Finance 30210: Managerial Economics Cost Analysis Heres the overall objective for the supply side Production Decisions Product Markets Factor Markets Supply/Demand Determines Factor prices (We have covered this) Factor Usage/Prices Determine Production Costs (We are here now) Demand determines markup over costs (Coming soon!) Primary Managerial Objective: Minimize costs for a given production level (potentially
subject to one or more constraints) Example: PG&E would like to meet the daily electricity demands of its 5.1 Million customers for the lowest possible cost Or Maximize production levels while operating within a given budget Example: Billy Beane and would like to maximize the production of the Oakland As while staying within payroll limits. The starting point for this analysis is to think carefully about where your output comes from. That is, how would you describe your production process is a function of Q F X 1 , X 2 , X 3 ,... Production Level One or more inputs A production function is an attempt to describe what inputs are involved in your production process and how varying inputs affects production levels Note: We are not trying to perfectly match reality we are only trying to approximate it!!!
Some production processes might be able to be described fairly easily: Sugar (Lbs) Your Time (Minutes) Q F L, S , W , C , T 8 Oz. Glasses of Lemonade Lemons Water (Gallons) Paper Cups With a fixed recipe for lemonade, this will probably be a very linear production process Lemonade recipe (per 8oz glass)
Squeeze 1 Lemon into an 8 oz glass Add 2 oz. of Sugar Add 8 oz. of Water Stir for 1 minute to mix 2 oz for each glass times 16 glasses = 2 lbs 1 Cup 1 F 1, 2,8,1,1 1 glass available for sale 1 Lemon per glass 8 ounces per glass 1 minute per glass to stir each 8 oz glass
In fact, we could write the production function very compactly: Lemonade recipe (per 8oz glass) Q X # of Lemonade Kits (one kit = 1 Lemon, 2oz. Sugar, 8 oz. Water, 1 Minute) Squeeze 1 Lemon into an 8 oz glass Add 2 oz. of Sugar Add 8 oz. of Water Stir for 1 minute to mix Q Slope = 1 X Q F X 1 , X 2 ,... Output = Wins
Inputs = Players 2 RS WP 2 2 RS RA Bill James used the following production function for wins RS = Runs Scored RA = Runs Given up 2 RS WP 2 2 RS RA RS = Runs Scored RA = Runs Given up W L
PCT RS RA 97 65 .599 867 657 867 2 WP .635 2 2 867 657 W L PCT
Average (Per Win): $2,079,361 Average (Per Run): $232,639 Total: $125,480,664 Average (Per Player): $5,228,361 Average (Per Win): $1,767,333 Average (Per Run): $191,866 0.9 RA = 657 RA = 756 0.8 0.7 .635 0.6 0.5 Yankees Cubs .4280.4 0.3 RS 2 WP 2
RS RA2 0.2 0.1 0 3 55 07 59 1 1 6 3 15 67 1 9 7 1 2 3 75 2 7 79 31 8 3 3 5 8 7 39 9 1 43 95 4 7 99 51 03 55 07 59 1 1 63 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1 0 1 0 11 11 1 2 13 13 1 4 1 4 15 15 RS = 654 RS = 867 Given their runs against, the Cubs needed 1000 runs scored to match the Yankees win percentage! To evaluate a players contribution to run production, numerous statistics are derived Runs Created On Base Percentage RC H W HBP OBP AB W HBP SF H = Hits W = Walks HBP = Hit by Pitch AB = At Bats SF = Sacrifice Flies
This was the single number in Moneyball H W TB AB W H = Hits W = Walks TB = Total Bases AB = At Bats We can then start comparing productivity to cost Derek Jeter ($15,729,365) 2011 2011 BA = .297 OBP RC Starlin Castro($567,000) 162 46 6 .355
546 46 6 5 162 46 212 75 546 46 BA = .307 OBP RC 207 35 6 .346 674 35 2 4 207 35 206 96 483 35 Some production processes might be more difficult to specify: How would you describe the production function for the business school? Q F X 1 , X 2 ,... Output(s) Input(s)
How would you describe the production function for the business school? What is the product of Mendoza College of Business? YOU ARE! Finance Degrees Undergraduate (BA) Accounting 1 Year MBA (MBA) Marketing 2 Year MBA (MBA) Management South Bend EMBA (MBA) Chicago EMBA (MBA) Masters of Accountancy (MA) Masters of Nonprofit Administration (MA) How would you describe
the production function for the business school? How would you characterize the inputs into Mendoza College of Business Facilities Classroom Space Office Space Conference/Meeting Rooms Equipment Information Technologies Communications Instructional Equipment Capital Inputs Personnel Faculty (By Discipline) Administrative Administrative Support Maintenance
Labor Inputs Staff How would you describe the production function for the business school? Have we left out an output? Notre Dame, like any other university, is involved in both the production of knowledge (research) as well as the distribution of knowledge (degree programs) Degrees Research F Capital , Labor Should the two outputs be treated as separate production processes? The next question would be: What is your ultimate objective? Degrees Research F Capital , Labor Is Notre Dame trying to maximize the quantity and quality of
research and teaching while operating within a budget? OR Is Notre Dame trying to minimize costs while maintaining enrollments, maintaining high research standards and a top quality education? Does it matter? The Notre Dame Decision Tree School of Architecture Under the golden dome, resources are allocated across colleges to maximize the value of Notre Dame taking into account enrollment projections, research reputation, education quality, and endowment/resource constraints College of Arts & Letters College of Business
School of Architecture School of Architecture Given the resources handed down to her, Dean Huang allocates resources across departments to maximize the value of a Business Degree and to maximize research output. Finance Department Management Department Marketing Department Accounting Department Graduate Programs Department chairs receive resources from Dean Woo and allocate those resources to
maximize the output (research and teaching) of the department Another issue has to do with planning horizon. Different resources are treated as unchangeable (fixed) over various time horizons It might take 5 years to design/build a new classroom building It could take 6 months to install a new computer network Now 6 mo 1 yr 2 yr 5 yr 10 yr It takes 1 year to hire a new faculty member
Tenured faculty are essentially cant be let go Shorter planning horizons will involve more factors that will be considered fixed From here on, lets keep things as simple as possible You produce a single output. There is no distinction as far as quality is concerned, so all we are concerned with is quantity. You require two types of input in your production process (capital and labor). Labor inputs can be adjusted instantaneously, but capital adjustments require at least 1 year Total Production Is a function of Q F K , L Capital (Fixed for any planning horizon under 1 year Labor (always adjustable)
Some definitions Q F K , L Marginal Product: marginal product measures the change in total production associated with a small change in one factor, holding all other factors fixed MPL Q L Q MPK K Average Product: average product measures the ratio of input to output Q APL L Q APK K Elasticity of Production: marginal product measures the change in total production associated with a small change in one factor, holding all other factors fixed
%Q MPL L %L APL %Q MPK K %K APK Over a short planning horizon, when many factors are considered fixed (in this case, capital), the key property of production is the marginal product of labor. Q F K , L MPL Q L For a given production function, the marginal product of labor measures how production responds to small changes in labor effort Q Fll ( K , L) 0
Q F ( K , L) Fll ( K , L) 0 F ( K , L) OR L Diminishing Marginal Returns: As labor input increases, production increases, but at a decreasing rate L Increasing Marginal Returns: As labor input increases, production increases, but at an increasing rate Consider the following numerical example: Q K .3L .0029 L 2 3
We start with a production function defining the relationship between capital, labor, and production Capital is fixed in the short run. Lets assume that K = 1 Q 1.3L2 .0029 L3 Suppose that L = 20. Q 1.3 20 2 .0029 203 96.8 Q K .3L2 .0029 L3 500 Increasing Marginal Returns Quantity 400 300 Decreasing Marginal Returns 200 Negative Marginal Returns
100 96.8 Labor 0 0 10 20 30 40 50 60 70 80 90 100
Maximum Production reached at L =70 Now, lets calculate some of the descriptive statistics Q K .3L2 .0029 L3 MPL Q L Q APL L Recall, K = 1 Labor (L) Quantity (Q) MPL APL Elasticity
1.4275 1.7674 MPL L APL The properties of the marginal product of labor will determine the properties of the other descriptive statistics MP hits a maximum at L = 35 1 Elasticity of production greater than one indicates MP>AP (Average product is rising) MPL L 1 APL Elasticity of
production less than one indicates MP
(l ) rk wl F k , l Q Rememberthis needs to be positive!! First Order Necessary Conditions l (l ) w Fl (k , l ) 0 Q F ( k , l ) w Fl (k , l ) F (k , l ) Q Marginal costs refer to changes in total costs when production increases rk wl MC Q With capital fixed, marginal costs are only influenced by labor decisions in the short run Average (Unit) costs refer to total costs divided by total production
rk wl rk wl AC Q Q Q Average fixed costs fall as output increases k Cost Minimization: Short Run is fixed (l ) rk wl F k , l Q w Fl (k , l ) F (k , l ) Q Recall that lambda measures the marginal impact of the constraint. In this case, lambda represents the marginal cost of producing more output Fll (k , l ) 0 Fll (k , l ) 0
Marginal costs are increasing Marginal costs are decreasing Fll (k , l ) 0 Marginal Cost vs. Average Cost Costs AC Minimum AC occurs where AC=MC MC y When AC is greater than MC, AC Falls When AC is less than MC, AC rises Fll (k , l ) 0 Marginal Cost vs. Average Cost
Costs AC MC y If production exhibits increasing marginal productivity, then Average Costs decline with production (it pays to be big!) Back to our example: Minimize costs for a given production level (potentially subject to on or more constraints) Lets imagine a simple environment where you can take the cost of labor as a constant. Suppose that labor costs $10/hr and that you have one unit of capital with overhead expenses of $30. You have a production target of 450 units: =1 Minimize 30 10 L Q K .3L2 .0029 L3 450 Objective Constraint
Q K .3L2 .0029 L3 450 With only one variable factor, there is no optimization. The production constraint determines the level of the variable factor. Quantity 450 Labor 450 Units of production requires 60 hours of labor (assuming that K=1) Lets imagine a simple environment where you can take the cost of labor as a constant. Suppose that labor costs $10/hr and that you have one unit of capital with overhead expenses of $30. You have a production target of 450 units: =1 Total Costs Minimize 30 10 L Q K .3L2 .0029 L3 450
Objective Constraint Solution: L = 60 Total Costs = 30 + 10(60) = $630 Average Costs = $630/450 = $1.40 Average Variable Costs = $600/450 = $1.33 Suppose that you increase your production target to 451. How would your costs be affected? If the marginal product of labor measures output per unit labor, then the inverse measures labor required per unit output w Q MC MPL MPL L Labor (L) 0 1
L wL w AVC Q APL We also know that the average variable cost is related to the inverse of average product Properties of production translate directly to properties of cost MCAVC. Average Variable Cost is Rising MC hits a minimum at L = 35
Labor Elasticity of production greater than one indicates MP>AP (Average product is rising) MPL L 1 APL Elasticity of production less than one indicates MP
Q K .3L2 .0029 L3 450 Objective Constraint We just minimized costs of one particular production target. Maximizing profits involves varying the production target (knowing that you will minimize the costs of any particular target). There should be one unique production target that is associated with maximum profits: Maximum Profits MR MC MR w MC MPL w MPL MR * MPL w Optimal Factor Use
Recall the alternative management objective: Maximize production levels while operating within a given budget Lets imagine a simple environment where you can take the cost of labor as a constant. Suppose that labor costs $10/hr and that you have one unit of capital with overhead expenses of $30. You have a production budget of $630: Total Output Maximize K .3L2 .0029 L3 Objective Available budget 30 10 L 630 Constraint 30 10 L 630 Just like before, there is no optimization. The budget constraint determines the level of
the variable factor. Cost 630 Labor $630 budget restricts you to 60 hours of labor (assuming that overhead = $30) Total Output Maximize K .3L2 .0029 L3 Objective Available budget 30 10 L 630 Constraint Now, if we were to think about altering the objective we would be
considering the effect on production of a $1 increase in the budget: Change in production Now, take the profit maximizing condition and flip it Q 1 MP L TC MC w 1 1 MR MC Change in Budget Both managerial objectives yield the identical result!!! MR * MPL w Optimal
Factor Use In the long run, we can adjust both inputs. Therefore, we need to look at how production changes as both factors adjust. Q K .3L2 .0029 L3 450 Labor L = 33 L = 13 Q 450 K=2 K = 30 Capital An isoquant refers to the various combinations of inputs that generate the same level of production In the long run, we need to think about the relative productivity of each
factor. Labor L TRS K L Q 450 Capital K The Technical rate of substitution (TRS) measures the amount of one input required to replace each unit of an alternative input and maintain constant production Q MPL L Recall some earlier definitions: MPK Marginal Product of Labor Q K
Marginal Product of Capital MPK TRS MPL Labor L If you are using a lot of capital and very little labor, TRS is small L Q 450 Capital K K A key property of production in the long run has to do with the substitutability between multiple inputs. l l
k ' l % k %TRS l k The elasticity of substitution measures curvature of the production function (flexibility of production) k Technical rate of Substitution measures the degree in which you can alter the mix of inputs in production. Consider a couple extreme cases: Perfect substitutes can always be can always be traded off in a constant ratio
Labor Perfect compliments have no substitutability and must me used in fixed ratios Labor Elasticity is Infinite Capital Elasticity is 0 Capital Cost Minimization: Long Run Min rk wl k ,l subject to k is variable F (k , l ) Q
(l , k ) rk wl F k , l Q Cost Minimization: Long Run (l , k ) rk wl F k , l Q First Order Necessary Conditions l (l , ) w Fl (k , l ) 0 k (l , ) r Fk (k , l ) 0 w r Fl (k , l ) Fk (k , l ) r Fk (k , l ) TRS w Fl (k , l ) Q F ( k , l ) Again, back to our example Lets imagine a simple environment where you can take the cost of labor and the cost of capital as a constant. Suppose that labor costs $10/hr and that capital costs $30 per unit. You have a production target of 450 units: Total Costs
Minimize 30 K 10 L Q K .3L2 .0029 L3 450 Objective Constraint Now we have two variables to solve for instead of just one! Consider two potential choices for Capital and Labor Q K .3L2 .0029 L3 450 L = 33 K=2 TC = 30*2 + 33*10 = $390 AC = $390/450 = $0.86 This procedure is relatively labor intensive L = 13 K = 30 TC = 30*30 + 13*10 = $1030
AC = $1030/450 = $2.29 This procedure is relatively capital intensive With more than one input, there should be multiple combinations of inputs that will produce the same level of output Minimize 30 K 10 L Q K .3L2 .0029 L3 450 Suppose that we lowered production by 1 unit by decreasing labor. What would happen to costs? Labor $10 Total Cost = 30*2 + 33*10 = $390 Average Cost = $390/450 = $.87 MC 33 w MPL 20
MC = $.50 Q 450 2 Capital Minimize 30 K 10 L Q K .3L2 .0029 L3 450 Now, lets increase production by one unit to get back to our initial production level by increasing capital $30 Pk MC MPk Labor 212 MC = $.50 33
By altering the production process slightly, we were able to maintain 450 units of production and save $0.36! MC = $.14 Q 450 2 Capital Here, we have too much labor. We can save costs by substituting capital for labor Pk 30 .14 MPk 212 w 10 .50 MPL 20
Here, we have too much capital. We can save costs by substituting labor for capital Labor Pk 30 1.11 MPk 27 33 w 10 .12 MPL 86 11 Q 450 2 15 Capital
Minimize 30 K 10 L Q K .3L2 .0029 L3 450 Pk 30 .28 MPk 106 Labor w 10 .27 MPL 36 Total Cost = 30*4 + 10*22 = $340 Average Cost = $.75 22 Q 450 4 Capital Short Run vs. Long Run
Minimize 30 K 10 L Solution: L = 60 (K Fixed at 1) Total Costs = 30 + 10(60) = $630 Average Costs = $630/450 = $1.40 w MC $2.13 MPL Q K .3L2 .0029 L3 450 Solution: L = 22, K = 4 Total Cost = 30*4 + 10*22 = $340 Average Cost = $.75 MC Pk w $.27 MPK MPL Long Run Average Cost will always be less than or equal short run average costs due to the increased
flexibility of inputs Each point on the long run average cost curve should represent the minimum of some short run average cost curve Average Cost SRAC SRAC SRAC SRAC LRAC $1.40 $0.75 Quantity 450 Suppose that the price of labor rises to $50 Minimize 30 K 50 L Solution: L = 60 (K Fixed at 1) Total Costs = 30 + 10(60) = $630 Average Costs = $630/450 = $1.40
Average Variable Costs = $600/450 = $1.33 w 10 MC $2.13 MPL 4.68 Q K .3L2 .0029 L3 450 In the short run, factor price changes cant be avoided without affecting the production target, so costs are very sensitive to factor price changes Solution: L = 60 (K Fixed at 1) Total Costs = 30 + 50(60) = $3,030 Average Costs = $3,030/450 = $6.73 w 50 MC $10.68 MPL 4.68
Suppose that the price of labor rises to $50 Minimize 30 K 50 L Q K .3L2 .0029 L3 450 Pk 30 .76 MPk 39 Labor w 50 .80 MPL 62 In the long run, if your production technique is flexible, you can avoid cost increases! Total Costs = 30(10) + 50(13) = $950 Average Costs = $630/450 = $2.11 Marginal Cost = $.80
22 13 Q 450 4 10 Capital Elasticity of substitution determines the response of costs to changes in input prices mc w l Low elasticity of substitution means that production is very inflexible
w l k Low price elasticity means that factor demands dont respond to factor prices Costs are very sensitive to factor price changes Elasticity of substitution determines the response of costs to changes in input prices mc w l k High elasticity of substitution means that
production is very flexible l High price elasticity means that factor demands respond significantly to factor prices w Costs are very insensitive to factor price changes As you expand production in the long run, you are adjusting both factors, so your costs will not depend on marginal products! Labor Q 600 22 Q 550 Q 500 Q 450
4 Capital In the long run, we are not looking for increasing or decreasing marginal returns, but instead, we are looking for increasing or decreasing returns to scale Recall the production function we have been working with. Q K .3L2 .0029 L3 1 Unit of capital and 20 units of labor generate 96.8 units of output. Q 1.3 20 2 .0029 203 96.8 Suppose we double our inputs Q 2.3 40 2 .0029 403 588 Doubling the inputs more than doubles production! We call this increasing returns to scale Increasing Returns to Scale F ( 2k ,2l ) 2 F (k , l ) Costs AC
MC y Marginal costs are always less than average costs Costs are decreasing (it pays to be big) Decreasing returns to Scale Costs F (2k ,2l ) 2 F (k , l ) MC AC y Marginal costs are always greater than average costs Costs are increasing (it pays to be small) Constant Returns to Scale F ( 2k ,2l ) 2 F (k , l ) Costs MC = AC y Marginal costs are always equal to average costs
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