the theory and estimation of cost the importance of cost in managerial decisions the definition and...
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The Theory and Estimation of Cost
• The Importance of Cost in Managerial Decisions• The Definition and Use of Cost in Economic Analysis• The Relationship Between Production and Cost• The Short Run Cost Function• The Long Run Cost Function• The Learning Curve• Economies of Scope• Economies of Scale: the Short Run Versus the Long Run• Supply Chain Management• Ways Companies Have Cut Costs to Remain Competitive
Learning Objectives
• Define the cost function and explain the difference between a short-run and a long-run cost function.
• Explain the linkages between the production function and the cost function.
• Distinguish between economic cost and accounting cost.
• Explain how the concept of relevant cost is used in the economic analysis of cost.
• Define short-run total cost, short-run variable cost, and total fixed cost and explain their relationship to each other.
Learning Objectives
• Define average cost, average variable cost, and average fixed cost and explain their relationship to each other in the short run. Do the same for average cost and average variable cost in the long run.
• Compare and contrast the short-run cost function and the long-run cost function and explain why economies of scale is considered to be a long-run phenomenon.
• Provide at least four reasons for the existence of economies of scale.
The Importance of Costin Managerial Decisions
• Ways to contain or cut costs over the past decade• Most common: reduce number of people on
the payroll
• Outsourcing components of the business
• Merge, consolidate, then reduce headcount
The Definition and Use of Cost in Economic Analysis
• Relevant cost: a cost that is affected by a management decision.
• Historical cost: cost incurred at the time of procurement.
• Opportunity cost: amount or subjective value that is forgone in choosing one activity over the next best alternative.
• Incremental cost: varies with the range of options available in the decision.
• Sunk cost: does not vary in accordance with decision alternatives.
The Relationship Between Production and Cost
• Cost function is simply the production function expressed in monetary rather than physical units.
• Assume the firm is a “price taker” in the input market.
The Relationship Between Production and Cost
• Total Variable Cost (TVC): the cost associated with the variable input, determined by multiplying the number of units by the unit price.
• Marginal Cost (MC): the rate of change in total variable cost.
• The law of diminishing returns implies that MC will eventually increase
MP
W
Q
TVCMC
The Relationship Between Production and Cost
• Plotting TP and TVC illustrates that they are mirror images of each other.
• When TP increases at an increasing rate, TVC increases at a decreasing rate.
The Short-Run Cost Function
• For simplicity the following assumptions are made:• The firm employs two inputs, labor and capital.• The firm operates in a short-run production period where
labor is variable, capital is fixed.• The firm uses the inputs to produce a single product.• The firm operates with a fixed level of technology.• The firm operates at every level of output in the most
efficient way.• The firm operates in perfectly competitive input markets and
must pay for its inputs at a given market rate. It is a “price taker” in the input markets.
• The short-run production function is affected by the law of diminishing returns.
The Short-Run Cost Function
• Standard variables in the short-run cost function:• Quantity (Q): the amount of output that a firm can
produce in the short run.• Total fixed cost (TFC): the total cost of using the
fixed input, capital (K)• Total variable cost (TVC): the total cost of using the
variable input, labor (L)• Total cost (TC): the total cost of using all the firm’s
inputs, L and K.TC = TFC + TVC
Total and Variable Costs
TC(Q): Minimum total cost of producing alternative levels of output:
TC(Q) = TVC(Q) + TFC
TVC(Q): Costs that vary with output.
TFC: Costs that do not vary with output.
$
Q
C(Q) = VC + FC
VC(Q)
FC
0
Fixed and Sunk Costs
FC: Costs that do not change as output changes.
Sunk Cost: A cost that is forever lost after it has been paid.
$
Q
FC
C(Q) = VC + FC
VC(Q)
The Short-Run Cost Function
• Standard variables in the short-run cost function:• Average fixed cost (AFC): the average per-unit cost of using
the fixed input K.AFC = TFC/Q
• Average variable cost (AVC): the average per-unit cost of using the variable input L.AVC = TVC/Q
• Average total cost (AC) is the average per-unit cost of using all the firm’s inputs.AC = AFC + AVC = TC/Q
• Marginal cost (MC): the change in a firm’s total cost (or total variable cost) resulting from a unit change in output.MC = TC/Q = TVC/Q
Average Total CostATC = AVC + AFCATC = C(Q)/Q
Average Variable CostAVC = VC(Q)/Q
Average Fixed CostAFC = FC/Q
Marginal CostMC = C/Q
$
Q
ATCAVC
AFC
MC
MR
The Short-Run Cost Function
• Important Observations• AFC declines steadily over the range of
production.
• When MC = AVC, AVC is at a minimum.
• When MC < AVC, AVC is falling.
• When MC > AVC, AVC is rising.
• The same three rules apply for average cost (AC) as for AVC.
The Short-Run Cost Function
• A reduction in the firm’s fixed cost would cause the average cost line to shift downward.
• A reduction in the firm’s variable cost would cause all three cost lines (AC, AVC, MC) to shift.
Fixed Cost
$
Q
ATC
AVC
MC
ATC
AVC
Q0
AFC Fixed Cost
Q0(ATC-AVC)
= Q0 AFC
= Q0(FC/ Q0)
= FC
Variable Cost
$
Q
ATC
AVC
MC
AVCVariable Cost
Q0
Q0AVC
= Q0[VC(Q0)/ Q0]
= VC(Q0)
$
Q
ATC
AVC
MC
ATC
Total Cost
Q0
Q0ATC
= Q0[C(Q0)/ Q0]
= C(Q0)
Total Cost
The Short-Run Cost Function
• Alternative specifications of the Total Cost function• Most commonly: specified as a cubic relationship
between total cost and output• As output increases, total cost first increases at a
decreasing rate, then increases at an increasing rate.
• Quadratic relationship• As output increases, total cost increases at an increasing
rate.
• Linear relationship• As output increases, total cost increases at a constant rate.
Cubic Cost Function
• C(Q) = f + a Q + b Q2 + cQ3
• Marginal Cost?• Memorize:
MC(Q) = a + 2bQ + 3cQ2
• Calculus:
dC/dQ = a + 2bQ + 3cQ2
An Example• Total Cost: C(Q) = 10 + Q + Q2
• Variable cost function:VC(Q) = Q + Q2
• Variable cost of producing 2 units:VC(2) = 2 + (2)2 = 6
• Fixed costs:FC = 10
• Marginal cost function:MC(Q) = 1 + 2Q
• Marginal cost of producing 2 units:MC(2) = 1 + 2(2) = 5
The Long-Run Cost Function
• In the long run, all inputs to a firm’s production function may be changed.
• Because there are no fixed inputs, there are no fixed costs.
• The firm’s long run marginal cost pertains to returns to scale.• First, increasing returns to scale.• As firms mature, they achieve constant returns, then
ultimately decreasing returns to scale.
The Long-Run Cost Function
• When a firm experiences increasing returns to scale:• A proportional increase in all inputs
increases output by a greater proportion.
• As output increases by some percentage, total cost of production increases by some lesser percentage.
The Long-Run Cost Function
• Economies of Scale: situation where a firm’s long-run average cost (LRAC) declines as output increases.
• Diseconomies of Scale: situation where a firm’s LRAC increases as output increases.
• In general, the LRAC curve is u-shaped.
Economies of Scale
LRAC
$
Q
Economiesof Scale
Diseconomiesof Scale
The Long-Run Cost Function
• Reasons for long-run economies• Specialization in the use of labor and capital.• Prices of inputs may fall as the firm realizes volume
discounts in its purchasing.• Use of capital equipment with better price-
performance ratios.• Larger firms may be able to raise funds in capital
markets at a lower cost than smaller firms.• Larger firms may be able to spread out promotional
costs.
The Long-Run Cost Function
• Reasons for Diseconomies of Scale• Scale of production becomes so large that it
affects the total market demand for inputs, so input prices rise.
• Transportation costs tend to rise as production grows.• Handling expenses, insurance, security, and
inventory costs affect transportation costs.
The Long-Run Cost Function
• In long run, the firm can choose any level of capacity.
• Once it commits to a level of capacity, at least one of the inputs must be fixed. This then becomes a short-run problem.
• The LRAC curve is an envelope of SRAC curves, and outlines the lowest per-unit costs the firm will incur over a range of output.
The Learning Curve• Learning Curve: line showing the relationship between labor
cost and additional units of output.• Downward slope indicates additional cost per unit declines as
the level of output increases because workers improve with practice.
• Measured in terms of percentage decrease in additional labor cost as output doubles.
Yx = Kxn
• Yx = Units of factor or cost to produce the xth unit• K = Factor units or cost to produce the Kth (usually first) unit• x = Product unit (the xth unit)• n = log S/log 2• S = Slope parameter
Economies of Scope
• Economies of Scope: reduction of a firm’s unit cost by producing two or more goods or services jointly rather than separately.
• Closely related to economies of scale.
Multi-Product Cost Function
• C(Q1, Q2): Cost of jointly producing two outputs.
• General function form:
22
212121, cQbQQaQfQQC
Economies of Scope
• TC(Q1, 0) + TC(0, Q2) > TC(Q1, Q2).
• It is cheaper to produce the two outputs jointly instead of separately.
• Example:• It is cheaper for Time-Warner to produce
Internet connections and Instant Messaging services jointly than separately.
Cost Complementarity
• The marginal cost of producing good 1 declines as more of good two is produced:
MC1Q1,Q2) /Q2 < 0.
• Example:• Cow hides and steaks.
Quadratic Multi-Product Cost Function• TC(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
• MC1(Q1, Q2) = aQ2 + 2Q1
• MC2(Q1, Q2) = aQ1 + 2Q2
• Cost complementarity: a < 0
• Economies of scope: f > aQ1Q2
TC(Q1 ,0) + TC(0, Q2 ) = f + (Q1 )2 + f + (Q2)2
TC(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
f > aQ1Q2: Joint production is cheaper
A Numerical Example:
• TC(Q1, Q2) = 90 - 2Q1Q2 + (Q1 )2 + (Q2 )2
• Cost Complementarity?
Yes, since a = -2 < 0
MC1(Q1, Q2) = -2Q2 + 2Q1
• Economies of Scope?
Yes, since 90 > -2Q1Q2
Supply Chain Management
• Supply Chain Management (SCM): efforts by a firm to improve efficiencies through each link of a firm’s supply chain from supplier to customer.
• Includes all internal and external activities required to fulfill a customer’s demand.
• Transaction costs are incurred by using resources outside the firm.
• Coordination costs arise because of uncertainty and complexity of tasks.
• Information costs arise because information is essential to the proper coordination of activities between the firm and its suppliers.
Supply Chain Management
• Ways to develop better supplier relationships• Strategic alliance: firm and outside supplier
join together in some sharing of resources.
• Competitive tension: firm uses two or more suppliers, thereby helping the firm keep its purchase prices under control.
Ways Companies Have Cut Costs to Remain Competitive
• The Strategic Use of Cost• Reduction in Cost of Materials• Using Information Technology to Reduce Costs• Reduction of Process Costs• Relocation to Lower-Wage Countries or
Regions• Mergers, Consolidation, and Subsequent
Downsizing• Layoffs and Plant Closings