between returns to scale and the average cost curve (assuming no fixed costs): If the production function is CRS in some region of y, then AC(y) is constant in that region. If the production function is IRS in some region of y, then AC(y) is decreasing in that region. (Often true at low levels of production due to specialization.) If the production function is DRS in some region of y, then AC(y) is increasing in that region. (Often true at high levels of production because need layers of management and harder to monitor employees.) A typical short run cost function in real world production processes has a fixed cost, F, as well as a variable cost, cv(y) or equivalently VC(y): c(y) = cv(y) + F Short run: AC(y) = c(y) / y = cv(y) / y + F / y = AVC(y) + AFC(y) Long run: AC(y) = c(y) / y = cv(y) / y = AVC(y) (because no fixed costs in the long run) Marginal cost: MC(y) = ∂c(y) /∂y =∂cv(y) /∂y (Note: This is a partial derivative because costs also depend on factor prices.) Key points to remember for drawing cost curves: ‐ AFC(0) and AC(0) equal infinity if there is a fixed cost (i.e., in the SR) ‐ AVC will usually decline at low y (due to IRS at low output) and will usually rise at high y (due to DRS at high output) ‐ AC will usually decline at low y (if there is a fixed cost, which is only in the SR) and will usually rise at high y (due to DRS at high output) ‐ MC(0) = AVC(0) ...
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