5-1 HOMOGENEOUS PRODUCTION FUNCTION

1
5-1 HOMOGENEOUS PRODUCTION FUNCTION

• For homogeneous production return to scale could
  easily be defined.
• f(tx1,tx2)tkf(x1,x2 )
• kgt1 , increasing return to sacle for small
  range.
• k1 , constant return to scale
• klt1 , decreasing return to scale
• just like the utility function , the homogeneous
  production function contain linear expansion path
  which means RTSx1x2f(x2/x1).
• the homothetic production function is a
  increasing transformation function of a linear
  homogeneous production function. As it is proved
  for the homothetic production utility
  function, for the homothetic production function
  the average cost function is independent from the
  level of production, and it is only a function
  the ratio of input price levels. This can be
  easily shown for the Cobb-Douglas production
  function.

q
x
x2
Expansion path
x1
2
5-1 HOMOGENEOUS PRODUCTION FUNCTION

• qf(x1,x2)
• If f(tx1,tx2) tkf(x1,x2) , then x1f1 x2f2
  kf(x1 , x2)
• If k1 , then x1f1 x2f2 q f(x1 , x2)
• (x1f1)/q (x2f2)/q 1
• (?q/ ?x1)(x1/q) (?q/ ?x2)(x2/q) 1
• ?x1,q ?x2,q 1 exhaustion theorem , or
• x1f1 x2f2 q marginal productivity theory
  of distribution
• two steps 1 each factor should receive its
  marginal productivity
• 2 all output should be
  exhausted.
• for k1, long run profit equals to zero.
• ?pq r1x1 r2x2 pq pf1x1 pf2x2 pq
  p(f1x1 f2x2)pq pq 0
• ?(t)pf(tx1,tx2) r1t x1 r2t x2 t pf(x1,x2)
  tr1x1 tr2x2 t?
• profit function is homogenous of degree one with
  respect to scale of
  
  production.
• If each factor is paid according to its value of
  marginal product , profit will be zero regardless
  of its scale of production.

3
5-1 HOMOGENEOUS PRODUCTION FUNCTION

• So when production function is homogeneous of
  degree one, the scale of production is not
  defined
• If ?gt0 , t (scale of production) can be
  increased forever.
• If ?lt0 the firm will go out of business.
• If ?0 scale can not be defined.
• Solution
• In order to use the exhaustion theorem results,
  considering the above difficulties
• 1 Production function defined not as homogeneous
  of degree one.
• 2- First and second order condition for profit
  maximization should exist.
• 3- maximum profit should equal to zero .
• ?pq r1x1 r2x2 0 , r1pf1 , r2pf2 ,
  
• ?pq pf1x1pf2x2 0 , qf1x1f2x2
  (exhaustion theorem)
• undefined scale of production, actually means
  non existence of the second order condition for
  profit maximization.

4
5-1 HOMOGENEOUS PRODUCTION FUNCTION

• x1f1 x2f2 q
• (f1x1f11x2f21)dx1 (f2 x1f12 x2f22 )dx2
  dq
• dq/dx1(dx20)f1f11x1 x2f21f1 , f11
  (-x2/x1)f21
• dq/dx2(dx10)f2f22x2 x1f12f2 , f22
  (-x1/x2)f12
• f11f22f122 f122 - f122 0(straight line.It
  should be greater than zero).
• However, constant return to scale assumption
  is needed in many cases what should be done we
  assume that
• 1-, the whole industry has a constant return to
  scale production function but the individual firm
  does not.
• 2-Scale of production is finite, in such a way
  that equates demand with supply in the whole
  industry .
• The long run cost function will have a special
  shape when production function is homogeneous .
  Suppose that x10 and x20 , relates to the one
  unit of the production level
• f (tx10,t x20)tk f(x10,x20) tk , so
  q tk
• c r1x10 r2x20 a ? cost of producing
  one unit

5
5-1 HOMOGENEOUS PRODUCTION FUNCTION

• Cat total cost of producing q units.
• qtk production function ,
• Caq1/k total cost function , AC TC/q
  aq(1-k)/k , MC(a/k)q(1-k)/k
• 5-2 C.E.S. PRODUCTION FUNCTION
• An special form of homogeneous production
  function is the one which has Constant Elasticity
  of Substitution.
• qAax1-? (1-a)x2 -? -1/ ?
  homogeneous of degree one.
• MPx1(?q/?x1) a/(A-?)(q/x1) ?1
  homogeneous of degree zero
• MPx2(?q/?x2) (1-a)/(A-?)(q/x2) ?1
  homogeneous of degree zero
• RTSx1 x2 ( MPx1)/( MPx2)a/(A-?)(q/x1)
  ?1/(1-a)/(A-?)(q/x2) ?1
• RTSx1 x2a/(1-a)(x2/x1) ?1 , if x1
  increases then RTS wil decrease
• s(f1f2)/(f12q) , f12(?1) a (1-a)q2?1/A2?
  ,
• s1/(?1) , ?(1- s)/ s
• since sgt0 , then ?gt-1 for concavity
  condition to prevail.

6
5-2 C.E.S. PRODUCTION FUNCTION

• 1 if s 0 then ? 8 ,
• RTS a/(1-a)(x2/x1) ?1
• when ? 8 , if x2gtx1 , RTS 8
  ,
• if x1gtx2 ,
  RTS 0
• 2 0lt slt1 , ? gt 0
• qAax1-? (1-a)x2 -? -1/ ?
• If qq0 , then , ax1-? (1-a)x2 -?
  q0/A K
• If x10 , x2 is not defined, if x1 8,
  ax1-? 0 , x2K/(1- a)-1/?
• If x20 , x1 is not defined, if x2 8,
  (1-a)x2 -? 0 , x1(K/a)-1/?

x2
RTS 8
RTS0
x1
x2
K/(1- a)-1/?x2
x1

• x1(K/a)-1/?

7
5-2 C.E.S. PRODUCTION FUNCTION

• 3 if s 1, ?0, q is not
  defined, using hopital rule we get
  Cobb-Douglass production function
• 4 if sgt1, , -1lt ? lt0 , ax1-? (1-a)x2
  -? q0/A K
• if x10 , then , x2K/(1- a)-?
• if x20 , then , x1(K/a)-?
• 5 s 8 , then ? -1 , ax1 (1-a)x2
  q0/A K
• if x10 , then , x2K/(1- a)
• if x20 , then , x1(K/a)

x2K/(1- a)-?
x1(K/a)-?
x2K/(1- a)
x1(K/a)
8
5-2 C.E.S. PRODUCTION FUNCTION

• Estimation of C E S
• RTSx1 x2a/(1-a)(x2/x1) ?1 (r1/r2)
• (x2/x1) a(r1/r2) s s 1/(1 ?)
  a(1-a)/ a s
• log(x2/x1)log a s log (r1/r2)

9
5-3 KHUN-TUCKER CONDITIONS

• First application
• qf(x1,x2)
• x1x11x12
• x12 x1 bought in the market with the market
  price equal to r1
• x11x1 produced by the firm with a production
  function x11g(x3)
• x3 input bought for the production of x1
  with a price equal to r3
• Max ? pq - TC pq (r1x12 r2x2
  r3x3)
• S.T. X11 g(x3)
• Max ? pf(x11x12, x2) r1x12 r2x2 r3x3
  ?g(x3) x11
• ??/?x11 ?x11 pf1 - ?0 ,
  x11 ?x110
• ??/?x12 ?x12 pf1 r1 0 ,
  x12 ?x120
• ??/?x2 ?x2 pf2 r2 0 ,
  x2 ?x20
• ??/?x3 ?x3 ? g(x3) r3 0 , x3
  ?x30
• ??/?? ??g(x3) x11 0 ,
  ? ??0
• Three cases could be mentioned

10
5-3 KHUN-TUCKER CONDITIONS

• 1- x1 is totally bought from the market
  x110 , x1x12
• x11 0 , ?x11 lt 0 , pf1 ?lt0 ,
  pf1 lt ?
• x12 ? 0 , ?x12 0 , pf1 r1 0 ,
  pf1 r1
• x30 , ?x3 lt 0 , ? g(x3) r3 lt0
  , ? lt r3 / g(x3) MCx1
• Pf1 r1 lt ? lt MCx1 , r1 lt MCx1
• 2 x1 is totally produced , nothing will be
  bought from the market.
• x11 ? 0 , ?x11 0 , pf1 ?0 ,
  pf1 ?
• x12 0 , ?x12 lt 0 , pf1 r1 lt 0 ,
  pf1 lt r1
• x3 ? 0 , ?x3 0 , ? g(x3) r3 0
  , ? r3 / g(x3) MCx1
• Pf1 ? MCx1 , pf1 lt r1 , ,
  r1gt MCx1
• 3- x1 is both produced and bought from the market
• x11 ? 0 , ?x11 0 , pf1 ?0 ,
  pf1 ?
• x12 ? 0 , ?x12 0 , pf1 r1 0 ,
  pf1 r1
• x3 ? 0 , ?x3 0 , ? g(x3) r3 0
  , ? r3 / g(x3) MCx1
• Pf1 ? MCx1 , pf1 r1 , ,
  r1 MCx1

11
5-3 KHUN-TUCKER CONDITIONS

• Second application
  
• qf(L, K) LL1 L2 L3 wage
  discrimination.
• If L1 L0 , wagew
• If L2 0.2 L0 , wage1.5w
• If L3 0.2 L0 , wage2w
• Max ?0 pf(L1L2L3,K) - wL1 - 1.5wL2 - 2wL3
  rk µ1(L0-L1) µ2(0.2L0-L2) µ3(0.2L0-L3)
• ??0/?L1 ?0L1 pfL w - µ1 0 ,
  L1 ?0L10
• ??0/?L2 ?0L2 pfL 1.5w µ2 0 , L2
  ?0L20
• ??0/?L3 ?0L3 pfL 2w µ3 0 ,
  L3 ?0L30
• ??0/?k ?0k pfk r 0 ,
  k ?0k0
• ??0/? µ1 ?0µ1 L0 L1 0 ,
  µ1 ?0µ10
• ??0/? µ2 ?0µ2 0.2L0 L2 0 ,
  µ2 ?0µ20
• ??0/? µ3 ?0µ3 0.2 L0 L3 0 ,
  µ3 ?0µ30
• Seven situations is possible

12
5-3 KHUN-TUCKER CONDITIONS

• 1- L1 L2 L3 0
• L1 0 , ?0L1lt0 , pfL- w - µ1 lt0 ,
• L0 L1 gt0 , µ10
  , pfL lt w
• 2- 0lt L1lt L0 , L2 L3 0 ,
• L1gt 0 , ?0L1 0 , pfL- w - µ1 0 ,
• L0 L1 gt0 , µ10
  , pfL w
• 3- L1L0 , L2 L3 0 ,
• L1gt 0 , ?0L1 0 , pfL- w - µ1 0 ,
• L0 L1 0 , µ1 gt 0
  , pfL gt w
• L2 0 , ?0L2 lt0 , pfL- 1.5 w µ2 lt0 ,
  
• 0.2 L0 L2 gt0 , µ2
  0 pfL lt1.5 w
• 4 - L1L0 , 0ltL2lt0.2 L0 , L30
• L1gt 0 , ?0L1 0 , pfL- w - µ1 0 ,
• L0 L1 0 , µ1 gt 0
• L2gt 0 , ?0L2 0 , pfL- 1.5 w µ2 0 ,
• 0.2 L0 L2 gt0 , µ2
  0 pfL 1.5 w

13
5-3 KHUN-TUCKER CONDITIONS

• 5 - L1L0 , L2 0.2 L0 , L30
• L1gt 0 , ?0L1 0 , pfL- w - µ1 0 ,
• L0 L1 0 , µ1 gt 0
• L2gt 0 , ?0L2 0 , pfL- 1.5 w µ2 0
  ,
• 0.2 L0 L2 0 , µ2
  gt0 1.5 wlt pfLlt2w
• 6 - L1L0 , L2 0.2 L0 , 0lt L3lt
  0.2 L0 pfL 2w
• 7 - L1L0 , L2 0.2 L0
  L3 0.2 L0 pfL gt 2w

14
5-4 Duality in production

• Max qf(x1,x2) cost function
  could be found from
• S.T. Cr1x1r2x2 production
  function
• Min Cr1x1r2x2
  production function could be found
• S.T. q0 f(x1,x2) from cost
  function (duality )
• xixi(r1,r2, q0) , or xixi(r1/r2 , q0)
• Cr1x1r2x2
• r1? f1 , r2? f2
• ( ?C/?r1)(?r1/?r1)x1 ( ?x1/?r1)r1 ( ?r2/?r1)x2
  ( ?x2/?r1)r2
• x1 ? f1 ( ?x1/?r1) ? f2 (
  ?x2/?r1)
• x1 ? f1 ( ?x1/?r1) f2 (
  ?x2/?r1)x1 ??q0/?r1x1
• ?C(q,r1,r2)/?r1 x1(q , r1, r2)
• ?C(q,r1,r2)/?r2 x2(q , r1, r2)
• From the above equations we could find q in
  terms of x1 and x2

15
5-4 Duality in production

• Example
• CA(r1ar2b)1/(ab)q1/(ab)
• A(ab)(aabb)-1/(ab)
• ?C/?r1 a/(ab)Aq1/(ab)(r2/r1)b/(ab) x1
  
• ?C/?r2 b/(ab)Aq1/(ab)(r2/r1)-a/(ab) x2
• x1ax2b q a/(ab)ab/(ab)b A(ab)
• q x1ax2b 1/a/(ab)ab/(ab)b A(ab)
  B x1ax2b

16
5-5 PRODUCTION UNDER UNCERTAINTY

• Basic idea
• Under certainty people know exactly what they are
  getting and how much utility they will yield.
  There are three types of uncertainty
• 1- some goods by their nature are games . like
  horse riding bets or insurance or stock market
  transaction. In these cases purchase does not
  guarantee a particular outcome.
• 2- Dealing with others, or uncertainty about the
  action of individuals.
• 3- Lack of understanding of information. Like
  information about weather condition . For this
  reason people are willing to pay for these kind
  of information.
• We only concentrate on the first type of
  uncertainty.
• Game x prizes of x1 , x2 , x3 , ..xn
• probability v1 , v2 , v3 ,
  ..vn

17
5-5 PRODUCTION UNDER UNCERTAINTY

• Game Flipping a coin x1 win 1( Head) ,
  x2loose 1(Tail)
• E(x) v1x1 v2x2 (1/2)(1) (1/2)( -1) 0
• If the player plays the game many times (n
  8) he will neither loose nor win.
  
• Game Flipping a coin x1 win 4( Head) ,
  x2loose 3(Tail)
• E(x) v1x1 v2x2 (1/2)(4) (1/2)( -3) 1/2
• If the player plays the game many times (n
  8) he will win 1/2.
• FAIR GAMES
• If cost of entry is equal to the expected value
  of the game, the game is called fair game. We
  expect that people accept the fair games. But ,
  the Petersburg paradox showed that this is not
  the case.
• .

18
5-5 PRODUCTION UNDER UNCERTAINTY

• Petersburg paradox
• Game Flipping a coin till head appears
• Xi represent the prize awarded when the head
  appears . The game could be played infinitely.
• amount of win probability of win
• x12 v11/2
• x222 v2
  (1/22)
• x323
  v3(1/23)
• .. .
• xn2n vn(1/2n) n
  8
• E(x)S vi xi ( 111..1) 8
• Is there any one who is willing to pay infinite
  amount of money for this game?
• Bernolli tried to solve the puzzle, by
  introducing the concept of
• Utility . He said that the utility of the win is
  important for decision making rather the the
  amount of the win by itself.

19
5-5 PRODUCTION UNDER UNCERTAINTY

• U(x)alog(xi)utility of the amount of win (x)
  probability of win
• u(x1)alog2
  v11/2
• u(x2)alog22
  v2 (1/22)
• u( x3)alog23
  v3(1/23)
• ..
  .
• u(xn)alog2n
  vn(1/2n)
• 
  n 8
• Eu(x)Sviu(xi)(1/2)alog2(1/22) alog22 ..
  (1/2n) alog2n
• alog2S(i/2i )2alog2
• As it is seen the expected utility of x is
  defined , but there is no upper bond on u(x). a
  could be very large, and the expected utility
  could be defined very large and unreasonable.
• Another attempt is made by J.V. Neuman, and O.
  Morgenstern by introducing the concept of
  decision theory under the uncertain situations,
  an attempt by these two to generalize some of the
  foundations of the theory of individual choice to
  cover uncertain situation.

20
5-5 PRODUCTION UNDER UNCERTAINTY

• Utility index
• The first attempt to in explaining the theory is
  to assign indices to utility under uncertain
  situations.
• Prizes x1, x2, x3 , .xn xn is
  the most preferred
• Prob. v1 , v2 , .vn U(x1) 0
  U(xn ) 1
• J.V. Neuman, and O. Morgenstern showed that there
  is a reasonable way to assign specific utility
  numbers to the different prizes available. What
  they proved is that a probability like ?i exist
  which makes the following relation holds
• U(xi) ?i u(xn)(1- ?i) u(x1)
• What this relation means is that, that there
  exist a probability such as ?i which makes the
  individual indifferent between the following two
  alternatives,(or both alternatives have the same
  satisfaction for the individual)
• 1- having xi with certainty,(with a satisfaction
  of u(xi)) and
• 2-a game winning xn with probability ?i and
  winning x1 with probability (1- ?i), with a
  satisfaction of ?i u(xn)(1- ?i) u(x1)

21
5-5 PRODUCTION UNDER UNCERTAINTY

• Suppose that we assign two numbers to u(xn) and
  u(x1)
• U(xn) 1 , and U(x1) 0
• U(xi) ?i u(xn)(1- ?i) u(x1) ?i
• for each i 1,2,.n , there exist a probability
  like ?i which is equal to the utility of having
  xi with certainty . So these probabilities could
  be considered as utility indices.
• Expected utility maximization
• Using the utility index it is possible to prove
  that a rational individual will choose among the
  uncertain situations based upon their expected
  value of utility to him.The one with the highest
  expected utility will be chosen.
• Suppose that individual faces two games
• Game 1 winning x2 with probability (q) ,
• winning x3 with probability (1-q),
• Game 2 winning x5 with probability (t),
• winning x6 with probability (1-t)
  

22
5-5 PRODUCTION UNDER UNCERTAINTY

• Eu(1)q u(x2)(1-q) u(x3)q ?2(1-q) ?3
• Eu(1)q?2 u(xn)(1- ?2) u(x1)(1-q) ?3
  u(xn)(1- ?3) u(x1)
• Eu(1)q ?2(1-q) ?3 u(xn)q (1- ?2) (1-q)
  (1- ?3) u(x1)
• If ?aq ?2(1-q) ?3, then q (1- ?2) (1-q)
  (1- ?3)(1- ?a)
• Eu(1) ?au(xn) (1- ?a)u(x1) ?a
• Eu(2)t u(x5)(1-t) u(x6)t ?5(1-t) ?6
• Eu(2)t ?5 u(xn)(1- ?5) u(x1)(1-t) ?6
  u(xn)(1- ?6) u(x1)
• Eu(2)t?5(1-t) ?6 u(xn)t (1- ?5) (1-t)
  (1- ?6) u(x1)
• If ?b t ?5 (1-t) ?6, then t (1- ?5)
  (1-t) (1- ?6)(1- ?b)
• Eu(2) ?b u(xn) (1- ?b )u(x1) ?b
• If Eu(1)gt Eu(2) , then ?agt?b
• Game one will be chosen because it yields a
  higher utility.

23
5-5 PRODUCTION UNDER UNCERTAINTY

• Risk Aversion
• Those games which has higher expected utility are
  less risky.
• Game 1 winning b with probability of 1/2
• winning -b (loosing b) with
  probability of 1/2
• Game 2 winning 2b with probability of 1/2
• winning -2b (loosing 2b) with
  probability of 1/2

U(x)
U(x)
U(w2b)
U(wb)
U(w)
U(w-b)
Eu(1)1/2u(wb)1/2 u(w-b)
Eu(2)1/2u(w2b)1/2 u(w-2b)
U(w - 2b)
x
W
Wb
W2b
W-b
W-2b
24
5-5 PRODUCTION UNDER UNCERTAINTY

• An individual who always rejects fair bets is
  said to be risk-averse . His marginal utility of
  income(wealth) is decreasing. He will always
  willing to pay a premium to insure himself
  against the risk. Change uncertain position to
  certain one.
• Production behavior
• There are two source of uncertainty for a
  producer
• price and quantity. At the first stage, suppose
  that price is not certain p1 , p2 , p3
  ,..pn price level
• v1 , v2 , v3 ,..vn
  probability
• ?ipiq c(q) profit level when price pi
• u(?i) utility obtained from the
  profit level when price is pi
• Eu(?)Sin vi u(?i) expected utility of
  profit derived in the uncertain situation .
• In the uncertain situations the producer should
  maximize the expected utility of profit, rather
  than maximizing the expected level of profits.
• d Sin vi u(?i)/dq 0 , Sin
  vi u(?i)(pi c(q))0

25
5-5 PRODUCTION UNDER UNCERTAINTY

• There could be three situations
• 1- individual is risk netural. He is indifferent
  between risky and non risky situations.
• d u(?i)/d?i 0 , u(?i)A is constant,
  u(?i)0
• v1A(p1 c(q)) v2A(p2 c(q)). vn A(pn
  c(q)0 , qq0
• 2- individual is risk averse. He prefers non
  risky situation to risky situation.
• d u(?i)/d?i lt 0 , u(?i)is decreasing ,
  u(?i) lt 0
• v1 u(?1)(p1c(q)) v2 u(?2)( (p2c(q)). vn
  u(?n)(pnc(q))0,, qq1
• and u(?1)gt u(?2)gt. u(?n), q1ltq0
• 3- individual is risk taker. He is prefers risky
  situation to non risky situation.
• d u(?i)/d?i gt 0 , u(?i)is increasing ,
  u(?i) gt 0
• v1 u(?1)(p1c(q)) v2 u(?2)( (p2c(q)). vn
  u(?n)(pnc(q))0 qq2
• , and u(?1)lt u(?2)lt. u(?n), q2gtq0

26
5-5 PRODUCTION UNDER UNCERTAINTY

• 2- suppose that price is known but quantity is
  not known. (future market in the agricultural
  products).
• quantity q1 , q2 , q3 ,.qn target
  q0, qifiq0
• fi percent of q0 which depends on climate.
• probability v1 , v2 , v3 , .vn
• ?ipfiq0 c(q0)
• Producer decision is to find the target
  production level
• d Sin vi u(?i)/dq 0 , Sin vi u(?i)(p fi
  c(q))0
• the same result will be obtained.

27
5-6 LINEAR PRODUCTION
x2
Expansion path

• 1-Fixed proportion input combination
• 2-constant return to scale.
• 1- one output q
• m inputs xi , I1,2,3,4.n
• one activity (production technique) j1
• xiaiq total input of xi necessary to
  produce total q
• aiinput of xi necessary to produce one unit
  of q
• qmin(xi/ai)
• suppose that a1 2 , and a2 5 ,
• q x1a1q x2a2q
• 1 2 5
• 2 4 10
• ..
• n 2n 5n
• Qmin(x1/2 , x2/5)min(8/2 , 10/5)2

q3
3
q2
2
q1
1
x1
1
2
3
Expansion path
x2
q2
10
q1
5
x1
2
4
8
28
5-6 LINEAR PRODUCTION

• 2- one output q
• m input xi i 1, 2, . m
• n activity j1, 2, .n
• q S1n qj
• xi Sj1n aij qj
• aij xi necessary to produce one unit of q
  in the jth activity.
• input of xi necessary to produce one unit of
  q Ai
• Ai (xi/q)Sj1n aij (qj/q) Sj1n aij ?j
  ?j qj/q
• qmin(xi/Ai)
• Suppose that there are three activities and
  two inputs

j1 J2 J3
X1 a111 a 12 2 a134
X2 a218 a225 a233

activity
input
29
5-6 LINEAR PRODUCTION
x2
J2
J1

• Not even it is possible to
• produce one unit of output
• with any of these activities,
• but also it is possible to
• produce one unit of q
• with the combination
• of these activities
• (xi/q) Sj1n aij ?j ,
• ?j qj/q , j1,2
• Suppose ?1 ?21/2 , q1
• X1qa11 ?1a12 ?2
• 1(1)(1/2)(2)(1/2)1.5
• X2qa21 ?1a22 ?2
• (1)(8)(1/2)(5)(1/2)6.5

16
Q2
E
J3
F
Q2
7.33
10
A
6.5
8
Q1
G
Q2
B
Q1
5
C
Q1
6.66
x1
1.5
2
4
6
8
10
1
30
5-6 LINEAR PRODUCTION

• Suppose ?21/3 ?32/3 , q2
• X1qa12 ?2a13 ?3
• 2(2)(1/3)(4)(2/3)6.66
• X2qa22 ?2 a23 ?3
• (2)(5)(1/3)(3)(2/3)7.33
• Only points on ABC line or EFG line are
  efficient. Any other point is not efficient.
  Efficient point ,(comparing to any other point),
  is the one that can produce the same level of
  output with minimum amount of inputs. In the same
  manner ith activity is not efficient if another
  activity like j, could be found which produce the
  same output with less output (either x1 or x2 or
  both should be lower).
• 3- s outputs qh h1,2,3,4..s
• m inputs xi i 1,2,3,4.
• one activity
• xi Sh1s aihqh
• aih input xi necessary to produce one unit of
  output h.

31
5-6 LINEAR PRODUCTION

• 4- h outputs qh h1,2,3,4,..s
• m inputs xi i 1,2,3.m
• n activity j1,2,3. N
• qh Sj1n ahj zj
• xi Sj1n bij zj
• ahjamount of output h in one unit of
  composite commodity basket in activity j .
• composite commodity basket basket of
  commodities contains from h outputs.
• Zj units of composite commodities produced in
  activity j
• bij input i used for the production of one
  unit of composite commodity in activity j .

32
5-6 LINEAR PRODUCTION

• Application of linear programming for linear
  activities
• Max TRp1q1 p2q2 pnqn original
• S.T. ai1q1 ai2q2 ..ainqn xi0 i
  1,2,3 ,.m
• L p1q1 p2q2 . ?1 x10 (a11q1a12q2a1nqn
  )
• ?2 x20
  (a21q1a22q2a2nqn)
• ?m xm0
  (am1q1am2q2amnqn)
• (?L/?q1)Lq1 p1 - ?1a11 ?2a21 - - ?mam1 0 ,
  q1 Lq10
• (?L/?q2)Lq2 p2 - ?1a12 ?2a22 - - ?mam2 0
  , q2 Lq20
• ..
• (?L/?qn)Lqn pn - ?1a1n ?2a2n - . ?mamn 0
  , qn Lqn0
• (?L/??1) L?1 x10 a11q1 a12q2-- a1nqn
  0 , ?1 L?1 0
• (?L/??2) L?2 x20 a21q1 a22q2-- a2nqn
  0 , ?2 L?2 0
• (?L/??m) L?m xm0 am1q1 am2q2-- amnqn 0
  , ?m L?m 0
• mn equations, mn unkowns (?1. ?m, q1,qn)
• Max TR Sj1n pjqj

33
5-6 LINEAR PRODUCTION

• Min TCr1x1r2x2..rmxm
  dual
• S.T. a1jr1a2jr2..amjrm pj j1,2,3.n.
• aijr1a2jr2..amjrm average cost of qj
• In the long run equilibrium in perfect
  competition , average cost could not be lower
  than price. If Acqj is lower than price, the
  optimum solution (minimum cost) for ri will be ri
  0 .
• Lr1x1r2x2rmxm µ1p10 (a11r1a21r2.
  am1rm)
• 
  .
• µnpn0 (a1n
  r1a2n r2. amn rm)
• (?L/?r1)Lr1x1 - µ1a11 - . - µna1n 0 ,
  r1Lr10
• (?L/?r2)Lr2x2 - µ1a21 - . - µna2n 0 ,
  r2Lr20
• ..
• (?L/?rm)Lrmxm- µ1am1 - . - µnamn 0 ,
  rmLrm0
• (?L/?µ1)Lµ1p1 - a11r1 - . - am1 rm 0 ,
  µ1Lµ10
• (?L/?µ1)Lµ2p2 - a12r1 - . - am2 rm 0 ,
  µnLµn0
• (?L/?µn)Lµnpn - a1n r1 - . -amn rm 0 ,
  µnLµn0
• mn equations, mn unknown , (r1 rm ,
  µ1µn)

34
5-6 LINEAR PRODUCTION

• Comparing original and dual F.O.C.
• Original parameterdual constraint pj
• Original constraint dual parameterxi
• ri ?i shadow price of xi
  TCSimrixiSjn pjqjTR
• qjµj shadow quantity for qj
  TCSimrixiSjn pjqjTR
• Example production of Car and Truck,
• price of car5000 , price of truck4000
• Resource Total required to
  produce required to produce
• available one unit Truck
  one unit of Car
• Labor 720 L.H. 1 L.H.
  2 L.H.
• Machine 900 M.H. 3 M.H.
  1 M.H.
• Steel 1800 T 5 Ton
  4 Ton
• Max TR4000T 5000C
• S.T. T 2C 720
  C300 , T 120
• 3 T C 900 max
  TR1980000
• 5 T 4C 1800

35
5-6 LINEAR PRODUCTION
Min 720pL900pk1800ps s.t. pL3pk5 ps 4000
2pLpk4 ps 5000 As it is clear , machine
constraint is not binding so Price of machine
is zero pk0 pL 5ps 4000
pL1500 2pL 4ps 5000 ps500
min
TC1980000
min TC maxTR
profit 0
Car
900
Machine
800
700
600
steel
500
400
TR4000T5000C
300
200
labor
100
Truck
100
200
300
400
500
600
700


36
H Q , CH 5 , PROBLEMS

• Q5-1 Each of the following production
  functions is homogeneous of degree one. In each
  case, derive the marginal products for x1 and x2
  and demonstrate that they are homogenous of
  degree zero (a) q(ax1x2 bx12 -
  cx22 )/(ax1bx2)
• (b) qAx1ax21-a bx1 cx2
• solution
• a (dq/dx1) (ax1-2bx1)(ax1bx2)-a(ax1x2
  bx12 - cx22 )/(ax1bx2)2
• MPx1(mx1,mx2)
• m2 (ax1-2bx1)(ax1bx2)-a(ax1x2 bx12 -
  cx22)/m2(ax1bx2)2 MPx1(x1,x2)
• (dq/dx2) (ax2 2cx2)(ax1bx2)-a(ax1x2
  bx12 - cx22 )/(ax1bx2)2
• MPx2(mx1,mx2)
• m2 (ax2-2cx2)(ax1bx2)-a(ax1x2 bx12 - cx22
  /m2(ax1bx2)2 MPx1(x1,x2)
• B (dq/dx1)aAx1a-1x21-a baA(x1/x2)a-1,
  MP(mx1,mx2)MP(x1,x2)
• (dq/dx2)aAx1ax2-a baA(x1/x2)a,
  MP(mx1,mx2)MP(x1,x2)

37
H Q , CH 5 , PROBLEMS

• Q5-2 An entrepreneur uses two distinct
  production processes to produce two distinct
  goods, Q1 and Q2. The production function for
  each good is CES, and the entrepreneur obeys the
  equilibrium condition for each. Assume that Q1
  has a higher elasticity of substitution and a
  lower value for the parameter than Q2 .
  Determine the input price ratio at which the
  input use ratio would be the same for both goods.
  Which good would have the higher input use ratio
  if the input price ratio were lower? Which would
  have the higher use ratio if the price ratio were
  higher?
• Solution the equilibrium conditions are as
  follows
• k1a1rs1 k2a2rs2 kiinput use
  ratio for i (x1/x2)i
• by assumption s1gt s2 and a1gt a2
  .
• if k1k2 , then a1rs1 a2rs2 , so r
  (a2/a1) r
• by assumption (s1 - s2)gt0 ,
• if rr , then k1k2 , and if rgt r , then
  k1gtk2 and vice versa

38
H Q , CH 5 , PROBLEMS

• Q5-3 An entrepreneur has the production function
  of the form
• qAx1ax21-a . She buys input and sells output
  at fixed prices, but is subject to a quota which
  allows her to purchase not more than x10 units of
  x1. She would have purchased more in the absence
  of quota. Use the Kuhn-Tucker analysis to
  determine the entrepreneurs conditions for
  profit maximization. What is the optimal relation
  between the value of the marginal product of each
  input and its price. What is the optimal relation
  between the RTS and the input price ratio.
• Solution
• Max ? pq r1x1 r2x2 ?(x10-x1)
• (??/?x1) pMPx1 r1 - ? 0 ,
  (??/?x1)x1 0
• (??/?x2) pMPx2 r2 0 ,
  (??/?x2)x2 0
• (??/? ?) (x10-x1) 0 ,
  ?(x10-x1) 0
• if x10, pMPx1lt r1 ? and if
  x1gt0, pMPx1 r1 ?
• if x20, pMPx2lt r2 and if
  x2gt0, pMPx2 r2
• r2pMPx2p(1-a)A(x1/x2)a
• r1pMPx1paA(x1/x2)a-1 - ?
• RTS (MPx1/MPx2)(r1 ?)/r2 gt r1 /r2

39
H Q , CH 5 , PROBLEMS

• Q5-4 Use Shepherd's lemma to find the
  production function that corresponds to the cost
  function C(r12(r1r2)1/2 r2)q and demonstrate
  that it is CES.
• Solution
• (?C/?r1)(1r2(r1r2)-1/2 )q
• (?C/?r2)(1r1(r1r2)-1/2 )q if r(r1/r2)
• (?C/?r1)(1r2(r1r2)-1/2 )q(1 r1/2)q
• (?C/?r2)(1r1(r1r2)-1/2 )q(1 r1/2)q
• (1 r1/2)q x1
• (1 r1/2) q x2 qx1x2/ (x1x2)
  ½(1/2)x1-1 (1/2)x2 1-1
• Q5-5 A farmer who sells at a fix price of 5
  dollars per unit and has the cost function
  C3.50.5 q2 , plans to maximizes profit under
  certainty. After planning she discovers that she
  can have a fertilizer applied that will increase
  her yield 40 percent with a probability of 0.25
  percent, 60 percent with a probability of 0.5 ,
  and 88 percent with a probability of 0.25 . Her
  utility function is U(?)1/2 . Determine the
  maximum amount that she is willing to pay for the
  fertilizer application. Contrast this amount with
  the expected value of the increase in her profit
  as a result of fertilizer application.

40
H Q , CH 5 , PROBLEMS

• Solution
• qtarget , P5 , C(q)3.5 0.5 q2 , U?
  1/2 , U(1/2) ?-1/2
• (v11/4 , d11.4) , (v21/2 , d21.6) ,
  (v31/4 , d31.88)
• EU(?)Si3 vi u( ?i ) Si3 vi u pdiq
  C(q) , ?i pdiq C(q)
• dEU(?)/dq Si3 vi u (?i) pdi C (q) 0
  
• qq0 , EU(?) Si3 vi u pdiq0 C(q0) U0
• no fertilizer situation
• pmc , q5 , ? pq TC 25 (3.5
  (0.5)(25) ) 9
• u (?)1/2 3
• U0 should be greater than 3 for the farmer to
  use fertilizer . The difference between these
  two (U0 3) is the farmers gain in
• terms of utility and maximum amount that he
  is willing to pay for fertilizer.
• d( E(?))/dq dv1 ?1 v2 ?2 v3
  ?31/4(5)(1.4q) - 3.5 0.5q2
• 1/2(5)(1.6q) - 3.5 0.5q2 1/4(5)(1.88q)
  - 3.5 0.5q2 /dq 0
• qA , ? E(?) ? , U U(?) , U can be
  compared with U0

41
H Q , CH 5 , PROBLEMS

• Q5-6 a linear production function contains
  four activities for the production of one output
  using two inputs. The input requirements per unit
  output are
• a111 a122 a133 a145
• a216 a225 a233 a242
• are any activities inefficient in the sense that
  there is no input price at which they would be
  used.

x2
J1
J2
Solution Comparing to point e point b is
not efficient. Since with the same amount of x1 ,
it uses less x 2 . In order to find the x1 and
x2 related to point e x1
q?a11(1-?)a13 , x2 q?a21(1-?)a23 x12
, q1 , a111 , a133 , ?1/2 x21(1/2)(6)
(1/2)(3)4.5
J3
a
b
6
e
5
4.5
J4
4
c
3
2
d
1
x1
1
2
3
4
5
6
42
H Q , CH 5 , PROBLEMS

• Q5-7 Each of the linear activities yields s
  outputs and uses m inputs as described by
• h outputs qh h1,2,3,4,..s
• m inputs xi i 1,2,3.m
• n activity j1,2,3. N
• qh Sj1n ahj zj xi Sj1n
  bij zj
• ahj the amount of output h in one unit of
  composite commodity in activity j .
• composite commodity basket of commodities
  contains from h outputs.
• Zj units of composite commodities produced in
  activity j
• bij input i used for the production of one
  unit of composite commodity in activity j .
• An entrepreneur possesses fixed quantities of
  each of the inputs. She desires to maximizes her
  total revenue from the sale of outputs at
  constant market prices. Formulate her
  optimization problem as a linear-programming
  system, and derive the dual programming system.

43
H Q , CH 5 , PROBLEMS

• Solution
• If ph is the price of qh , total income is equal
  to Y
• ahjamount of output h in one unit of composite
  commodity in activity j .
• composite commodity basket of commodities
  contains from h outputs.
• Zj units of composite commodities produced in
  activity j
• bij input i used for the production of one
  unit of composite commodity in activity j
• Y Shs ph qh Shs ph Sj1n ahj zj , so
  maximization problem is
• max Y Shs ph Sj1n ahj zj total revenue
• S.T. xi Sj1n bij zj input constraint
  (decision variable Zj).
• the dual formulation is
• Min CSim ri xi total cost
• S.T. Sim bij ri yj (yj Shs ph ahj revenue
  from one level of Zj , Zj 1 )
• Sim bij ricost of producing one level of Zj
  (decision variableri)
• Sim bij ri cost of production of one unit of
  composite commodity in activity j.
• yj ph Sj1n ahj Sj1n ph ahj income
  received from the sale of one unit of composite
  commodity in activity j .