A Closer Look at Production and Costs Chapter 7 Appendix A Closer Look at Production and Costs © 2006 Thomson/South-Western

The Production Function and Efficiency The production function identifies the maximum quantities of a particular good or service that can be produced per time period with various combinations of resources, for a given level of technology

Exhibit 11: Production Function The effects of adding additional labor to an existing amount of capital can be seen by starting with any level of capital and reading across the table. When 1 unit of capital and 1 unit of labor are employed, the firm produces 40 units of output per month. If the amount of labor is increased by 1 unit with the amount of capital held constant, output increases to 90 units and marginal product of labor is 50 units If 1 unit of capital is combined with 7 units of labor, the firm can produce 290 units of output per month. Several combinations of labor and capital yield 290 units of output per month 90 – 40 = 50

Exhibit 11: Production Function A Firm’s Production Function Using Labor and Capital: Production per Month Units of Capital Units of Labor Employed per Month Employed per Month 1 2 3 4 5 6 7 1 40 90 150 200 240 270 290 2 90 140 200 250 290 315 335 3 150 195 260 310 345 370 390 4 200 250 310 350 385 415 440 5 240 290 345 385 420 450 475 6 270 320 375 415 450 475 495 7 290 330 390 435 470 495 510 The marginal product of labor first rises, showing increasing marginal returns from labor, and then declines, showing diminishing marginal returns Similarly, the marginal product of capital also reflects first increasing marginal returns from capital, then diminishing marginal returns Different combinations of resources yield the same rate of output; for example, several combinations of labor and capital yield 290 units of output per month.

Exhibit 12: Isoquants 10 a h f b g 5 Units of capital per month e Q Combinations that yield 290 units of output are presented as points a, b, c, and d Connecting these points forms an isoquant, Q1 Isoquant shows all the technologically efficient combinations of two resources, such as labor and capital, that produce a certain rate of output  along Q1, output remains constant at 290 units, but the combination of resources varies from much capital and little labor to little capital and much labor 10 a h f b g 5 Units of capital per month e Q (475) 3 c Q (415) d 2 Q (290) 1 5 10 Units of labor per month

Properties of Isoquants Different isoquant for every output rate Isoquants farther from the origin indicate higher rates of output Along a given isoquant, the quantity of labor employed is inversely related to the quantity of capital employed  isoquants have negative slopes Isoquants do not intersect Isoquants are usually convex to the origin  any isoquant gets flatter as we move down along the curve

Marginal Rate of Technical Substitution The absolute value of the slope of the isoquant is the marginal rate of technical substitution, MRTS, between two resources The MRTS is the rate at which labor substitutes for capital without affecting output  when much capital and little labor are used, the marginal productivity of labor is relatively great and the marginal productivity of capital relatively small  one unit of labor will substitute for a relatively large amount of capital

Exhibit 12: Isoquants 10 a Units of capital per month b 5 c d Q (290) As more labor and less capital are used, the marginal product of labor declines and the marginal product of capital increases In moving from point a to b, 1 unit of labor substitutes for 2 units of capital  MRTS between points a and b equals 2 From c to d, 2 units of labor substitute for 1 unit of capital  MRTS between c and d equals 1/2 10 a Units of capital per month b 5 c d Q (290) 1 5 10 Units of labor per month

Marginal Rate of Technical Substitution The extent to which one input substitutes for another is directly linked to the marginal productivity of each input For example, between points a and b, 1 unit of labor replaces 2 units of capital, yet output remains constant  labor’s marginal product, MPL – the additional output resulting from an additional unit of labor – must be twice as large as capital’s marginal product, MPC MRTS = MPL / MPC

Isocost Lines We now turn to the combination of resources that should be employed to minimize the cost of producing a given rate of output Suppose a unit of labor costs the firm $1,500 per month, and a unit of capital costs $2,500  TC = (w x L) + (r X C) TC = $1,500L + $2,500C

Exhibit 13: Firm’s Isocost Lines w r $1,500 $2,500 10 Slope = – — = – ——–– = – 0.6 Isocost line identifies all combinations of capital and labor the firm can hire for a given total cost There would be an isocost line for every possible level of total cost Units of capital per month 5 TC=$22,500 TC=$19,500 TC=$15,000 5 10 15 Units of labor per month

Isocost Lines At the point where the isocost line meets the vertical axis, the quantity of capital that can be purchased equals the total cost divided by the monthly cost of a unit of capital  TC / r Where the isocost line meets the horizontal axis, the quantity of labor that can be purchased equals the total cost divided by the monthly cost of a unit of labor  TC / w

Isocost Line The slope of the isocost line is given by Slope of isocost line = -(TC/r)/(TC/w) = -w/r In our example, the absolute value of the slope of the isocost line is w /r = 1,500 / 2,500 = 0.6  the monthly wage is 0.6, or six tenths of the monthly cost of a unit of capital  hiring one more unit of labor, without incurring any additional cost, the firm must employ 0.6 fewer units of capital

Exhibit 14: Optimal Combinations of Inputs 10 TC = $19,000 The profit-maximizing firm will operate at point e, where the isoquant is just tangent to the isocost line of $19,000 At point e, the isoquant and isocost line have the same slope  MRTS = ratio of input prices  MRTS = w / r = 1,500 / 2,500 = 0.6 Units of capital per month 5 e Q2 (415) 5 10 Units of labor per month

Expansion Path Imagine a set of isoquants representing each possible rate of output, and given the relative cost of resources, we can then draw isocost lines to determine the optimal combination of resources for producing each rate of output The points of tangency in Exhibit 15 show the least-cost input combinations for producing several output rates

Exhibit 15: Expansion Path Output rate Q2 can be produced most cheaply using C units of capital and L units of labor Connecting the tangency points forms the firm’s expansion path  the lowest long run total cost for each rate of output The firm’s long-run average cost curve indicates, at each rate of output, the total cost divided by the rate of output Expansion path and the long-run average cost curve are alternative ways of portraying costs in the long run T C 4 T C 3 T C 2 Expansion path Units of capital per month T C 1 d c b Q 4 C a Q 3 Q 2 Q 1 L L' Units of labor per month

Exhibit 15: Expansion Path We can use this exhibit to distinguish between short-run and long-run adjustments in output. The firm producing Q2 at point b requires C units of capital and L units of labor If, in the short run, the firm wants to increase output to Q3, the only way is by increasing the quantity of labor to L', which requires moving to point e Point e is not the cheapest way to produce Q3 in the long run because it is not a tangency point In the long run, capital is variable and if the firm wishes to produce Q3, it should minimize total cost by adjusting from point e to point c Units of capital per month T C 4 T C 3 T C 2 Expansion path T C 1 d c b e Q 4 C a Q 3 Q 2 Q 1 L L' Units of labor per month

Expansion Path If the relative prices of resources change, the least-cost resource combination will also change  the firm’s expansion path will change For example, if the price of labor increases, capital becomes relatively less expensive  the efficient production of any given rate of output will therefore call for less labor and more capital