Types of Costs 10/7/2014. Explicit Costs: Costs that involve a direct monetary outlay. Implicit Costs: Costs that do not involve outlays of cash.

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1 Cost and Cost Minimization Types of Costs Explicit Costs: Costs that involve a diect monetay outlay. Implicit Costs: Costs that do not involve outlays of cash. Example: money that an ailine can get by enting, athe than actually using, its on plane. Oppotunity Costs: The value of the best altenative that is fogone hen anothe altenative is chosen Example: aise Aluminum had to plants, one in Tacoma and anothe in Spokane in 000. It had initially signed a long tem electicity contact at a pice of $3 megaatt/hou. But the pice of electicity as $1000 pe megaatt/hou in 001. What did aise Aluminum do? Shut don the smeltes (at least a fe days) and sell the electicity in the open maket. (othe fims, like Tea Industies, poducing poe, did the same). Hence, the oppotunity cost of a megaatt/hou in 001 as not $3, but $1000. Types of Costs Sunk Costs (unecoveable): Costs that have aleady been incued and cannot be ecoveed. Example: The ental a fim pays fo the building it uses, if the lease contact pohibits subletting. Non sunk Costs (ecoveable): Costs that ae incued only if a paticula decision is made. Example: Building a factoy ($ 5 million) Befoe it is built: All is non sunk Afte it is built: A potion might be sunk (unecoveable) 1

2 Falling into the Sunk Cost Fallacy Application 7.3 Conside the folloing condition A: Condition A you paid $10.95 to see a movie (o Pay TV.) Afte 5 minutes, you ae boed and the movie seems petty bad Ho much time do you keep atching the movie? 0 min, 10, 0, 30, until the end of the movie. Expeiment ith/ithout A: Senio citizens: Same amount of time ith/ithout A College Students: Moe time ith A than ithout, so they fell into sunk cost fallacy teating the $10.95 as a non sunk cost, hile it as aleady sunk Cost Minimization ong Run: The peiod of time that is long enough fo the fim to vay the quantities of all its inputs as much as it desies. Shot Run: The peiod of time in hich at least one of the fim s quantities cannot be changed. Example: 1) Restauant: is vaiable, is fixed ) Scientific lab: is fixed, is vaiable Shot un costs Cheat sheet 1. Vaiable and nonsunk: ΔQ costs Vaiable But if Q = 0, then costs = 0 Nonsunk Example: labo and a mateials. Fixed and nonsunk: Q no change in costs. Fixed But if Q = 0 then costs = 0 Nonsunk Example: Heating 3. Fixed and sunk: Q no change in costs Fixed But if Q = 0 then costs>0 Sunk Example: motgage payment ease that cannot be sublet Cost Minimization ingedients: Isocost and Isoquant. Isocost line: The set of combinations of labo and capital that yield the same total cost fo the fim TC= + hee : pice of labo (age) : pice of capital (inteest ate) Example: TC TC1000 TC 3000 TC Then, 50 (vetical intecept) 0 TC (hoizontal intecept) 10 =10 =0

3 Moe on the Isocost line TC = + Since is the vetical axis, e solve fo to obtain TC=, o, TC hee TC denotes the vetical intecept of the Isocost line, and denotes the slope of the isocost line. Example (cont.) TC isocost line of, 10, 0 implies an Cost Minimization We ant to minimize TC eaching a given output (isoquant). This is gaphically epesented by pushing the isocost line donads until it eaches the isoquant epesenting the output e must be poducing, Q 0. Points E and F also poduce output Q 0, but at a highe cost TC 1 Cost Minimization Cost minimization Poblem To find the tangency point (point A) Slope of isoquant=slope of isocost line MRTS, MP MP MP MP Additional output pe dolla spent on labo = additional output pe dolla spent on capital At Point E: Slope of isoquant < Slope of isocost MP MP MP MPk MP MPk (Hence, inceasing labo is still optimal) Reach a given output q f (,), hee q Q Min + minimize isocost line., Subject to q f (,) isoquant (, ; ) q(, ) F.O.C.s f 0 MP f 0 MP q f(, ) 0 q f(, ) MP } MP MP MP 0 Tangency beteen the Isocost line and the Isoquant! 3

4 Example Poduction function Q 50 Hence, MP 5 and MPk 5 Input pices ae $5 and $0. a) What is the cost minimizing combination of and that eaches an output Of Q units? 1/ 5 1/ 1/ MP Tangency: 1/ 1/ 1/ MP Not Done Yet! 4k= We also kno that the cost minimizing combination of and must lie on the isoquant Q0 1000, that is, We no have a system of to equations ith to unknons: } If e plug *=10 into =4, e obtain the optimal value of, Hence, the cost minimizing combination of inputs is: 10 and 40 Quey #1 A fim has a Cobb Douglas poduction function fo its inputs of capital and labo. The fim is cuently paying $10 pe labo hou and $5 pe machine hou. The fim is cuently at an efficient poduction level, employing an equal numbe of machines and okes. What can e infe about the maginal poductivities of capital and labo at this point? a) MP = MP b) MP = MP c) MP = MP d) MP =.5MP Quey #1 Anse Anse C The cost minimizing condition: MRTS, = MP / MP MP / MP = (/) Input pices: = $10 and = $5 So, the tangency condition fo cost minimization entails MP / MP = ($10/$5) Coss multiplying, e obtain 5(MP ) = 10(MP ) Simplifying, MP = MP Pages

5 Cone Point Poblem Hee, the optimal solution doesn t have a tangency beteen an isocost line and an isoquant cuve. Cone solutions aise hen inputs ae pefect substitutes, i.e., Q=a+b Isocost line is flatte than the Isoquant: MP MP eaanging: MP MP MP MP hich implies the fim ants to use labo alone Example of Cone solutions: P oduction Function : Q 10 Whee MP 10and MPk P iceof labo : 5 pe unit P iceof capital : pe unit Fim ish to poduce : Q 00 units Using the pevious figue e obseved that the optimal combination is a cone solution. MP Why? Because, that is 10 MP 5 MP 10 MP Altenatively, note that 1 5 So that the maginal poduct pe dolla of labo exceeds the maginal poduct pe dolla of capital (>1), then the fim ill substitute labo pe capital until it uses no capital (=0). In the hoizontal axis of the above figue. Quey # Then, the quantity of labo must satisfy Q= 10 +, hee e kno = 0 then eaching the isoquant Q=00 units implies 00=10+x0, o 00= Summaizing, the fim uses = 0 okes and = 0 units of capital. (cone point) Suppose in a paticula poduction pocess that capital and labo ae pefect substitutes so that thee units of labo ae equivalent to one unit of capital. If the pice of capital is $4 pe unit and the pice of labo is $1 pe unit, the fim should a) employ capital only. b) employ labo only. c) use thee times as much capital as labo. d) use thee times as much labo as capital. 5

6 Quey # Anse Anse B In this paticula case, e have a cone point solution. The pice of labo is $1/Unit, hile that of capital is $4/ Unit. In addition, e ae infomed that thee units of labo, 3, ae equivalent to one unit of capital, i.e., 3= Because these inputs ae pefect substitutes, this fim can minimize its cost by spending $3/Unit on labo athe than spending $4 / Unit on capital. (Remembe that 3 Units of abo as equal to 1 Unit of Capital) Pages Compaative statics: An incease in ages Δ 1. An incease in ages fom 1 to, pivots the isocost line inads, fom C 1 to C.. To still each isoquant Q 0, the fim cannot keep spending TC 0, it must incu a lage cost TC 1 >TC 0. (Paallel shift of the isocost line outads, fom C to C 3 ). An Incease in Wages An incease in the pice of labo (Δ) poduces an inad pivoting of the isocost line (steepe isocost line) But the fim must still each Q=100 units! They d bette incu lage TC! (shift isocost outad) Compaing A and B: Then the cost minimizing amount of labo must go don ( ) and the cost minimizing quantity of capital must go up (), fom point A to B. A incease in, hen the Cost minimizing pai as at a kink (, ) No change in the combination of and befoe/afte the 6

7 Quey #3 Suppose capital and labo ae pefect complements fo a paticula poduction pocess. If the pice of labo inceases, holding the pice of capital and the level of output constant, the fim should a) use moe capital and less labo. b) use moe labo and less capital. c) use the same amounts of capital and labo. d) eliminate all use of labo. Quey #3 Anse Anse C This fim has a fixed popotions poduction function, Q=min{a,b}. Hence, inputs ae used in specific atios, and An incease in the pice of labo does not cause the fim to substitute capital fo labo. If the pice of capital and the level of output ae held constant, the fim ould continue to use the same amount of both labo and capital. Pages Compaative Statics () change in eachable output Compaative Statics () change in eachable output inputs ΔQ in and ΔQ Δ, but (infeio input) Expansion Path: A line that connects the costminimizing input combinations of (,) as the quantity of output, Q inceases, holding input pices constant. Nomal input: An input hose cost minimizing quantity inceases as the fim poduces moe output. The fim s expansion path ill have a positive slope. Infeio input: An input hose cost minimizing quantity deceases as the fim poduces moe output. The fim s expansion path ill have a negative slope. 7

8 Can both inputs be infeio? NO! abo Demand Cuve abo Demand Cuve : A cuve that shos ho the fim s cost minimizing quantity of labo vaies ith the pice of labo. 1) ΔW fom =$1 (at A) to =$ (at B), labo usage deceases. This is depicted in A and B, espectively, in the top figue, and A and B in the bottom figue of labo demand fo Q=100. abo Demand ) When e incease output fom Q= 100 & Q=00, If dem shifts outad, then is a nomal input (as depicted in the figue). If dem shifts inads, then is an infeio input. Pice of labo in A and C is the same, e only change output fom Q=100 to Q=00 Cost minimizing input combination vaies fom A to C in the top figue, hich implies a shift fom A to C in the bottom figue Can abo demand be vetical? Yes, When both inputs ae used in fixed popotions, e sa that age changes don t affect the cost minimizing input combination. (Remembe the figue of ight angled isoquants?). Hence, labo demand ould be insensitive to ages: 8

9 Q 50 10/7/014 Finding the abo Demand Algebaically Conside a Cobb Douglas poduction function Q 50 Fom the tangency condition beteen isoquant and isocost, e obtain MP l MP o MP MP Q Q (0.5)(50) (0.5)(50) MP Hence, MP implies o solving fo, Finding abo Demand Algebaically Plugging the above expession,, into the poduction function Q 50 e obtain Q Q Q Q 50 We just found the demand cuve fo capital, i.e. capital demand. Finding abo Demand Algebaically Q Plugging the above esult, into 50 Q 50 Q Q hich descibes the demand cuve fo capital, i.e., the labo demand. Finding abo Demand Algebaically Note that: 1) Capital demand, Q 50, is Deceasing in Inceasing in Inceasing in Q ) abo demand, Q 50 Inceasing in, is Deceasing in Inceasing in Q Since an ΔQ poduces an incease in the demand of both and, both inputs ae nomal (not infeio). 9

10 Pice Elasticity of Demand fo abo: The pecentage change in the cost minimizing quantity of labo ith espect to a 1 pecent change in the pice of labo., * 100 % * 100 % 1% in ages in the fim's labo demand of % Hence, it depends on the slope of the demand cuve fo labo.,w The pice elasticity of the demand fo labo depends on the elasticity of substitution, σ beteen to inputs ( and ): fom Ch. 6 Δ W CES ith =.5 Δ W CES ith = o measues such slope W A change in has almost no effect on The same change in induces a geat change in In both cases dops fom $ to $1 (a 50% dop), but % Incease in labo demand in figue (a), and labo only inceases fom 4.6 to 5. 17% Incease in labo demand in figue (b), and labo inceases a lot: fom. to 5. Similaly fo the pice elasticity of the demand fo capital: *100% k, *100% Intepetation: 1%in inteest ates() inthefim's labo demandfo capitalof a of, % It depends on the slope of the demand cuve fo capital, 10

11 Pice elasticities of input demand fo manufactuing industies in Alabama Capital Poduction abo Nonpoduction abo Electicity Textiles Pape Chemicals Metals Conside the textile industy (fist o): The 0.50 in the second cell implies that a 1% incease in the age ate fo poduction okes only entails a 0.5% decease in the demand fo labo of the typical textile fim in Alabama. (abo demand is athe insensitive to labo). All but one of the pice elasticities of input demand ae beteen 0 and 1, suggesting that industies do not aggessively educe thei demand of the input hose pice became elatively moe expensive. Cost Minimization in the Shot Run Fixed Capital In the long un the fim modifies and in ode to each Q 0. Solution: Point A In the Shot un is fixed at If the fim must each output level of Q 0, it must use F, incuing a lage cost, i.e., a highe isocost. Cost Minimization in the Shot Run Example: conside the Cobb Douglas poduction function Q 50 If is fixed at in the shot un, then the costminimizing is found by solving fo, Q Q 50 Q,500,500 This is the demand fo labo in the shot un, hee is fixed. Exta pactice: eaning by Doing execise 7.6 (3 inputs). We go ove this execise next. Thee inputs eaning by Doing 7.6 Conside the Cobb Douglass poduction function Q M, hee denotes labo, capital, and M a mateials. Hence, the maginal poducts ae: MP MP k MPM M Assume that input pices ae =1, =1, m=1 a) If the fim ants to poduce Q=1, hat is the cost minimizing input combination *, *, M*? MP 1/ 1 1 MP 1/ 1 MP 1/ 1 M 1 M MP m 1/ M 1 M ==M 11

12 Using ==M in the poduction function yields: = M= Theefoe, =16, hich entails that =16 and M=16 b) If capital is fixed at 4 units, i.e., 4 units, hat is the cost minimizing input combination (*,M*)? MP MP M 1/ 1 M 1 M m 1/ M 1 Plugging that infomation into the poduction function, e obtain: =4 (fixed) M= 5 5 Hence, since =5, then M=5, hile the fixed amount of capital emains 4. c) What if no e fix the amount of capital at 4, and the amount of labo at 9 okes? 1 3 M M 7 M 49 M 9 4 Hence, M=49, hile the to othe fixed inputs emain at 4 and 9. Summay of the Cost Minimization Poblem ith 3 Inputs ong un cost minimization fo Q=1 Shot un cost minimization fo Q=1 hen =4 abo, Capital, Mateials, M Minimized Total Cost $ $54 Shot un cost minimization fo Q=1 hen =4 and = $6 1