Chapter 8 Cost Mc GrawHillIrwin Copyright 2008 by

Chapter 8 Cost Mc. Graw-Hill/Irwin Copyright © 2008 by The Mc. Graw-Hill Companies, Inc. All Rights Reserved.

Main Topics ¢Types of cost ¢What do economic costs include? ¢Short-run cost: one variable input ¢Long-run cost: cost minimization with two variable inputs ¢Average and marginal costs ¢Effects of input price changes ¢Economies and diseconomies of scale 8 -2

Types of Cost ¢ Firm’s total cost is the expenditure required to produce a given level of output in the most economical way ¢ Variable costs are the costs of inputs that vary with output level ¢ Fixed costs do not vary as the level of output changes, although might not be incurred if production level is zero ¢Avoidable versus sunk costs 8 -3

Production Costs: An Example Table 8. 1: Fixed, Variable, and Total Costs of Producing Garden Benches 8 -4 Number of Benches Produced per Week Fixed Costs (per Week) Variable Cost (per Week) Total Cost (per Week) 0 $1, 000 $0 $1, 000 33 $1, 000 500 1, 500 74 $1, 000 2, 000 132 $1, 000 2, 000 3, 000

Economic Costs ¢Some economic costs are hidden, such as lost opportunities to use inputs in other ways ¢Example: Using time to run your own firm means giving up the chance to earn a salary in another job ¢An opportunity cost is the cost associated with forgoing the opportunity to employ a resource in its best alternative use 8 -5

Short Run Cost: One Variable Input ¢ If a firm uses two inputs in production, one is fixed in the short run ¢ To determine the short-run cost function with only one variable input: ¢Identify the efficient method for producing a given level of output ¢This shows how much of the variable input to use ¢Firm’s variable cost = cost of that amount of input ¢Firm’s total cost = variable cost + any fixed costs ¢ Can be represented graphically or mathematically 8 -6

Figure 8. 1: Variable Cost from Production Function 8 -7

Figure 8. 2: Fixed, Variable, and Total Cost Curves ¢ Dark red curve is variable cost ¢ Green curve is fixed cost ¢ Light red curve is total cost, vertical sum of VC and FC 8 -8

Long-Run Cost: Cost Minimization with Two Variable Inputs ¢In the long run, all inputs are variable ¢Firm will have many efficient ways to produce a given amount of output, using different input combinations ¢Which efficient combination is cheapest? ¢Consider a firm with two variable inputs K and L, and inputs and outputs that are finely divisible 8 -9

Isocost Lines ¢ An isocost line connects all input combinations with the same cost ¢ If W is the cost of a unit of labor and R is the cost of a unit of capital, the isocost line for total cost C is: ¢ Rearranged, ¢ Thus the slope of an isocost line is –(W/R), the negative of the ratio of input prices 8 -10

Isocost Lines, continued ¢ Isocost lines closer to the origin represent lower total cost ¢ A family of isocost lines contains, for given input prices, the isocost lines for all possible cost levels of the firm ¢ Note the close relationship between isocost lines and consumer budget lines ¢Lines show bundles that have same cost ¢Slope is negative of the price ratio 8 -11

Sample Problem 1: ¢Plot the isocost line for a total cost of $20, 000 when the wage rate is $10 and the rental rate is $40. ¢How does the isocost line change if the wage rises to $20?

Least-Cost Production ¢ How do we find the least-cost input combination for a given level of output? ¢Find the lowest isocost line that touches the isoquant for producing that level of output ¢ No-Overlap Rule: The area below the isocost line that runs through the firm’s least-cost input combination does not overlap with the area above the Q-unit isoquant ¢ Again, note the similarities to the consumer’s problem 8 -13

Garden Bench Example, Continued ¢In the long run, Naomi and Noah can vary the amount of garage space they rent and the number of workers they hire ¢An assembly worker earns $500 per week ¢Garage space rents for $1 per square foot per week ¢Inputs are finely divisible 8 -14

Figure 8. 7: Least-Cost Method, No-Overlap Rule Example Square Feet of Space, K A 2500 2000 D 1500 B 1000 Q = 140 C = $3500 C = $3000 1 2 3 4 5 6 Number of Assembly Workers, L 8 -15

Interior Solutions ¢ A least-cost input combination that uses at least a little bit of every input is an interior solution ¢ Interior solutions always satisfy the tangency condition: the isocost line is tangent to the isoquant there ¢Otherwise, the isocost line would cross the isoquant ¢Create an area of overlap between the area under the isocost line and the area above the isoquant ¢This would not minimize the cost of production 8 -16

Least-Cost Production and MRTS ¢ Restate the tangency condition in terms of marginal products and input prices: ¢ Slope of isoquant = -(MRTSLK) ¢ MRTS = ratio of marginal products ¢ Slope of isocost lines = -(W/R) ¢ Thus the tangency condition says: ¢ Marginal product per dollar spent must be equal across inputs when the firm is using a least-cost input combination 8 -17

Least-Cost Input Combination ¢ How can we find a firm’s least-cost input combination? ¢ If isoquant for desired level of output has declining MRTS: ¢ Find an interior solution for which the tangency condition formula holds ¢ That input combination satisfies the no-overlap rule and must be the least-cost combination ¢ If isoquant does not have declining MRTS: ¢ First identify interior combinations that satisfy the tangency condition, if any ¢ Compare the costs of these combinations to the costs of any boundary solutions 8 -18

Sample Problem 2: ¢Suppose the production function for Gadget World is Q = 5 L 0. 5 K 0. 5. The wage rate is $25 and the rental rate is $50. What is the least-cost combination of producing 100 gadgets? 200?

The Firm’s Cost Function ¢ To determine the firm’s cost function need to find least-cost input combination for every output level ¢ Firm’s output expansion path shows the least -cost input combinations at all levels of output for fixed input prices ¢ Firm’s total cost curve shows how total cost changes with output level, given fixed input prices 8 -20

Figure 8. 10: Output Expansion Path and Total Cost Curve 8 -21

Average and Marginal Cost ¢ A firm’s average cost, AC=C/Q, is its cost per unit of output produced ¢ Marginal cost measures now much extra cost the firm incurs to produce the marginal units of output, per unit of output added ¢ As output increases: ¢ Marginal cost first falls and then rises ¢ Average cost follows the same pattern 8 -22

Cost, Average Cost, and Marginal Cost Table 8. 3: Cost, Average Cost, and Marginal Cost for a Hypothetical Firm 8 -23 Output (Q) Tons per day Total Cost (C) (per day) Marginal Cost (per day) Average Cost (per day) 0 $0 $0 $0 1 1, 000 2 1, 800 900 3 2, 100 300 700 4 2, 500 400 625 5 3, 000 500 6 3, 600 600 7 4, 300 700 614 8 5, 600 1, 300 700

AC and MC Curves ¢ When output is finely divisible, can represent AC and MC as curves ¢ Average cost: ¢Pick any point on the total cost curve and draw a straight line connecting it to the origin ¢Slope of that line equals average cost ¢Efficient scale of production is the output level at which AC is lowest ¢ Marginal cost: ¢Firm’s marginal cost of producing Q units of output is equal to the slope of its cost function at output level Q 8 -24

Figure 8. 16: Relationship Between AC and MC ¢ AC slopes downward where it lies above the MC curve ¢ AC slopes upward where it lies below the MC curve ¢ Where AC and MC cross, AC is neither rising nor falling 8 -25

Marginal Cost, Marginal Products, and Input Prices ¢ Intuitively, a firm’s costs should be lower the more productive it is and the lower the input prices it faces ¢ Formalize relationship between marginal cost, marginal products, and input prices using the tangency condition: 8 -26

More Average Costs: Definitions ¢ Apply idea of average cost to firm’s variable and fixed costs to find average variable cost and average fixed cost: ¢ Since total cost is the sum of variable and fixed costs, average cost is the sum of AVC and AFC: 8 -27

Average Cost Curves ¢Fixed costs are constant so AFC is always downward sloping ¢At each level of output the AC curve is the vertical sum of the AVC and AFC curves ¢Average cost curve lies above both AVC and AFC at every output level ¢Efficient scale of production exceeds output level where AVC is lowest 8 -28

Figure 8. 18: AC, AVC, and AFC Curves 8 -29

Figure 8. 20: AC, AVC, and MC Curves 8 -30

Effects of Input Price Changes ¢ Changes in input prices usually lead to changes in a firm’s least-cost production method ¢ Responses to a Change in an Input Price: ¢When the price of an input decreases, a firm’s leastcost production method never uses less of that input and usually employs more ¢For a price increase, a firm’s least-cost input production method never uses more of that input and usually employs less 8 -31

Figure 8. 21: Effect of an Input Price Change ¢ Point A is optimal input mix when price of labor is four times more than the price of capital ¢ Point B is optimal when labor and capital are equally costly 8 -32

Short-run vs. Long-run Costs ¢ In the long run a firm can vary all inputs ¢ Will choose least-cost input combination for each output level ¢ In the short run a firm has at least one fixed input ¢ Produce some level of output at least-cost input combination ¢ Can vary output from that in short run but will have higher costs than could achieve if all inputs were variable ¢ Long-run average variable cost curve is the lower envelope of the short-run average cost curves ¢ One short-run curve for each possible level of output 8 -33

Figure 8. 24: Input Response over the Long and Short Run 8 -34

Figure 8. 25: Long-run and Shortrun Costs 8 -35

Figure 8. 26: Long-run and Shortrun Average Cost Curves 8 -36

Economies and Diseconomies of Scale ¢ What are the implications of returns to scale? ¢ A firm experiences economies of scale when its average cost falls as it produces more ¢Cost rises less, proportionately, than the increase in output ¢Production technology has increasing returns to scale ¢ Diseconomies of scale occur when average cost rises with production 8 -37

Figure 8. 28: Returns to Scale and Economies of Scale 8 -38

Sample Problem 3 (8. 12): Noah and Naomi want to produce 100 garden benches per week in two production plants. The cost functions at the two plants are and , and the corresponding marginal costs are MC 1 = 600 – 6 Q 1 and MC 2 = 650 – 4 Q 2. What is the best output assignment between the two plants?