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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 zeroAvoidable versus sunk costs
FC is avoidable if it is =0 when Q=0FC is sunk if it is >0 when Q=0
Production Costs: An Example
Table 8.1: Fixed, Variable, and Total Costs of Producing Garden Benches
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 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 waysExample: 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 outputThis shows how much of the variable input to useFirm’s variable cost = cost of that amount of inputFirm’s total cost = variable cost + any fixed costs
Can be represented graphically or mathematically
Figure 8.1: Variable Cost from Production Function
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)=2Lthe firm needs Q/2 units of labor to
produce Q units of outputif 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 variableFirm 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
- A and B costs the same- D is cheaper - What are other costs combinations??
A
D
B
W=$500 and r=$1
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 linesLines show bundles that have same costSlope 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 outputNo-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
Figure 8.7: Least-Cost Method, No-Overlap Rule Example
Q = 140
Square Feetof Space, K
1 2 3 4 5 6
500
1000
1500
2000
2500
Number of Assembly Workers, L
B
A
C = $3500
D
C = $3000
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 thereOtherwise, the isocost line would cross the isoquantCreate an area of overlap between the area under
the isocost line and the area above the isoquantThis 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
High School L
Q=100, slope= -(MPH/MPC)= -1
slope= -(MPH/MPC) > -1
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
No Output
L
K
Output Expansion Path
Output Expansion Path
C=$2000 C=$4000 C=$7000
Q=100 Q=200 Q=300
Lumpy Inputs
1
D E F
TC
C=$2000
C=$4000
C=$7000
Q=100 Q=200 Q=300
TC Curve
C
Q
C=$1000
F
E
D
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
Cost, Average Cost, andMarginal Cost
Table 8.3: Cost, Average Cost, and Marginal Cost for a Hypothetical Firm
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 1,000 1,000
2 1,800 800 900
3 2,100 300 700
4 2,500 400 625
5 3,000 500 600
6 3,600 600 600
7 4,300 700 614
8 5,600 1,300 700
8-23
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 originSlope of that line equals average costEfficient scale of production is the output level at
which AC is lowestMarginal 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 curvesAverage cost curve lies above both AVC and
AFC at every output levelEfficient scale of production (Qe)exceeds
output level where AVC is lowest
Figure 8.18: AC, AVC, andAFC Curves
Figure 8.20: AC, AVC, andMC Curves
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
Figure 8.25: Long-run and Short-run Costs
Figure 8.26: Long-run and Short-run Average Cost Curves
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 moreCost rises less, proportionately, than the increase in
outputProduction technology has increasing returns to
scale
Diseconomies of scale occur when average cost rises with production
Figure 8.28: Returns to Scale and Economies of Scale
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?
Read: 8.9