Cost Minimization and Cost Curves. Beattie, Taylor, and Watts Sections: 3.1a, 3.2a-b, 4.1
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1 Cost Minimization and Cost Curves Beattie, Talor, and Watts Sections: 3.a, 3.a-b, 4.
2 Agenda The Cost Function and General Cost Minimization Cost Minimization ith One Variable Input Deriving the Average Cost and Marginal Cost for One Input and One Output Cost Minimization ith To Variable Inputs
3 Cost Function A cost function is a function that maps a set of inputs into a cost. In the short-run, the cost function incorporates both fied and variable costs. In the long-run, all costs are considered variable. 3
4 Cost Function Cont. The cost function can be represented as the folloing: C c(,,, n * + * + + n * n Where i is the price of input i, i, for i,,, n Where C is some level of cost and c( is a function 4
5 Cost Function Cont. When there are fied costs, the cost function can be represented as the folloing: C c(, 3,, n * + * + TFC Where i is the price of the variable input i, i, for i and Where i is the price of the fied input i, i, for i 3, 4,, n Where TFC 3 * n * n When inputs are fied, the can be lumped into one value hich e usuall denote as TFC Where C is some level of cost and c( is a function 5
6 Cost Function Cont. Suppose e have the folloing cost function: C c(,, 3 5* + 9* + 4* 3 If 3 as held constant at 4, then the cost function can be ritten as: C c(, 4 5* + 9* + 56 Where TFC in this case is 56 6
7 Cost Function Cont. The cost function is usuall meaningless unless ou have some constraint that bounds it, i.e., minimum costs occur hen all the inputs are equal to zero. 7
8 Standard Cost Minimization Model Assume that the general production function can be represented as f(,,, n. Min. r. t.,,..., n n n subject to : f (,,..., n 8
9 Cost Minimization ith One Variable Input Assume that e have one variable input ( hich costs. Let TFC be the total fied costs. Assume that the general production function can be represented as f(. Min. r. t. + TFC subject to : f ( 9
10 Cost Minimization ith One Variable Input Cont. In the one input, one output orld, the solution to the minimization problem is trivial. B selecting a particular output, ou are dictating the level of input. The ke is to choose the most efficient input to obtain the output. 0
11 Eample of Cost Minimization Suppose that ou have the folloing production function: f( 6 - You also kno that the price of the input is $0 and the total fied cost is 45. Min r. t. subject to : f ( 6
12 Eample of Cost Minimization Cont. Since there is onl one input and one output, the problem can be solved b finding the most efficient input level to obtain the output. ± ± ± ± (( 6 ( 6 ( 0 6 6
13 Eample of Cost Minimization Cont. Given the previous, e must decide hether to use the positive or negative sign. This is here economic intuition comes in. The one that makes economic sense is the folloing: 3 9 3
14 Cost Function and Cost Curves There are man tools that can be used to understand the cost function: Average Variable Cost (AVC Average Fied Cost (AFC Average Cost (ATC Marginal Cost (MC 4
15 Average Variable Cost Average variable cost is defined as the cost function ithout the fied costs divided b the output function. AVC AVC AVC f ( APP 5
16 Average Fied Cost Average fied cost is defined as the cost function ithout the variable costs divided b the output function. AFC AFC TFC TFC f ( 6
17 Average Total Cost Average total cost is defined as the cost function divided b the output function. It is also the summation of the average fied cost and average variable cost. ATC ATC ATC + TFC + TFC TFC + f ( f ( 7
18 Marginal Cost Marginal cost is defined as the derivative of the cost function ith respect to the output. To obtain MC, ou must substitute the production function into the cost function and differentiate ith respect to output. MC TC( TVC( f MC dtc( d * ( is inverse of MPP f * f dtvc( d ( + TFC ( f ( 8
19 Eample of Finding Marginal Cost Using the production function f( 6 -, and a price of 0, find the MC b differentiating ith respect to. To solve this problem, ou need to solve the production function for and plug it into the cost function. This gives ou a cost function that is a function of. 9
20 0 Eample of Finding Marginal Cost Cont. d dc MC d dc MC d dc MC 9 5 *0(9 ( *0( c( C 9 0(3 c( C Plugging this into the cost function gives : 9 3 6
21 Notes on Costs MC ill meet AVC and ATC from belo at the corresponding minimum point of each. Wh? As output increases AFC goes to zero. As output increases, AVC and ATC get closer to each other.
22 Production and Cost Relationships Summar Cost curves are derived from the phsical production process. The to major relationships beteen the cost curves and the production curves: AVC /APP MC /MPP
23 Product Curve Relationships When MPP>APP, APP is increasing. This implies that hen MC<AVC, then AVC is decreasing. When MPPAPP, APP is at a maimum. This implies that hen MCAVC, then AVC is at a minimum. When MPP<APP, APP is decreasing. This implies that hen MC>AVC, then AVC is increasing. 3
24 4 Product Curve Relationships Cont. '( ( 0 '( ( 0 '( ( 0 ] ( [ '( ( ( f f f f f f f f f d davc f AVC > < > < > < > <
25 5 Product Curve Relationships Cont. AVC MC APP MPP APP MPP MPP APP f f f f > < > < > < > < > < > < '( ( '( (
26 Eample of Eamining the Relationship Beteen MC and AVC Given that the production function f( 6 -, and a price of 0, find the input(s here AVC is greater than, equal to, and less than MC. To solve this, eamine the folloing situations: AVC MC AVC > MC AVC < MC 6
27 Eample of Eamining the Relationship Beteen MC and AVC Cont. AVC 0 f ( MC MPP AVC MC
28 Eample of Eamining the Relationship Beteen MC and AVC Cont. AVC MC MPP AVC > MC > > f ( 0 0 > >
29 Eample of Eamining the Relationship Beteen MC and AVC Cont. AVC > 0 f ( MC MPP AVC < MC 0 0 < >
30 Revie of the Iso-Cost Line The iso-cost line is a graphical representation of the cost function ith to inputs here the total cost C is held to some fied level. C c(, + 30
31 3 Finding the Slope of the Iso- Cost Line d d C C C C +
32 Eample of Iso-Cost Line Suppose ou had $000 to spend on the production of lettuce. To produce lettuce, ou need to inputs labor and machiner. Labor costs ou $0 per unit, hile machiner costs $00 per unit. 3
33 Eample of Iso-Cost Line Cont. Given the information above e have the folloing cost function: C c(labor, machiner $0*labor + $00*machiner 000 0* + 00* Where C 000, labor, machiner 33
34 Eample of Iso-Cost Line Graphicall X 0 (/0*
35 Finding the Slope of the Iso- Cost Line d d 0 35
36 Notes on Iso-Cost Line As ou increase C, ou shift the iso-cost line parallel out. As ou change one of the costs of an input, the iso-cost line rotates. 36
37 Input Use Selection There are to as of eamining ho to select the amount of each input used in production. Maimize output given a certain cost constraint Minimize cost given a fied level of output Both give the same input selection rule. 37
38 Cost Minimization ith To Variable Inputs Assume that e have to variable inputs ( and hich cost respectivel and. We have a total fied cost of TFC. Assume that the general production function can be represented as f(,. Min. r. t., + subject to : f (, + TFC 38
39 39 First Order Conditions for the Cost Minimization Problem ith To Inputs ( 0, ( , (,, ( Γ Γ Γ Γ f MPP f MPP f f TFC λ λ λ λ λ λ λ
40 Implication of MRTS Slope of Iso-Cost Line Slope of iso-cost line - /, here is the cost of input and is cost of input. MRTS -MPP /MPP This implies MPP /MPP / Which implies MPP / MPP / This means that the MPP of input per dollar spent on input should equal MPP of input per dollar spent on input. 40
41 Eample of Cost Minimization ith To Variable Inputs Suppose ou have the folloing production function: f(, 0 ½ ½ Suppose the price of input is $ and the price of input is $4. Also suppose that TFC 00. What is the optimal amount of input and if ou ant to produce 0 units. 4
42 Eample of Cost Minimization ith To Variable Inputs Cont. Summar of hat is knon:, 4, TFC 00 0 ½ ½ 0 Min. r. t., subject to :
43 43 Eample of Cost Minimization ith To Variable Inputs Cont. Solution done in class ,, ( Γ Γ Γ Γ λ λ λ λ λ
44 44 Solving Eample Using Ratio of MPP s Equals Absolute Value of the Slope of the Cost Function ( 0(4 3.Plugintoproduction function Set. and MRTS MPP MPP MRTS
45 Solving Eample Using MRTS from the Isoquant and Setting it Equal to the Slope of the Cost Function. Find Isoquant.Find MRTS MRTS 3.Set 00 MRTS Solve for using
46 Final Note on Input Selection It does not matter hether ou are tring to maimize output given a fied cost level or minimizing a cost given a fied output level, ou ant to have the iso-cost line tangent to the isoquant. This implies that ou ill set the absolute value of MRTS equal to the absolute value of the slope of the iso-cost line. 46
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