A Closer Look at Production and Costs CHAPTER 7 Appendix © 2003 South-Western/Thomson Learning

The Production Function and Efficiency The way in which resources can be combined to produce output can be summarized by a firm’s production function 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 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 If 1 unit of capital is combined with 7 units of labor, the firm can produce 290 units of output per month. The firm produces the maximum possible output given the combination of resources employed  that same output could not be produced with fewer resources  production is technologically efficient 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  marginal product of labor is 50 units

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 By reading across the table, you will discover that the marginal product of labor first rises, showing increasing marginal returns from labor, and then declines, showing diminishing marginal returns Similarly, by holding the amount of labor constant and following down the column, you find that the marginal product of capital also reflects first increasing marginal returns from capital, then diminishing marginal returns Notice that 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 provides another perspective on this information

Exhibit 12: Isoquants 10 a h Units of capital per month f b g 5 e Q Combinations that yield 290 units of output are presented as points a, b, c, and d. These points can be connected to form an isoquant Q1 10 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 a h Units of capital per month f b g 5 e Q (475) 3 c Q (415) d 2 Q (290) 1 5 10 There will be a different isoquant for every quantity of output. Units of labor per month

Properties of Isoquants There is a different isoquant for every output rate the firm could possibly produce with isoquants farther from the origin indicating 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

Properties of Isoquants Isoquants do not intersect. Since each isoquant refers to a specific rate of output, an intersection would indicate that the same combination of resources could, with equal efficiency, produce two different amounts of output 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 Thus, 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) However, as more labor and less capital are used, the marginal product of labor declines and the marginal product of capital increases  it takes more labor to make up for a one-unit reduction in capital In moving from point a to b along Q1, 1 unit of labor substitutes for 2 units of capital  the MRTS between points a and b equals 2 10 a Units of capital per month b 5 For example, in moving from point c to point d, 2 units of labor substitute for 1 unit of capital  MRTS between points c and d equals ½. 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

Marginal Rate of Technical Substitution Anywhere along the isoquant, the marginal rate of technical substitution of labor for capital equals the marginal product of labor divided by the marginal product of capital, which also equals the absolute value of the slope of the isoquant MRTS = MPL / MPC

Marginal Rate of Technical Substitution If labor and capital were perfect substitutes in production, the rate at which labor substituted for capital would remain fixed along the isoquant  the isoquant would be a downward sloping straight line Summary Isoquants farther from the origin represent higher rates of output Isoquants slope downward Isoquants never intersect Isoquants are bowed toward the origin

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 Isocost line identifies all combinations of capital and labor the firm can hire for a given total cost h t n o w r $1,500 $2,500 m 10 Slope = – — = – ——–– = – 0.6 r e p l a t i p 3 such isocost lines for three different levels of total cost are shown here. In reality, there would be an isocost line for every possible level of total cost a c 5 TC=$22,500 f o TC=$19,500 s TC=$15,000 t i n U 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 Thus, 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

Choice of Input Combinations Exhibit 14 brings together the isoquants and the isocost lines. The profit maximizing firm wants to produce its chosen output at the minimum cost  it tries to find the isoquant closest to the origin that still touches the isoquant

Exhibit 14: Optimal Combinations of Inputs The profit maximizing firm will operate at point e, where the isoquant is just tangent to the isocost line of $19,000 h t n 10 TC = $19,000 o m At point e, the isoquant and isocost line have the same slope  the marginal rate of technical substitution equals the ratio of input prices r e p l a t i p a 5 c e MRTS = w / r = 1,500 / 2,500 = 0.6  the firm adjusts resource use so that the rate at which one input can be substituted for another in production – the marginal rate of technical substitution – equals the rate at which one resource can be exchanged for another in resource markets, -- w / r. f o Q2 (415) s t i n U 5 10 Units of labor per month

Expansion Path If we 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 The least-cost input combination for producing each rate of output is shown by the point of tangency between the isoquant and the isocost line. For example, output rate Q2 can be produced most cheaply using C units of capital and L units of labor T C h t 4 n T o C m The line formed by connecting these tangency points is the firm’s expansion path  the lowest long-run total cost for each rate of output. For example, the firm can produce output rate Q2 for TC2, output rate Q3 for TC3, and so on. 3 r e T C p 2 l Expansion path a t i T p C a 1 d c f c o b s Q t 4 i C a Q n Similarly, the firm’s long-run average cost curve indicates, at each rate of output, the total cost divided by the rate of output  the expansion path and the long-run average cost curve are alternative ways of portraying costs in the long run 3 U Q 2 Q 1 L L' Units of labor per month

Exhibit 15: Expansion Path We can also 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. T C h t 4 n T o C m 3 r e T Suppose that in the short run, the firm wants to increase output to Q3. C p 2 l Expansion path a t i T p C Since capital is fixed in the short run, the only way to produce Q3 is by increasing the quantity of labor to L', which requires moving to point e. a 1 d c f c o b s e Q t 4 i C a Q n 3 U Q 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 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