Download 2017 - Segment 1.3 - Theory and Estimation of Cost

Managerial Economics
Theory and Estimation of
Costs
Aalto University
School of Science
Department of Industrial Engineering and Management
January 10 – 26, 2017
Dr. Arto Kovanen, Ph.D.
Visiting Lecturer
Production costs
 There are alternative types of costs
 Opportunity cost is the value of forgone opportunity
(could be indirect and implicit)
 Out-of-pocket cost
 Incremental cost varies by the range of options available
 Sunk cost does not vary by the scale of production
 Firm’s cost structure reflects its production process,
driven by production technology and input prices
 Usually firms are “price takers” in the market for
inputs, but not always (e.g., only employer in town)
 Total variable costs (TVCs) are associated with the
variable inputs, such as labor, materials, and so on
Production costs (cont.)
 Firms try to minimize costs of production regardless of
market conditions!
 Total fixed costs (TFCs) are associated with the use of
fixed inputs (some may be variable in the long run)
 Total costs (TCs) represent the total of TVC and TFC
 Marginal cost (MC) is the change in total cost
associated with a change in output
MC = dTC/dQ = dTVC/dQ + dTFC/dQ
 Relationship between marginal product and marginal
cost: when MP is falling, MC is increasing (due to the
diminishing returns)
Production costs (cont.)
 Formally, this can be expressed as follows:
MC = dTVC/dQ = w*dL/dQ = w*(1/MP) where MP is the
marginal product of labor input
 Short-run cost function tells what is the minimum cost
necessary to produce a particular level of output
 Average total cost (AC) is the average cost per unit of
all inputs
 Average variable cost (AVC) is the per-unit cost of using
variable inputs
 Average fixed cost (AFC) is the per-unit cost of fixed
inputs
Production costs (cont.)
Production costs (cont.)




Exercise: plot TC, TVC, AC, AVC and MC curves
Analyze the relationship between MC and AVC
What can we say about AFC in this table?
How does a change in the cost of fixed input affect the
position of AC curve?
 How does a change in the unit cost of variable input
affect the position of MC, AC, and AVC curves?
 Example: unit cost of labor declines.
Production costs (cont.)
Production costs (cont.)
 In the long-run, (almost) all inputs can be varied and
there are no fixed costs
 Long-run production costs are influenced by
economies of scale
 If IRTS, then costs increase at a decreasing rate
 If CRTS, then costs increase at a constant rate
 If DRTS, then costs increase at an increasing rate
 We can calculate
 Long-run marginal cost (LRMC)
 Long-run average cost (LRAC), which is u-shaped,
reflecting the different stages of economies of scale
Production costs (cont.)
 While the short-run average cost curves are derived
from the law of diminishing returns, the long-run
cost curve gets its shape from return to scale,
characterized by the underlying long-run production
function
 The shape of the long-run average cost curve varies
by industry (due to variations in return to scale)
 There is no a priori reason to assume that the LRATC
is u-shaped (e.g., CRTS has a horizontal LRATC curve)
 Minimum efficient scale (MES) is the lowest per unit
cost of production in the long-run
Production costs (cont.)
 In the short-run, total cost function is the following:
TC = PKK0 + w*L = TFC + TVC
where the capital stock is fixed while the labor input is
variable
 Example: David was a computer programmer at the
ICM and earned $120,000 per year. Now David has
established his own consultancy. His monthly fixed
costs, including rent, property, casualty and health
insurance are $5,000. David’s monthly variable cost
includes wages/salaries, telephone, maintenance,
office supplies, internet , ... , and total $20,000.
Production costs (cont.)
 Assuming that David does not pay himself a salary,
what are the total monthly out-of-pocket costs?
 What are the firm’s explicit monthly accounting and
economic costs?
 Can we say anything about the following with this
information:
 Is David’s company profitable?
 How many hours David should work to minimize his
costs?
Production costs (cont.)
 Duality of firm’s optimization problem:
 Minimizing PKK + wL subject to g(K, L)] ≥ Q*
 Maximize g(K, L)] subject to PKK + wL ≤ c*
 Both lead to the same first-order conditions
except for the Lagrangian (λ; measures the
impact on the objective when the constraint
is relaxed)
 Write the Lagrangian equation for both and
solve it!
Production costs (cont.)
 In the short run firm should continue producing as long as
unit price covers average variable cost (P > AVC) – WHY?
 In the long run firm has to cover all costs
 Hence it should continue to operation only if it expects to
earn positive economic profit in the long run
 A firm suffering persistent economic losses should close
 Example. Firm’s cost function is C = 270 + 30Q + 0.3Q2. It
faces demand P = 50 – 0.2Q. What is optimal level of Q in
the short and long run?
 The same principle applies to multiple production cases,
i.e., each product should have a positive contribution to
the firm’s profits
Production costs with several
products






Suppose a firm produces two goods: Q1 and Q2
Total fixed cost is $2.4 million per year
Prices of goods are P1 = $10.00 and P2 = $6.50
Average variable costs are AVC1 = $9.00 and AVC2 = $4.00
Let us assume that Q1 = 1.2 million and Q2 = 0.6 million
Is it profitable to produce both goods in the short and
long run?
 What if P2 falls to $5.50?
 What if P2 falls to 3.50?
Economies of scope and cost
complementaries
 Economies of scope exist when the total cost of
using the same production facility to produce two or
more goods is less than that of producing these
goods at separate facilities
TC (Q1, 0) + TC(0, Q2) > TC(Q1, Q2)
 Example: It may be less expensive for Ford Motor
Company to produce cars and trucks using a single
assembly line than do it in two separate facilities
 Cost complementarities exist when MC of Q1 is
lowered by increasing the production of Q2
Economies of scope (cont.)
 Example. A company produces two products (SUVs
and light trucks) in the same manufacturing plant
 Its costs are described by the following function:
TC(Q1, Q2) = 25 + Q12 + 4*Q22 - 5*Q1*Q2
 Calculate AC and MC for each product?
 Do cost complementarities exist? Show where?
 Discuss economies of scope for the firm.
 If the firm sells its division producing Q1, how much it
will cost to produce 10 units of Q2?
 What is the total cost of producing 5 units of Q1 and
10 units of Q2 after divesture?
Learning effect
 Learning in industry means that as the production
volume increases the unit cost of production falls
 As more times a task has been performed, the less
time it requires on each subsequent iteration – this is
learning
 Learning curves vary widely by industry
 Learning curves can be used for many purposes:
 Internal: forecasting labor usage, scheduling,
costing
 External: supply-chain dynamics
 Strategic: evaluation of company performance
Learning effect (cont.)
 Arithmetic approach: each time production doubles,
labor per unit declines by a constant (learning) factor
 That is, time to produce the n-th unit = T*Ln
 E.g., 1st unit requires 100 hours. If learning factor is
0.8, 2nd unit requires 80 hours, 4th unit 64 hours, ….
 Logarithmic approach: Tn = t1*(Nc) where c is the
slope of the learning curve
 Learning curve coefficient approach: Tn = T1*c where
the coefficient c is given for each level of production
 Reasons for learning effect
 Over time labor learns to do things more efficiently
 Better use of equipment, changes in input mix
 Product redesign
Example of firm’s decisions









The firm produces a single product or service
The firm services a single market
Its objective is to maximize profit
The task is to determine how much to produce and at
what price to sell
There is no uncertainty
Let us assume that the demand is declining in price
Q = 8.5 – 0.05*P
Draw the demand curve
Rearrange as follows: P = 170 – 20* Q
This is the firm’s “inverse demand equation”
Example (cont.)
 Total revenue of the firm is given by:
P*Q = 170*Q – 20*Q2
 Question to ask is what level of output maximizes the
firm’s revenues?
d(P*Q)/dQ = 170 – 40*Q = 0
Q* = 170/40 = 4.25 = 4 units
 What is the shape of the revenue function?
 There is, however, costs associated with production
 Let us assume that C = 100 + 38*Q
where 100 = fixed cost (unrelated to Q)
38 = variable cost (related to Q)
Example (cont.)
 We see that total costs (C) is linear, rising function of Q
 Profit is the difference between revenues and costs
 In this example, total profits (π) can be written as
π = P*Q – C = 170*Q – 20*Q2 - (100 + 38*Q)
 Maximization of profits with respect to Q means
dπ/dQ = 170 – 40*Q – 38 = 0
Q* = (170 – 38)/40 = 132/40 = 3.3 = 3 units
 Taking into account production costs, it is optimal for
the firm to produce 3 units
 Profits = P*Q – C = (170 – 20*3)* 3 – (100 – 38*3)
= 330 – 100 – 114 = 116
Sensitivity analysis
 We have assumed full certainty about the parameter of
the example
 In reality, however, conditions can changes
 Sensitivity analysis helps assess the impact of changes
in the key parameters
 Increased overhead/fixed costs, rising from 100 to 112
 Increased material costs, rising from 38 to 46
 Increased demand. The new demand is given by
Q = 10 – 0.05* P
 How would production and profits be affected by
these changes?
Sensitivity analysis (cont.)
 Let’s us further assume that one-half of the demand is
coming from abroad
 What implications will this have for firm’s operations?
 How would changes in the exchange rate would affect
firm’s production decision?
 What about having a portion of inputs imported from
abroad?
 How would depreciation of the U.S. dollar against the
foreign currency (say, the Euro) affect the producer’s
output, sales at home and abroad, and profitability?