1. the analytics of constrained optimal decisions microeco nomics spring 2016 costs, profit maximization & supply curve ………….1introduction ………….3 main concepts: an overview session one ………….4fixed, variable and total costs ………….5 average costs ………….6marginal cost ………….7 profit function ………….8profit maximization ……….10 break even analysis …………11shut-down analysis: the short-run ……….12 individual supply curve ……….13 market supply curve ………14exit decision: long-run analysis ……….15 entry decision: long-run analysis ……….18 key points

2. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |1 cost structure: the aluminum industry ► What type of costs does an enterprise incur? ► Does the cost structure differ across industries? ► Does the cost structure differ across firms within the same industry?  The supply chain analysis is still a useful (though sometimes considered outdated) framework to understand the cost structure for a firm/industry. We start with a fairly simple example from the aluminum-producing industry. The flow-diagram on the left “maps” the costs of a typical aluminum smelter (the table on the right). inputs productionfinal product ● alumina ● raw materials ● electricity ● plant power ● consumables ● maintenance ● labor ● freight ● general & adm. Cost type($/mt) alumina369 raw materials125 electricity316 plant power10 consumables70 maintenance50 labor150 freight45 general & adm.75 Total1,210 Figure 1. Cost structure for a typical aluminum smelter  The costs listed on the table are “average” costs as they are expressed in “ $ per metric tone”, i.e. cost per unit produced. How would these costs change for different levels of production/output?  We will devote a considerable part of this section of the course to categorize different types of costs (variable, fixed, average, marginal) and understand what role(s) these costs play into profit maximization decision and developing firm’s supply curve.

3. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |2 cost structure: the airline industry ► What type of costs does an enterprise incur? ► Does the cost structure differ across industries? ► Does the cost structure differ across firms within the same industry?  The cost structure differs across industries depending on the production drivers along the supply chain. The diagrams below offers a glimpse at the cost structure for a few US Network Carriers and Value Carriers for 2014/2015.  Easy to observe that the cost structure differs even across firms within the same industry. The analysis we propose in the remaining of this section of the course should be designed in a way that incorporates this feature and explains its impact on market outcome. Figure 2. Main drivers of costs in the airline industry Cost per Available Seats Mile (cents) Source: Oliver Wyman, Airline Economic Analysis, 2015-2016

4. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |3 main concepts: costs, profit and supply curve ► fixed, variable and total costs ► average and marginal costs We identify costs that vary with the level of production and calculate per unit costs of production. Why is it useful to calculate the “per unit” cost of production ? ► profit maximization ► break-even and shut-down prices No doubt every firm’s objective is to maximize its profit. For a given price P what is the production level Q(P) that would provide the maximum profit ? While profit maximization is desirable, sometimes firms exit the market arguing that the price level is too low and losses are impossible to carry over. How low is the price level for which the exit is preferable ? ► firm level capacity ► market-level capacity aggregation The analysis up to this point provides us with the quantity produced (supplied) by a firm for a given price P. If there are more firms, how do we aggregate production across firms ? costs profit supply

5. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |4 costs typology fixed costs All costs that do not vary with the level of production are called fixed costs – these are constant with respect to the level of Q (production or output): FC = constant. variable costs All costs that vary with the level of production are called variable costs – these are a function of Q (production or output): VC = VC ( Q ) total costs All costs incurred by the firm to produce: it is the sum of variable and fixed costs TC ( Q ) = VC ( Q ) + FC  The shape of the cost functions is specific to the firm/industry. Three example are shown below. TC VC FC output ( Q ) TC VC FC output ( Q ) TC VC FC output ( Q ) VC = b 1 Q + b 2 Q 2 + b 3 Q 3 TC = b 1 Q + b 2 Q 2 + b 3 Q 3 + FC VC = b 1 Q + b 2 Q 2 TC = b 1 Q + b 2 Q 2 + FC VC = b 1 Q TC = b 1 Q + FC

6. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |5 costs typology average cost For any cost category we can define the corresponding average cost by dividing the dollar cost by the output: ► average fixed cost : AFC ( Q ) = FC / Q ► average variable cost: AVC ( Q ) = VC ( Q )/ Q ► average total cost: ATC ( Q ) = TC ( Q )/ Q = [ VC ( Q ) + FC ]/ Q = AVC ( Q ) + AFC ( Q )  The shape of the average cost functions is specific to the firm/industry and is completely defined by the functional form of the variable and total costs. ATC AVC output ( Q ) AVC output ( Q ) ATC AVC output ( Q ) AVC = b 1 + b 2 Q + b 3 Q 2 ATC = b 1 + b 2 Q + b 3 Q 2 + FC / Q AVC = b 1 + b 2 Q ATC = b 1 + b 2 Q + FC / Q AVC = b 1 ATC = b 1 + FC / Q ATC

7. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |6 costs typology marginal cost We define the marginal cost as the incremental cost necessary to produce one more unit of output, i.e. the change in total cost induced by a change in output: ► marginal cost : MC ( Q ) =  TC /  Q Since the fixed costs are constant, the change in total costs is equal exactly with the change in variable costs thus, as  TC =  VC : MC ( Q ) =  VC /  Q  The shape of the marginal cost function is specific to the firm/industry and is completely defined by the functional form of the variable and total costs. AVC output ( Q ) ATC AVC output ( Q ) ATC MC = AVC output ( Q ) MC = b 1 + 2 b 2 Q + 3 b 3 Q 2 AVC = b 1 + b 2 Q + b 3 Q 2 VC = b 1 Q + b 2 Q 2 + b 3 Q 3 MC MC = b 1 + 2 b 2 Q AVC = b 1 + b 2 Q VC = b 1 Q + b 2 Q 2 MC MC = b 1 AVC = b 1 VC = b 1 Q ATC

8. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |7 profit maximization profit function Let’s start with the profit function, i.e. for a given output Q and price P the profit is revenuetotal cost ► The total cost can be written as: i. TC ( Q ) = ATC ( Q )  Q (Use ATC ( Q ) = TC ( Q )/ Q ) The profit function can be written: ii. TC ( Q ) = AVC ( Q )  Q + FC (Use TC ( Q ) = VC ( Q ) + FC and the definition AVC ( Q ) = VC ( Q )/ Q ) The profit function can be written: ► These two ways to write the profit function are equivalent, but each emphasizes how each cost component affects the profit. Notice that for the linear case ( MC ( Q ) = MC ) – for which both the average variable cost and the marginal costs are constant – the profit function becomes iii. if MC ( Q ) = MC The profit function can be written:

9. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |8 profit maximization  Let’s consider a situation in which the marginal cost is increasing and the price level is fixed (the firm can sell any quantity it wishes at this price level). The firm can produce at most Q max (maximum production capacity.)  Start with the top diagram on the right. Would the firm produce the first unit? For that first unit, the firm receives a price P and incur a cost MC (1) < P. Thus, by producing the first unit, the profit increases.  Would the firm produce the second unit? For that second unit, the firm receives a price P and incur a cost MC (2) < P. Again, by producing the second unit, the profit increases.  In the top diagram, the best choice (to maximize profit) is Q *. Why would the firm never produce more than Q *?  For the bottom diagram on the right the firm would keep increasing it’s output until it reaches its maximum capacity. 1 MC ( Q ) Q max P 2 Q*Q* MC ( Q ) Q max P Q * = In the short-run, the firm continues producing until the price equals the marginal cost or the maximum capacity is reached. The optimal output Q* satisfies either P = MC ( Q *) or Q * = Q max profit maximization Figure 3. Profit maximization

10. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |9 profit maximization: graphical analysis AVC Q MC ATC Q* P ATC ( Q* ) AVC ( Q* )  Let’s start with a price level P > min ATC. To maximize profit the firm will choose the output Q * such that P = MC ( Q *). Figure 5. Graphical analysis of profit maximization  The diagram on the right provides the graphical solution to the profit maximization problem.  Once the optimal output is determined we can calculate the profit corresponding to this output as ► The blue area at the top is the profit ► The orange area in the middle is the fixed cost FC. ► The red area at the bottom is the variable cost VC ( Q *) Remark. The sum of the three areas is the total revenue P  Q *.

11. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |10 break-even analysis  As a reminder, for a given output Q and price P, the profit is revenuetotal cost  For the purpose of illustration, let’s consider the situation in the diagram on the right. The average total cost is minimized for some output, call it Q be, thus min ATC = ATC ( Q be )  If the price level is above min ATC, the firm can make a positive profit (by producing for example Q be ).  If the price level is below min ATC, the firm cannot make a positive profit (the best it can do is to produce Q be but even for this output the profit is negative). ATC ( Q ) Q be minATC = ATC ( Q be ) As long as P > min ATC the firm makes a positive profit, however for P < min ATC the firm makes negative profit. The break-even price level is defined as the level of price for which the maximum possible profit is exactly zero, thus P break-even = min ATC ATC increases as the firm chooses any other output (larger or smaller) than Q be break-even Figure 4. Break-even analysis

12. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |11 shut-down analysis: the short-run  Again, for a given output Q and price P, the profit is revenuetotal cost  The firm has two options : exit or continue to produce. ► If the firm exits (shuts-down) then, in the short-run, the fixed cost is still incurred and the firm makes a profit of ► If the firm continues to produce it makes a profit (given a price level P and output Q ):  Comparing the two profits we can draw the following conclusion: as long as the firm can choose an output Q for which P > AVC ( Q ) then the firm is better off producing. As long as P > min AVC the firm covers at least part of the fixed costs. If P < min AVC the firm losses even more than its fixed costs thus the best decision would be to shut-down. The shut- down price level is defined as the level of price for which the firm cannot cover any of its fixed costs: P shut-down = min AVC shut-down

13. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |12 individual supply curve P break-even P shut-down Figure 6. Supply curve, profit maximization, break-even and shut-down analysis AVC Q MC ATC variable costs fully covered fixed costs fully covered variable costs fully covered fixed costs partially covered choose Q * > 0 such that either P = MC ( Q *) or Q * = Q max variable costs partially covered no fixed costs covered positive profit range negative profit range choose Q * = 0 P break-even P shut-down AVC Q MC ATC supply curve The supply curve associates to each possible price level the profit maximizing output. In particular:  for P  P shut-down, Q * satisfies P = MC ( Q *) or Q * = Q max  for P < P shut-down, Q * satisfies Q * = 0 Remark. For P  P shut-down the supply curve coincides with the MC curve.

14. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |13 market supply curve Q  We managed to derive the supply curve for one firm. How do we determine the aggregated (market) supply cure when there are more than one firm participating in the market?  The key point here is really the supply curve definition: for each possible price level the supply curve associates the profit maximizing output. Derivation of the market supply curve could be done in two steps: ► for each price determine the optimal (profit maximizing) output for each firm i, let this output be Q i ( P) ► add individual outputs to determine the total supply for a given price P : Q marke t ( P ) = Q 1 ( P) + Q 2 ( P) +… + Q n ( P) supply curve  MC Q max Firm 1 Q supply curve  MC Q max Firm 2 Q market supply curve 2 Q max Market Figure 7. From individual firms’ supply curves to market supply curve: the case of a market with two identical firms  The aggregated supply is the horizontal sum of individual supply curves. Notice that the market supply capacity is the sum of individual firms’ capacity.

15. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |14 exit decision: long-run analysis AVC Q MC ATC P break-even P shut-down  The analysis we conducted so far assumed that once the firm shuts down (basically set Q = 0) it will still incur the fixed costs. This allowed us to derive the shut-down price.  Moreover, we concluded that the firm will keep producing even if it incurs losses as long as P  min AVC. But for how long can the firm sustain these losses? Not forever.  It is conceivable that once production ends the fixed costs will be incurred only for a limited period of time (think of rent). We call the period over which the fixed costs cannot be avoided as the short-run. Once the fixed costs become avoidable the long-run begins.  How does the behavior of the firm changes in the long-run compared to short-run? Figure 8. Short-run and long-run supply curve AVC Q MC ATC P shut-down  The only change is that in the long-run the firm would rather shut-down for any price level below min ATC. Remark. It is sometimes difficult to accept that the firm should shut-down in the long-run if the price is below minATC.  What of the price returns above minATC?  The interpretation is really “ if in the long-run, i.e. forever, the price remains below minATC then the firm should shut-down ”.

16. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |15 entry decision: long-run analysis  The previous section discussed the exit (or shut-down) decision for firms in the long-run. We really assumed that firms are already in the market… How do firms decide to entry in the market?  We will consider here a fairly simple setup: ► in order to setup a fully operational firm an initial investment I 0 is required; once this amount is paid the firm incurs the usual variable and fixed costs and generates a per-period profit for each period the firm operates. ► the per-period opportunity cost for the investment is r (think of this as the interest rate that the firm could have received each period if instead of investing in setting up the firm the amount would have been deposited at a bank paying a per- period interest rate r )  How do we analyze this investment problem?  We will use the same approach we did when we analyzed the exit/shut-down decision: we compared the benefits from all available alternatives. The alternatives are analyzed below: ► If the firm is set up the per-period maximum profit is ► If the firm investment opportunity is not pursued the per-period flow is  We can draw the following conclusion: as long as in the long-run the price P is such that the firm is better off setting up the firm.

17. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |16 entry decision: long-run analysis  The entry condition can be easily detailed to.  The long-run price should satisfy: cost after entry cost of entry  The term FRATC ( Q ) stands for the full reinvestment average total cost and it represents the sum of operating costs (once the firm is set up) and the cost of setting up the firm.  The above inequality simply states that the long-run market price should cover both the per unit of output cost of operating the firm and the per unit of output cost of setting up the firm. Remark. Notice that FRATC > ATC and thus the long-run entry price will always be at least as high as the long-run exit price (which is equal to min ATC ). entry decision As long as P  min FRATC the investment in firm is more profitable. The lowest long-run price for which investment in the firm is a better alternative is given by P entry = min FRATC = min[ ATC ( Q ) + r  I 0 / Q ]

18. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |17 entry decision: long-run analysis  Graphically the shut-down and entry price are show below. AVC Q MC ATC ( long-run ) P exit FRATC ( short-run ) P shut-down ( long-run ) P entry per unit variable costs per unit fixed costs per unit investment costs price in this range covers all operational and investment costs Figure 9. Entry and exit thresholds Remark. The MC curve passes through the minimum of all “average” type curves.

19. microeconomic s the analytics of constrained optimal decisions lecture 1 costs, profit maximization & supply curve  2016 Kellogg School of Management lecture 1 page |18 key points ► The starting point of the microeconomic analysis is to consider a simplistic model of the market :  firms are relatively small and act as price takers producing homogenous goods (cannot differentiate through quality, characteristics, etc)  firms are assumed to maximize profit – this generates firm ‘s individual supply curve  the “price taking” and “homogenous goods” assumptions allow us to construct the market supply curve as the sum of the individual firms’ supply ► Some analytical aspects must be accepted:  understand the typology of costs - fixed, variable and total - average and marginal  understand the connection between costs and decisions to entry/exit in the short-run and long-run ► It seems that “there is a rule for each case” in terms of entry/exit and short-/long-run decisions … In fact there is a simple, very intuitive, general rule :  For each decision there are two alternatives to consider. Carefully consider what is the profit in each alternative… the choice of one alternative over the other is dictated by which alternative offers a higher profit.  In all the “cases” we considered, the profit for one of the alternative depended on the market price and comparing this profit with the profit under the alternative we derived the several “benchmark prices” for entry/exit in the short-/long-run scenarios… But the logic was always the same!