Calculus in Business. By Frederic A. Palmliden December 7, 1999

Transcription

1 Calculus in Business By Frederic A. Palmliden December 7, 999

2 Optimization Linear Programming Game Theory

3 Optimization The quest for the best Definition of goal equilibrium: The equilibrium state is defined as the optimum position for a given economic unit and in which the said economic unit will be deliberately striving for attainment of that equilibrium.

4 There are always many alternatives when any kind of an economic project is to be carried out. One or more will however be more desirable.

5 Most common criterion in economics Maimizing Minimizing

6 Definition of an objective function: A function whose dependent variable represents the object of maimization or minimization and in which the set of independent variables indicates the objects whose magnitudes can be chosen with a view to optimizing.

7 If the first derivative of f () at a point 0 is f '( then the value 0) 0 of the function at this point will be : a relative maimum a relative minimum neither

8 Concept of the second derivative and higher orders derivative d n y d n

9 If f '( 0) 0 then the value of the function at that point will be -a relative maimum if -a relative minimum if f ( 0 f ( 0 ) 0 ) 0

10 A firm must choose the output level such that MC=MR R=R() total revenue function C=C() total cost function Objective function ( ) R( ) C( )

11 First condition for a maimum d d d 0 d ( ) R( ) C( ) 0 iff R ( ) C( ) or MR MC

12 Second condition d d ( ) R( ) C( ) 0 iff R( ) C( )

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14 An objective function for a multiproduct firm ) ( ) (, ), ( z y y z y y z y z y y z y z f f f f y f z z

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16 For a maimum For a minimum 0,, 0 y yy yy y f f f f f f f 0,, 0 y yy yy y f f f f f f f

17 Eample R C P 0 P 0 Note: C 4

18 Profit function: P P P P C R

19 4 P 0 4 Solutions: P 0 4P0 P0 4P0 P, 5 5 0

20 Second condition: Hessian H 4 4 H 4 0 H 5 0

21 Linear Programming Technique that allows decision makers to solve maimization and minimization problems where there are certain constraints that limit what can be done.

22 The objective function is a linear function of the variables to be determined. The values of these variables must satisfy certain constraints, which are in the form of inequalities.

23 Production planning eample: Profit pair batch of cotton cloth finished is $.00 with process, $0.90 with process, $.0 with process 3. Process uses 3 machine-hours of finishing capacity per batch of cotton cloth processed, process uses.50 machine-hours and process 3 uses 5.5 machine-hours. Process uses 0.40 hours of labor per batch of cotton cloth processed, process uses 0.50 hours, and process 3 uses 0.35 hours. 6,000 machine-hours per week 600 hours per week

24 Problem: maimize Constraints: ,, ,

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30 In this kind of problem the optimal solution will generally entail the use of no more processes than there are constraints , ,

31 Game Theory A game is a competitive situation in which two or more persons pursue their own interests and no person can dictate the outcome

32 The rules of the game A strategy A player s payoff

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35 A Nash equilibrium is a set of strategies such that each player believes that it is doing the best it can given the strategy of the other player(s).

36 Selection in Dynamic Entry Games The Model i {,} N ( i) the N-stage game in which firm i is the initial leader.

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39 Assumption A: the functions v v,, C C, are continuous on X. Assumption B: the functions v,v are strictly increasing on X. Assumption C: the functions C,C Assumption D: the functions are strictly decreasing on X. are nonincreasing on X. ( C v ),( C v)

40 Symmetric Firms v C v( l v C ) 0 v 3 C 3

41 Average cost pricing C( L ) 0

42 The firm s OEPP L C( n ) n n v( k ) for n,..., N

43 In the unique perfect equilibrium of any symmetric entry game i {,} N (i) with firm i maintains with the OEPP { N, N,..., } Total rent: N n v( n )

44 Firm plays In and Firm plays In its payoff is s payoff is, payoff maimized at i N i j 0 ) ( ) ( N n n i v C i N n n i v v ) ( ) ( 0 ) (, N N N i v

45 i N i Firm plays In and Firm enters, its total profit is C( d i ( i j i ) ) 0 N n v( n ) 0

46 If firm played i j plays Out, G(j) is plays In and N and its OEPP v( N ) N n v( n ) 0

47 Optimal choice for firm i In and N Merci beaucoup