, 1

4

..., .. . ..

4.

: : . . , .

4. 1. : .

1.1. : , . :

1.1. : (R) (Q):: . 4.1.

1.1. :

, (R)

, (Q)

1.2. ( ) ( ) : Q = F(R1,R2,Rm), Rj j

1.2. , , : R(Q) = F-1

1.2. - - : Q(R1, R2, , Rm) = A R1a1 R2a2 Rmam,

: ;Rj j; aj , j .

1.2. -

:Q(R1, R2,, Rm)= min {a1R1, a2R2, amRm},

: Rj j; aj j.

4. 2. . ,

2.1. j- , (constant) , : Rk = const, if k j

2.1.

, , (variable) :Rj = Fj-1(Q)

2.1. : j- TPj (total product) , j- j-

2.1. : TPJ , ( ) TPj > 0

2.1. : j- APj (average product) , j-

2.1. : j :APj (Rj) = TPj / Rj = Q(Rj,R^) / Rj > 0

, , () APj

2.1. : j- MPj (marginal product) , j-

2.1. : j :MPj (Rj) = TPj / Rj = Q(Rj,R^) / Rj

:MPj (Rj) = TPj (Rj) TPj (Rj-1) = = Q(Rj,R^) Q(Rj-1,R^)

2.1. : , , () MPj

2.2. , -: Q = F(L,K), : K=K^= const : (K) ; (L) : Q = F(L,K^) = F(L)

LLTPLAPLMPLQQ L* L0 L** L** L* L01234 . 4.2. , (L)

tg = AP(L0) = MP (L0) (I)1

2.2. , (I): tg = AP(L0) = MP (L0) : (L0, TPL(L0)) TPL(L0) / L0 = APL(L0) = tg (L0, TPL(L0)) [TPL(L0)] / L = MPL(L0) = tg APL(L0) = MPL(L0) = tg

2.2. , (II): : APL(L0) = MPL(L0)

2.2. , (II):APL = (L) L0, F.O.C.: APL(L0) / L = 0APL / L = (TPL / L) / L = = (TPL / L)L-1 + TPL[(L-1) / L] = = MPLL-1 TPLL-2 = L-1[ MPL APL ] = 0 F.O.C. , MPL = APL

2.2. , ():L* L L** , : MPL 0; MPL / L < 0

2.2. , 1234

2.2. , = :VMPL = wVMPL : VMPL = pQMPL

2.2. , :

() :VMPL = pQMPL; MPL(L,K^) = w / pQ; L = (MPL-1) pQ / w

LDL = MPL-1MPLQ L*= L3 L0 L**234. 4.3. 1(w/pQ)4(w/pQ)3(w/pQ)2(w/pQ)1 L1 L22.2. , :

2.2. , : , :w ( )K^ - ()MPL - PQ -

2.2. , : ( )

4.

3. .

3.1.

3.1. ( ):max Q (R1, R2, Rm )TC0 pR1 R1 pR2 R2 pRm Rm = 0 (1.)Rj 0, j=1,m

3.1. ( ):min (pR1 R1+ pR2 R2 + + pRm Rm) Q* - Q (R1, R2, Rm ) = 0 (2.) Rj 0, j=1,m

3.1. : () ( )

3.1. (m+1) : (m+1)

3.1. : L(R1, R2, Rm,) max (1.*) , (TC0)

3.1. :(R1, R2, Rm ,) min (2.*)

3.1. (T): max [ T(Q (R1, R2, Rm )] (3.) Rj 0

3.1. ( ) (3.) :T(Q (R1, R2, Rm)) = = TR(Q(R1, R2, Rm)) TC(Q(R1, R2, Rm )) = = pQ Q (R1,R2, Rm ) p1R1 p2R2 pmRm

3.1. (3.) m , ( F.O.C.) T(Q (R1,R2,Rm)) Rj

3.1. (3.): T(Q()) / R1 = pQ [Q() / R1] p1 = 0 T(Q()) / R2 = pQ [Q() / R2] p2 = 0 T(Q()) / Rm = pQ [Q() / Rm] pm = 0

3.1. (m) (m) , R* = (R1*, R2*, Rm*)

[(1.), (2.), (3.)] : R* = (R1*, R2*, Rm*)

3.2. , , (Rj>0) : MP1(R*) / PR1 = MP2(R*) / PR2 = = MPm(R*) / PRm = = 1/

, (Rk 0), Rk* = 0, : MPk(R*) / PRk < ( )

4. 4.

4. , (L) (K) I : II : ( )III :

4.1. , , , (L,K, Q(L,K))

4.1. ( ) , ()

4.1. LKQQQ. 4.4. (Qa, Qb, Qc)

4.1. () , ,

4.1. : , , , -

4.1. : , :

4.1. : LKQQQ. 4.5. ,

4.1. LKQ1Q2 . 4.6. - Q(L,K) = aLK

4.1. LKQ1Q2. 4. 7. Q(L,K) = aL + bKtg = a/b

4.1. LKQ1L*L**K**K*. 4.8. Q1:L* L L**K* K K**

4.1. LKQ1Q2 . 4.9.

Q = min {aL, bK}tg = a / b

4.1. LKQ1T1T2Q2. 4.10. :

T1: Q = min {aL, bK}T2: Q = min {cL, dK}Q^ = T1 + (1-)T2

tg = a/btg = c/d

4.1. : : , ( )

4.1. : MRTS

- ( ) MRTSji

4.1. : MRTSMRTSji , j i :MRTSji = Rj / Ri < 0 Q=const

: MRTSKL = K / L

4.1. : MRTSKL : dQ = 0 (1) . : dQ = dQL + dQK= = L MPL + K MPK = 0 (2)

4.1. : MRTSKL (2), , : MRTSKL = MPL / MPK (3)

|MRTSKL| = (L,K), : /L < 0, /K > 0: MRTSKL

4.1. : MRTSKL LQKAB . 4.11.

MRTSKLA = tg MRTSKLB = tg

4.2. , (TC0)

..,

4.2.

( ) :TC0 p1RR1 p2RR2 pmRRm 0

4.2. , :TC0 wL rK 0, : w ( )r ( )

4.2. :K = TC0 / r (w/r)L ()

4.2. LTC0/w K

TC0/r

. 4.12.

tg = w / r = MRSEKL

4.2. ( ) MRSEKL : , MRSEKL = tKL = - w /r = tg

4.3. : LKQ3Q2Q1GEMATC/wTC/r B

. 4.13. () ( )

4.3. : LKQ*EATC0/wTC0/r . 4.14. () ( )

BCG

4.

5.

5.1. , () . , ( , )

5.1.

, , Q** = F(L**, K^)

5.1. K^ : L = L0, L0 - , APL

5.1. ( , ) ,

5.1. L L** K K^ E1 A2 = E2 E3 A1 A3 Q1 Q2 Q3 . 4.15. ( -)

tg = w / rQ** A**

5.1. L L** K K^ E1 A2 = E2 E3 A1 Q1Q2=Q** Q3 . 4.16. ( )

tg = - w / rtg = MPL/ MPK

5.1. : (K* < K^) (K** > K^)

5.2. , ,

5.2.

: :

5.2.

( , ) ,

5.2.

, , MRTSKL (. 4.17.)

5.2. L K A2 A1 3 Q1 Q2 Q3 . 4.17.

tg = MRTSA tg = MRTSB 2 1 A3

5.2. LKQ2Q1E1E2

. 4.18.

tg = w / r = MRTSKLE

5.2. : MRSEKL = - w / r = tg

, k, , , Ek : RTSKL = tg = MRSEKL

5.2. - : Q = ALK, tg = K / L = (w) / (r)

5.3. : K*: L* = (w) : (r) = const () , ( - )

5.3. () , ( - )

5.3. () , : ()

5.3. , , k :Q = F(Rm) = k F(Rm), : k = ; Rm = (R1,R2,Rm)

5.3. () :

5.3. (1.) () (IRS - Increasing Returns to Scale )

, : > ; k > 1

5.3. (2.) () (CRS - Constant Returns to Scale )

, : = ; k = 1

5.3. (3.) () (DRS - Decreasing Returns to Scale )

, : < ; k < 1

5.3. ( k ) : k,

5.3. L K Q1Q2= 2 Q1 Q3 = 3 Q1 . 4.19. k (k > 1 IRS)

L1L2L3 K1K2K3L2 < 2L1L3 < 3L1

K2 < 2K1K3 < 3K1

5.3. L K Q1Q2= 2 Q1 Q3 = 3 Q1 . 4.20 k (k = 1 CRS)

L1L2L3 K1K2K3L2 = 2L1L3 = 3L1

K2 = 2K1K3 = 3K1

5.3. L K Q1Q2= 2 Q1 Q3 = 3 Q1 . 4.21. k (k < 1 DRS)

L1L2>2L1L3>3L1 K1K2>2K1K3>3K1

5.4. : ( )

5.4. (SE) , : ,

5.4. (OE) , : ( ) : pR > 0 OE < 0; pR < 0 OE > 0

5.4.

5.4. LTC0/w2

TC0/w1

TC0/rQkQmE2E1 SEKE3 SELOEKK. 4.22. 1 2tg = - w1/r tg = - w2/r OEL

5.5. ,

5.5. : , ,

5.5. : : (. . 4.23. 4.24.) ( )

5.5. LKQ2Q1

. 4. 23 . K*/ L* = const = tg Q^1Q^2MRTSKL = const

5.5. LKQ2Q1

. 4.24. () K*/ L* = const = tg Q^1Q^2|MRTSKL^| < |MRTSKL|

5.5. LKQ2Q1

. 4.25. () K / L = const = tg Q^1Q^2|MRTSKL^| > |MRTSKL|

5.5. :(1) (2) ,