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Extended Cost Example 2

Extended Cost Example 2. Marginal and average product curves The marginal product of input 1 is the partial derivative of the production function with respect to the first input. The average product of input 1 is the ratio of output to input 1.

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Extended Cost Example 2

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  1. Extended Cost Example 2

  2. Marginal and average product curves • The marginal product of input 1 is the partial derivative of the production function with respect to the first input. • The average product of input 1 is the ratio of output to input 1. • Given the short-run production function y = f(z1, 16) = 2(z1)0.25 • the marginal and average product functions are

  3. Units of output Units of output AP1 AP1 MP1 MP1 Z1 Z1 • Examples of graphs of marginal and average product curves

  4. Given the fixed level of z2, let Z*(y) denote the amount of input 1 needed to produce output y (i.e., f(Z*(y), z2) = y). Then • AP1 (Z*(y))Z*(y) = y, • the short-run variable cost to produce y is w1Z*(y), and the short-run average variable cost to produce y is • Thus there is an inverse relationship between SAVC and AP1. Combining this with the relationship between any marginal and corresponding average functions, we see that SMC >SAVC if and only if MP1 <AP1. • This statement must be interpreted carefully, since the cost functions are functions of output while the product functions are functions of the variable input. • The output y for the cost curve corresponds to input level Z*(y) for the product curves.

  5. Units of output $ SMC AP1 MP1 SAVC Z*(y) y output Z1 • Graphs of short-run cost and product curves. • Note output y for the cost curve corresponds to input level Z*(y) for the product curves.

  6. Relationship between long-run and short-run cost curves • Fixthe input prices, w1 and w2. • We will use the notation C(y), MC(y), AC(y), z1 (y), and z2(y) to denote the long-run cost, long-run marginal cost, long-run average cost, and the two conditional factor demands, respectively. • We will use the notation STC(y; z2), SMC(y; z2), and SATC(y; z2) to denote the short-run total cost, short-run marginal cost, and short-run average total cost, respectively, where each short-run function depends on both the output level, y, and the level of the fixed factor, z2. • Some simple observations provide the insights necessary to understand the relationship between the long-run and short-run cost curves.

  7. 1) In the long-run, it is always possible to mimic what was done in the short-run, so it cannot be cheaper to produce y in the short-run than in the long-run, i.e., • STC(y; z2) ≥ C(y) for every z2 and y. • Since the average cost functions are obtained by dividing the total cost functions by the same output level, a similar condition holds for average cost functions: • SATC(y; z2) ≥ AC(y) for every z2 and y.

  8. (2) If, in the short-run, the fixed factor is at the optimal (long-run) level for producing a certain output level, then the short-run cost of producing that output level is no more than the long-run cost (since the variable factor may also be set at the (long-run) optimal level for producing that output). • Combined with observation (1), this yields • STC(y; z2(y)) = C(y) for every y. • Once again a similar condition holds for the average cost functions: • SATC(y; z2(y)) = AC(y) for every y.

  9. STC(y;1) STC(y;2) $ STC(y;3) C(y) output • Together, these two observations show that the long-run cost curve is what is called the “lower envelope” of the short-run cost curves. This is illustrated for an example in which there are only three possible values for z2, 1, 2, and 3. For each output level, the long-run cost is the minimum of the three possible short-run costs for producing that output.

  10. STC(y;z2(y*)) $ C(y) y* output • A similar description applies when z2 is not constrained to only three possible values. • [The shape shown for the long-run cost function is not general. The long-run cost function must satisfy two properties: in the long-run the firm can get out of business, so C(0) = 0; and long-run cost must increase as output increases. Beyond that, different production functions could lead to almost any shape.]

  11. The key features are that • any short-run total cost curve must lie on or above the long-run cost curve, • the curves touch at output levels (y* in the figure) for which the short-run fixed factor matches the (long-run) conditional factor demand, • if the cost curves don’t have kinks, the short-run curve is tangent to the long-run curve where they touch. • Again a similar result holds for the average cost curves: • the long-run average cost curve is the “lower envelope” of the short-run average total cost curves, • the two are tangent when they touch.

  12. (3) Since the slope of a cost curve is the corresponding marginal cost, the fact that the short-run curve is tangent to the long-run curve where they touch means that • SMC(y;z2(y)) = MC(y) for every y • i.e., when the short-run fixed factor is at the optimal level for producing output y, the short-run and long-run marginal costs are equal at y.

  13. (4) Not only must the curves be tangent where they touch, but also • the short-run curve must be more sharply “curved upward” at that point since it can never lie below the long-run curve. • Since the second derivative with respect to output determines the curvature, and the second derivative with respect to output is the “derivative of the first derivative”, i.e., the derivative of the corresponding marginal cost with respect to output, this means • i.e., short-run marginal cost is increasing at least as fast as long-run marginal cost at the point of tangency of the cost curves. For the average cost curves, the short-run curve must be more sharply “curved upward” than the long-run curve at points at which they are tangent.

  14. Combining these observations with other things we know (e.g., marginal is below average when average is decreasing, marginal equals average at the minimum of average, and marginal is above average when average is increasing), we can demonstrate the relationship between short-run and long-run marginal and average cost curves. • Starting with a U-shaped long-run average cost curve, AC(y), and the corresponding long-run marginal cost curve, MC(y), consider an output level, y*.

  15. SMC(y;z2(y**)) SMC(y;z2(y*)) $ MC(y) SATC(y;z2(y*)) AC(y) SATC(y;z2(y**)) y** output y* • When z2 = z2(y*), the corresponding short-run average total cost is tangent to the long-run average cost at y = y*. Since long-run average cost is declining at y*, short-run average total cost must be declining there as well. That means the minimum of short-run average total cost must be at an output larger than y*.

  16. SMC(y;z2(y**)) SMC(y;z2(y*)) $ MC(y) SATC(y;z2(y*)) AC(y) SATC(y;z2(y**)) y** output y* • The other point is the minimum point of the short-run average total cost curve, since the marginal and average are always equal where average is minimized.

  17. SMC(y;z2(y**)) SMC(y;z2(y*)) $ MC(y) SATC(y;z2(y*)) AC(y) SATC(y;z2(y**)) y** output y* • A similar analysis is carried out at y** in the figure. Since long-run average cost is increasing at y**, short-run average total cost (for z2 = z2(y**)) must also be increasing at y**, and the minimum point for this short-run average total cost must occur at an output less than y**.

  18. Sample problems • 1. Consider a short-run production function with one variable input, x. When the output is 100, for the corresponding level of the variable input, the average product of the variable input is 20 and the marginal product of the variable input is 10. The per-unit price for the variable input is $500. • a. What is the level of the variable input when output is 100? • b. Approximately how much extra variable input would be needed to raise output from 100 to 101? • c. What is the short-run marginal cost at output 100? • d. Is the short-run marginal cost greater than, equal to, or less than short-run average variable cost at output 100? Explain.

  19. 2. Consider a firm with production function q=x(y0.5) where x and y are the amounts of the two inputs and q is the amount of output. • What is the marginal rate of technical substitution at input bundle (x, y) =(5, 4)? • What is the marginal rate of technical substitution at a general input bundle (x,y)? • If input prices are p = 6 and p =9, what is the optimal input bundle to produce output 3? • If input prices are p = 6 and p = 9, what is the firm’s lowest long-run cost of producing output 3? • If input prices are p = 6 and p= 9, for a general output, q, what is the relationship between the optimal amounts of inputs x and y used to produce q? • Suppose in the short run the amount of the second input is fixed at y =4. Find the total product, marginal product, and average product as functions of x. • Suppose in the short run the amount of the second input is fixed at y =4 and the input prices are p = $16 and p = $5. Find all seven short-run cost functions.

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