Ka'idar Uzawa

Uzawa's theorem, wanda aka fi sani da tsayayyen ci gaban jihar, wani tsari ne a cikin Ci gaban tattalin arziki wanda ke gano nau'in aiki na Canjin fasaha don cimma hanyar ci gaba mai daidaituwa a cikin tsarin ci gaban Solow-Swan da Ramsey-Cass-Koopmans. Masanin tattalin arziki na Japan Hirofumi Uzawa ne ya tabbatar da shi a 1961.

Wani nau'i na gaba ɗaya na ka'idar ya ƙunshi sassa biyu. Na farko ya bayyana cewa, a karkashin zato na yau da kullun na samfurin Solow-Swan da Ramsey, idan babban birnin, saka hannun jari, amfani, da fitarwa suna ƙaruwa a ƙimar ƙimar ƙima, waɗannan ƙimar dole ne su zama daidai. Sashe na biyu ya tabbatar da cewa, a cikin irin wannan hanyar ci gaba mai daidaituwa, aikin samarwa, 

Y = 

F ~ 

(Abin da ya fi dacewa da shi)  

A ~ 

, K , L)  

{\displaystyle Y={\tilde {F}} ({\tilda {A}},K,L) } 

(A ina 

A 

{\displaystyle A} 

fasaha ce, 

K 

{\displaystyle K} 

babban birni ne, kuma 

L 

{\displaystyle L} 

aiki ne), ana iya sake rubuta shi ta yadda canjin fasaha ke shafar fitarwa kawai a matsayin ma'auni a kan aiki (watau 

Y = F (K , A L)  

{\displaystyle Y=F (K,AL) } 

) dukiyar da aka sani da karuwar aiki ko canjin fasaha na Harrod. 

Ka'idar Uzawa ta nuna iyakancewar samfuran Solow-Swan da Ramsey. Daidaita zaton daidaitattun ci gaba a cikin irin waɗannan samfuran yana buƙatar cewa canjin fasaha ya zama karuwar aiki. Sabanin haka, aikin samarwa wanda ba zai iya wakiltar tasirin fasaha ba a matsayin karuwar ma'aikata ba zai iya samar da hanyar daidaitaccen ci gaba ba.

## Bayani
A cikin wannan shafin, maɓallin a kan mai canji zai nuna abin da aka samo game da lokaci (watau X ̇ (t) Yanayi d X (t) d t {\displaystyle {\dot {X}} (t) \equiv {dX (t) \over dt}} Rashin amfani da shi Har ila yau, yawan ci gaban mai canzawa X (t) {\displaystyle X (t) } za a nuna shi g X Sanya X ̇ (t) X (t) {\displaystyle g_{X}\equiv {\frac  (t) }{X (t) }}} .

**Ka'idar Uzawa**

Ana samun wannan sigar a cikin Acemoglu (2009) kuma an daidaita ta daga Schlicht (2006):

Misali tare da aikin samarwa 

Y (t) = 

F ~ 

(Abin da ya fi dacewa da shi)  

A ~ 

(t), K (t), L (t))  

{\displaystyle Y (t) ={\tilde {F}} ({\til de {A}} (t),K (t),L (t)) } 

, inda 

F ~ 

: 

R 

+ 

2 

× 

A 

→ 

R 

+ 

{\displaystyle {\tilde {F}}:\mathbb {R} _{+}^{2}\times {\mathcal {A}}\to \mathbb {S} _{&}} 

da kuma 

A ~ 

(t) ∈ 

A 

{\displaystyle {\tilde {A}} (t) \in {\mathcal {A}}} 

yana wakiltar fasaha a lokacin t (inda 

A 

{\displaystyle {\mathcal {A}}} 

wani sashi ne na son rai na 

R 

N 

{\displaystyle \mathbb {R} ^{N}} 

ga wasu lambobi na halitta 

N 

{\displaystyle N} 

Rashin amfani da shi Ka ɗauka cewa 

F ~ 

{\displaystyle {\tilde {F}}} 

yana nuna dawowar yau da kullun zuwa sikelin a 

K 

{\displaystyle K} 

da kuma 

L 

{\displaystyle L} 

. Girman babban birnin a lokacin t an ba shi ta hanyar

\dot{K}(t)=Y(t)-C(t)-\delta K(t)

inda 

δ 

{\displaystyle \delta } 

shine raguwar farashin kuma 

C (t)  

{\displaystyle C (t) } 

shine amfani a lokacin t.

Ka yi la'akari da cewa yawan jama'a yana ƙaruwa a hankali. 

L (t) = bayyana (n t) L (0)  

{\displaystyle L (t) =\exp (nt) L (0) } 

, kuma cewa akwai wani lokaci 

T 
0 

{\displaystyle {\dot {Y}} (t) /Y (t) =g_{Y}>0} 

, 

K ̇ 

(t)  

/ 

K (t) = 

g 

K 

>
0 

{\displaystyle {\dot {K}} (t) /K (t) =g_{K}>0} 

, da kuma 

C ̇ 

(t)  

/ 

C (t) = 

g 

C 

>
0 

{\displaystyle {\dot {C}} (t) /C (t) =g_{C}>0} 

. Sa'an nan kuma

1.  

g 

Y 

= 

g 

K 

= 

g 

C 

{\displaystyle g_{Y}=g_{K}=g _{C}} 

; da kuma

2. Akwai aiki 

F: 

R 

+ 

2 

→ 

R 

+ 

{\displaystyle F:\mathbb {R} _{+}^{2}\to \mathbb {B} _{&}} 

wannan yana da daidaituwa na digiri 1 a cikin muhawara guda biyu kamar haka, ga kowane 

t ≥ T 

{\displaystyle t\geq T} 

, ana iya wakiltar aikin samar da duka kamar yadda 

Y (t) = F (K (t), A (t) L (t))  

{\displaystyle Y (t) =F (K (t),A (t) L (t)) } 

, inda 

A (t) ∈ 

R 

+ 

{\displaystyle A (t) \in \mathbb {R} _{+}} 

da kuma 

GS ta shafi 

A ̇ 

(t)  

/ 

A (t) = 

g 

Y 

- n 

{\displaystyle g\equiv {\dot {A}} (t) /A (t) =g_{Y}-n} 

.

## Shafin hujja

### Lemma 1
**Tabbacin**: Lura cewa ga kowane 

Z (t)  

{\displaystyle Z (t) } 

, 

g 

Z 

= 

Z ̇ 

(t)  

Z (t)  

= 

d a cikin Z (t)  

d t 

{\displaystyle g_{Z}={\frac  (t) }{Z (t) }}={\ frac {d\ln Z (t) }{dt}}} 

. Saboda haka, 

g 

X 

α 

Y 

= 

d 

d t 

Ci (X (t)  

)  

α 

Y (t) ] = α 

d Xi X (t)  

d t 

+ 

d a cikin Y (t)  

d t 

= α 

g 

X 

+ 

g 

Y 

{\displaystyle g_{X^{\alpha }Y}={\frac {d}{dt}}\ln[ (X (t)) ^{\alfa }Y (t) ]=\alpha {\frac { d\ln X (t) }{dt}}+{\fraC {d\ln Y (t) }{dt}},=\alfa g_{Y}} 

.

### Tabbacin ka'idar
Mun fara nuna cewa ci gaban saka hannun jari 

I (t) = Y (t) − C (t)  

{\displaystyle I (t) =Y (t) -C (t) } 

dole ne ya daidaita yawan ci gaban babban birnin 

K (t)  

{\displaystyle K (t) } 

(watau 

g 

Na 

= 

g 

K 

{\displaystyle g_{I}=g_{K}} 

) 

* Kyakkyawan aiki
* Lissafin ci gaba

## Manazarta