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

2. 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

3. 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

4. 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
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.

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. Exhibit 13: Firm’s Isocost Linesw 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

12. 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

13. 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

14. Exhibit 14: Optimal Combinations of Inputs10 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

15. 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

16. 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

17. 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

18. 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