Question

A production function at time 𝑡 is given by
\(Y_t=A_tK^a_tL^{1-a}_t\) , 𝛼 ∈ (0, 1), 𝛼 ≠ 0.5, 
where 𝑌 is output, 𝐾 is capital, 𝐿 is labour and 𝐴 is the level of Total Factor Productivity. Define per capita output as \(𝑦_𝑡 ≡ \frac{𝑌_𝑡}{ 𝐿_𝑡} \)and capital-output ratio as \(𝑘_𝑡 ≡\frac{ 𝐾_𝑡}{ 𝑌_𝑡}\) . For any variable 𝑥𝑡 , denote \(\frac{𝑑𝑥_𝑡}{ 𝑑𝑡}\) by 𝑥̇ . The per capita output growth rate is

• A. \(𝛼\)\(\frac{y}{y}=\frac{1}{(1-α)}\frac{A}{A}+\frac{α}{(1-α)}\frac{k}{k}\)
• B. \(𝛼\)\(\frac{y}{y}=\frac{α}{(1-α)}\frac{A}{A}+\frac{1}{(1-α)}\frac{k}{k}\)
• C. \(𝛼\)\(\frac{y}{y}=(1-α)\frac{A}{A}+α\frac{k}{k}\)
• D. \(𝛼\)\(\frac{y}{y}=α\frac{A}{A}+(1-α)\frac{k}{k}\)

Answer 1

The Correct Option is A

Given:
\[ Y_t = A_t K_t^{\alpha} L_t^{1-\alpha},\qquad \alpha\in(0,1),\ \alpha\neq 0.5, \] per–capita output \(y_t\equiv Y_t/L_t\). Also the capital–output ratio is defined here as \[ k_t \equiv \frac{K_t}{Y_t}. \] You asked for the growth rate \(\dfrac{\dot y_t}{y_t}\). Below are two commonly used, equivalent expressions and short derivations. 

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1) Expression in terms of \(A,K,L\) (standard form) Start with the per–worker form: \[ y_t=\frac{Y_t}{L_t}=A_t\left(\frac{K_t}{L_t}\right)^{\alpha}. \] Take logarithms and differentiate: \[ \ln y_t = \ln A_t + \alpha\ln\!\left(\frac{K_t}{L_t}\right) = \ln A_t + \alpha\big(\ln K_t - \ln L_t\big). \] Differentiate w.r.t. time: \[ \frac{\dot y_t}{y_t}=\frac{\dot A_t}{A_t}+\alpha\!\left(\frac{\dot K_t}{K_t}-\frac{\dot L_t}{L_t}\right). \] This is the standard decomposition: growth of per-capita output = TFP growth + α × growth of capital per worker. 

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2) Equivalent expression in terms of the capital–output ratio \(k_t=K_t/Y_t\) From the definitions, \[ k_t=\frac{K_t}{Y_t}\quad\Rightarrow\quad \frac{K_t}{L_t}=k_t\frac{Y_t}{L_t}=k_t y_t. \] Substitute into \(y_t=A_t(K_t/L_t)^{\alpha}\): \[ y_t = A_t (k_t y_t)^{\alpha} = A_t\,k_t^{\alpha}\,y_t^{\alpha}. \] Rearrange: \[ y_t^{1-\alpha}=A_t\,k_t^{\alpha}. \] Take logs and differentiate: \[ (1-\alpha)\frac{\dot y_t}{y_t}=\frac{\dot A_t}{A_t}+\alpha\frac{\dot k_t}{k_t}. \] Hence \[ \boxed{\displaystyle \frac{\dot y_t}{y_t}=\frac{1}{1-\alpha}\,\frac{\dot A_t}{A_t}+\frac{\alpha}{1-\alpha}\,\frac{\dot k_t}{k_t}. } \]  

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Summary (both equivalent): \[ \boxed{\frac{\dot y_t}{y_t}=\frac{\dot A_t}{A_t}+\alpha\Big(\frac{\dot K_t}{K_t}-\frac{\dot L_t}{L_t}\Big) \quad\text{(or)}\quad \frac{\dot y_t}{y_t}=\frac{1}{1-\alpha}\frac{\dot A_t}{A_t}+\frac{\alpha}{1-\alpha}\frac{\dot k_t}{k_t} } \] Use the first form when you have growth rates of \(K\) and \(L\); use the second when you have the capital–output ratio \(k\).