Cost Functions. Definitions of Costs. Economic Cost. [See Chap 10]

Transcription

1 Cost Functions [See Chap 0]. Definitions of Costs Economic costs include both implicit and explicit costs. Explicit costs include wages paid to employees and the costs of aw mateials. Implicit costs include the oppotunity cost of the entepeneu and the capital used fo poduction. Economic Cost The economic cost of any input is its oppotunity cost: the emuneation the input would eceive in its best altenative employment 3

2 Model Fim poduces single output, q Fim has N inputs {z, z N }. Poduction function q f(z, z N ) Monotone and quasi-concave. Pices of inputs {, N }. Pice of output p. 4 Fim s Payoffs Total costs fo the fim ae given by total costs C z + z Total evenue fo the fim is given by total evenue pq pf(z,z ) Economic pofits (π) ae equal to π total evenue - total cost π pq - z - z π pf(z,z ) - z - z 5 Fim s Poblem We suppose the fim maximizes pofits. One-step solution Choose (q,z,z ) to maximize π Two-step solution Minimize costs fo given output level. Choose output to maximize evenue minus costs. We fist analyze two-step method Whee do cost functions come fom? 6

3 COST MINIMIZATION PROBLEM. 7 Cost-Minimization Poblem (CMP) The cost minimization poblem is min z + z s.t. f ( z, z) q and z, z Denote the optimal demands by z i (,,q) Denote cost function by C(,,q) z (,,q) + z (,,q) Poblem vey simila to EMP. Output constaint binds if f(.) is monotone. 0 8 CMP: Gaphical Solution Given output q, we wish to find the lowest cost point on the isoquant z C Isocost line ae paallel with a slope of / : C 3 C q C < C < C 3 Z 9 3

4 CMP: Gaphical Solution The minimum cost of poducing q is C z C C 3 This occus at the tangency between the isoquant and the total cost cuve z C q The optimal choice is (z,z ) z z 0 CMP: Lagangian Method Set up the Lagangian: L z + z + λ[q - f(z,z )] Find the fist ode conditions: L/z - λ(f/z ) 0 L/z - λ(f/z ) 0 L/λ q - f(z,z ) 0 Cost-Minimizing Input Choices Dividing the fist two conditions we get: f / z MRTS f / z The cost-minimizing fim equates the MRTS fo the two inputs to the atio of thei pices. Equivalently, the fim equates the bang-pebuck fom each input f / z f / z 4

5 Intepetation of Multiplie Note that the fist ode conditions imply the following: λ f f The Lagange multiplie descibes how much total costs would incease if output q would incease by a small amount. 3 The Fim s Expansion Path The fim can detemine the costminimizing combinations of z and z fo evey level of output The set of combinations of optimal amount of z and z is called the fim s expansion path. 4 The Fim s Expansion Path The expansion path is the locus of costminimizing tangencies Z E The cuve shows how inputs incease as output inceases q q q 0 Z 5 5

6 The Fim s Expansion Path The expansion path does not have to be a staight line the use of some inputs may incease faste than othes as output expands depends on the shape of the isoquants The expansion path does not have to be upwad sloping. 6 Example: Symmetic CD Poduction function is symmetic cobbdouglas: q z γ z γ The Lagangian fo the CMP is L z + z + λ[q - z γ z γ ] 7 Example: Symmetic CD FOCs fo a minimum: L/z - λz (γ-) z γ 0 L/z - λz γ z (γ-) 0 Reaanging yields z z. Using the constaint qz γ z γ, / / z (, q and z(, q Substituting, the cost is c γ / γ / (, z + z ( ) q / γ / 8 6

7 Example: Pefect Complements Suppose q f(z, z ) min(z,z ) Poduction will occu at the vetex of the L-shaped isoquants, z z. Using constaint, z z q Hence cost function is C(,,q) z + z ( + )q 9 COST FUNCTIONS. 0 Total Cost Function The cost function shows the minimum cost incued by the fim is C(,,q) z (,,q) + z (,,q) Cost is a function of output and input pices. When pices fixed, sometimes wite C(q) 7

8 Aveage Cost Function The aveage cost function (AC) is found by computing total costs pe unit of output C(, aveage cost AC(, q Maginal Cost Function The maginal cost function (MC) equals the exta cost fom one exta unit of output. C(, maginal cost MC(, q 3 Pictue # Concave poduction function. 4 8

9 Pictue # Non-concave poduction function 5 Pictue #3 Non-concave poduction function. Fixed cost of poduction. 6 Cost Function: Popeties. c(,,q) is homogenous of degee in (, ) If pices double constaint unchanged, so cost doubles.. c(,,q) is inceasing in (,,q) 3. Shepad s Lemma: c(,, q) zi (,, q) i If ises by, then c(.) ises by z (.) Input demand also changes, but effect second ode. 4. c(,,q) is concave in (, ) 7 9

10 Cost Function: Concavity and Shepad s Lemma At, the cost is c(, ) z + z c(, ) c(, ) c pseudo c(, ) If the fim continues to buy the same input mix as changes, its cost function would be C pseudo Since the fim s input mix will likely change, actual costs will be less than C pseudo such as C(,,q) 8 Cost Function: Popeties 5. If f(z,z ) is concave then c(,,q) is convex in q. Hence MC(q) inceases in q. Concavity implies deceasing etuns. Moe inputs needed fo each unit of q, aising cost. 6. If f(z,z ) is exhibits deceasing (inceasing) etuns then AC(q) inceases (deceases) in q. Unde DRS, doubling inputs poduces less than double output. Hence aveage cost ises. 7. AC(q) is inceasing when MC(q) AC(q), and deceasing when MC(q) AC(q). If MC(q) AC(q) then cost being dagged up. When AC(q) minimized, MC(q)AC(q). 9 Aveage and Maginal Costs Aveage and maginal costs MC is the slope of the C cuve MC AC If AC > MC, AC must be falling min AC If AC < MC, AC must be ising Output 30 0

11 3 Can Costs Look Like This? Left: When AC minimized, MCAC. Right: If no fixed costs ACMC fo fist unit. If fixed costs, AC fo fist unit. 3 Input Demand: Popeties. z i (,,q) is homogenous of degee 0 in (, ) If pices double constaint unchanged, so demand unchanged.. Symmety of coss deivatives Uses Shepad s Lemma 3. Law of demand Uses Shepad s Lemma and concavity of c(.) z c c z 0 c z 33 SHORT-RUN VS. LONG-RUN.

12 Shot-Run, Long-Run Distinction Costs may diffe in the shot and long un. In the shot un it is (elatively) easy to hie and fie wokes but elatively difficult to change the level of the capital stock. Suppose fim wishes to aise poduction Can t change capital stock Hies moe wokes. Capital/Labo balance no longe optimal. High poduction costs. 34 Time Fames In vey shot un, all inputs ae fixed. In shot un, some inputs fixed with othes ae flexible. In medium un, all inputs ae flexible but fim cannot ente/exit. Fixed costs ae sunk. In long un, all facto ae flexible and fim can exit without cost. 35 Example: f(z,z )(z -) /3 (z -) /3 Cobb-Douglas poduction but fist unit of each input is useless. In long un, L z + z + λ[q - (z -) /3 (z -) /3 ] FOC becomes (z -) (z -). Using constaint, demands ae / / 3/ 3/ z (, q + and z(, q Long-un cost function (,, ) ( ) / / γ c q z + z q + ( + ) with c(,,0)

13 Example: f(z,z )(z -) /3 (z -) /3 In medium un, statup cost of ( + ) is sunk. Cost function is thus (,, ) ( ) / / γ c q z + z q + ( + ) with c(,,0) Example: f(z,z )(z -) /3 (z -) /3 In shot un, z is fixed at z. The constaint in the CMP becomes q (z -) /3 (z -) /3 Reaanging, 3 q z ' + z Cost function is 3 q c (, z + z' + + z' z ' In vey shot un, (z,z ) fixed so output fixed. 38 z Shot-Run Total Costs When z is fixed at z, the fim cannot equate MRTS with the atio of input pices z q q q 0 z ' Z Z z 39 3

14 Relationship between Shot- Run and Long-Run Costs Total costs SC (z ) SC (z ) C SC (z ) The long-un C cuve can be deived by vaying the level of z q q q Output 40 Shot-Run Maginal and Aveage Costs The shot-un aveage total cost (SAC) function is SAC total costs/total output SC/q The shot-un maginal cost (SMC) function is SMC change in SC/change in output SC/q 4 4