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Chapter 8. Cost. 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
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Chapter 8 Cost
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
FC is avoidable if it is =0 when Q=0 • FC is sunk if it is >0 when Q=0
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
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
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
if SR production function is Q=F(L)=2L • the firm needs Q/2 units of labor to produce Q units of output • if w=$15, then variable cost function: VC(Q)=wL, or 15(Q/2) • if sunk fixed costs=$100, the firm’s total cost is:C(Q)=100+15(Q/2) • Worker-out-problem 8.1
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
Figure 8.5: Isoquant Example While All these input combinations are associated with efficient production methods, their costs are not all equal A B C
Figure 8.5: Isoquant Example W=$500 and r=$1 - A and B costs the same - D is cheaper - What are other costs combinations?? A D B
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 The level of K associated with each level of L on ISOC line
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
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 isoquant • Again, note the similarities to the consumer’s problem
Garden Bench Example, Continued • In the long run, the producer 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
Square Feet of Space, K Figure 8.7: Least-Cost Method, No-Overlap Rule Example A D 2500 2000 B 1500 Q = 140 C = $3000 C = $3500 1000 500 1 2 3 4 5 6 Number of Assembly Workers, L
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
Boundary Solutions • That’s if the least cost input combination excludes some inputs • Such inputs may not be used (not productive compared to other inputs)
College Edu L Q=100, slope= -(MPH/MPC)= -1 slope= -(MPH/MPC) > -1 High School L A Least cost combination A, (MPH/MPC) > (WH/WC)
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
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
Sample Problem 2: • Suppose the production function for Gadget World is Q = 5L0.5K0.5. 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
Figure 8.10: Output Expansion Path and Total Cost Curve 8-21
Lumpy Inputs Output Expansion Path C=$2000 C=$4000 C=$7000 Q=100 K Q=200 Q=300 D E F 1 Output Expansion Path No Output L
TC Curve TC C F C=$7000 E C=$4000 D C=$2000 C=$1000 Q=100 Q Q=300 Q=200
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
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
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
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:
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:
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 (Qe)exceeds output level where AVC is lowest
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 least-cost 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
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
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
Figure 8.24: Input Response over the Long and Short Run In SR, increasing Q from 140 to160: Shift from B-F In LR, Shift from B-D A is least C combination Thus: CLR<CSR
Figure 8.24: Input Response over the Long and Short Run In SR, decreasing Q from 140 to120: Shift from B-E In LR, Shift from B-A A is least C combination Thus: CLR<CSR
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
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 MC1 = 600 – 6Q1 and MC2 = 650 – 4Q2. What is the best output assignment between the two plants?
