Cost Theory and Applications

Topics to be covered Relevant Cost Cost functions and cost elasticity Cost estimation Economies of Scope Learning Curve

1. Relevant Costs- costs that vary with alternatives 2. Sunk costs- costs incurred regardless of alternative. Costs that cannot be recovered (no opportunity value) 3. Incremental cost- costs that change with a change in activity level (output level, investment level, etc.) · Proper cost measure (long run v. short run) must be tied to the duration of the planning horizon. · long run- all inputs are variable. · short run- one or more inputs are fixed. Consider variable, incremental and marginal costs Ignore sunk costs Variable costs depends on planning horizon

Rail vs Motor Carrier AFC AVC AFC Price RailMotor Minimum Prices should be based on average variable or marginal costs. Of course, actual prices can be higher. In 1980, Congress defined relevant cost as variable costs. Example of Defining Relevant Costs

What costs are relevant in multi product firms? Proposed Madison-Chicago train service? –Chicago-Madison Train Variable cost Fixed Cost Hiawatha Line- common line costs –Midwest territory- common territory costs Amtrak System- common system costs All services on a train should cover their direct costs + common train costs. All trains on a line should cover their direct costs + common line costs All lines in a territory should cover their direct costs + common territory costs All territories in the system should cover their direct cost + common system costs. Trains Lines Territories

Deriving cost functions Suppose Price of labor = $40 and Price of capital = $10/250hp Short Run: Y=1 (HP=250), vary X Q STCAvg Cost 1 10(1) + 40(1) = (1) + 40(2) = (1) + 40(3) = (1) + 40(4) = (1) + 40(5) = (1) + 40(6) = (1) + 40(7) = (1) + 40(8) = (1) + 40(9) = Long Run Suppose Optimal X = Y 1 10(1) + 40(1) = (2) + 40(2) = (3) + 40(3) = (4) + 40(4) = (5) + 40(5) = (6) + 40(6) = (7) + 40(7) = (8) + 40(8) = 4006 LRTCSRTC Q Q Total Cost $/period

Short and Long Run Total Costs SRTC1 SRTC2 LRTC Quantity $/Yr LAC=LRTC/Q LMC = dLRTC/dQ AFC=TFC/Q SAC = SRTC/Q SAVC= TVC/Q SMC= dSRTC/dQ TFC TVC

Unit Cost Curves $/Q Quantity SAV1 SMC1 SMC2 SAC2 LAC LMC

2. Long and Short Run Cost Concepts- a. long run: all inputs can be varied Returns to Scale v. long run costs b. Short run: certain inputs are fixed per time period Average fixed cost = AFC = total fixed cost/Q Average variable cost = SAVC = total variable cost/Q Average total cost = SAC = total cost/Q Incremental cost = marginal cost = SMC = d total cost/dQ

3. Cost Elasticity = dC/dQ Q/C = Marginal cost/ average cost a. if cost elasticity < 1, economies of scale in LR economies of utilization in SR b. if cost elasticity > 1, diseconomies of scale in LR diseconomies of utilization in SR c. If long run cost elasticity = short run cost elasticity, firm has efficient size plant for that output d. Railroad Example: Long Run Short Run Small Railroads Large Railroads Small roads have too little output Large roads have excess capacity SMC SAC LAC=LMC Small RRLarge RR

Unit Cost vs. Cost Elasticity SRMC2 SRAC2 SRMC1 SRAC1 LRAC $/Q QUANTITY

Empirical Cost Estimation Approaches A. Accounting: may involve the following 1. separate fixed and variable costs 2. assign variable portion to output measures, 3. calculate unit costs by dividing assigned cost by output units 4. For a particular product or service, multiple the unit costs by the respective number of units output for that product B. Engineering C. Survey or Survivor Techniques D. Statistical

Engineering Technique Example: Oil Pipeline Throughput = f(diameter of pipe, horsepower of engines driving fluids, number of pumping stations) T = H D / or T = k H.37 D 1.73 where T = throughput H = horsepower D = diameter of pipe

Statistical A. Long Run v. Short Run Cross sectional v. Time series data B. Requirements 1. Output Matching - example of deferred maintenance in RR 2. Uniform production with a time period 3. no technological change- might add a time variable to the regression equation 4. no changes in factor prices or inflation 1. deflate by a price index 2. reconstruct costs based on future prices and historic input and output levels 3. include factor prices in the cost function

Long Run and Short Run Costs Chrysler Ford $/Q Output (Q)

Technical Change Add a time term to the Cost Function ln Cost = b 0 + b 1 ln Q + b 2 t + e Q = output t = time b 2 = the percentage change in cost per year

Adjust for Factor Prices Deflate by a price index –Cost / CPI = b 0 + b 1 Q Include factor prices in the Cost Function –ln Cost = b 0 + b 1 ln Q + b 2 ln P L + b 3 ln P F + b 4 ln P K + e Reconstitute Costs based on future prices

Production with Two Outputs-- Economies of Scope Examples: – Chicken farm--poultry and eggs – Automobile company--cars and trucks – University--Teaching and research

Production with Two Outputs-- Economies of Scope The degree of economies of scope measures the savings in cost can be written: – C(Q 1 ) is the cost of producing Q 1 alone – C(Q 2 ) is the cost of producing Q 2 alone – C(Q 1 Q 2 ) is the joint cost of producing both products

Production with Two Outputs-- Economies of Scope Interpretation: – If SC > 0 -- Economies of scope – If SC < 0 -- Diseconomies of scope

Example: Economies of Scope in the Trucking Industry Issues – TruckLoad versus Less-than-TruckLoad – Direct versus indirect routing – Length of haul

Example: Economies of Scope in the Trucking Industry Empirical Findings – Results SC = for reasonably large firm SC = for very large firms – Interpretation Combining partial loads at an intermediate location lowers cost management difficulties with very large firms.

Many network industries exhibit economies of scope, density or scale From Shapiro and Varian, Information Rules One of the most fundamental features of information goods is that their cost of production is dominated by “first copy costs.” In the language of economics, the fixed costs of production are large, but the variable cost of reproduction are small. This cost structure leads to substantial economies of scale: the more you produce, the lower your average cost of production. But there is more to it than just economies of scale, the fixed costs and the variable costs of producing information have a special structure. The dominant component of the fixed costs of producing information are sunk cost, cost that are not recoverable if production is halted. … Sunk costs generally have to be paid up front, before commencing production. The variable cost of information production also have an unusual structure: the cost of producing an additional copy typically does not increase, even if a great many copies are made. These cost structures characterize the airline industry and others but the information goods is an extreme example. Lessons in Pricing: Don’t get greedy. Play tough

Airline Industry Hubs with Density Economies LTL truckers tried it but found it provided too slow service in regional markets Old Route Structures Hub and Spoke

Network Effects Demand Based Value depends upon how many others in the group use the product.