Question:

Let the production function be given by
$ Y_t = A_t K_t ^{\alpha} H_t ^{\beta} 𝐿_t^{1βˆ’\alphaβˆ’\beta}$
where, at time t, $Y_t$ is output, $A_t$ is level of Total Factor Productivity, $K_t$ is physical capital, H is human capital, and L is labour. $\alpha$ = 1/5 and $\beta$ = 2/5 If the growth rate of $Y_t$ equals 10 percent, the growth rate of $K_t$ equals 5 percent, the growth rate of $H_t$ equals 5 percent, and the growth rate of $L_t$ equals 10 percent, then the growth rate of $A_t$ is

Updated On: Feb 10, 2025
  • 2 percent
  • 3 percent
  • 5 percent
  • 10 percent
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The Correct Option is B

Solution and Explanation

Calculating the Growth Rate of At in a Cobb-Douglas Production Function

Step 1: Cobb-Douglas Growth Rate Formula

The growth rate of output (gYt) in a Cobb-Douglas production function is given by: 

gYt = gAt + Ξ±gKt + Ξ²gHt + (1 βˆ’ Ξ± βˆ’ Ξ²)gLt

Step 2: Substituting Given Values

We are given:

  • gYt = 10%
  • Ξ± = 1/5, Ξ² = 2/5
  • gKt = 5%, gHt = 5%, gLt = 10%

Substituting into the formula:

10% = gAt + (1/5 Γ— 5%) + (2/5 Γ— 5%) + (2/5 Γ— 10%)

Step 3: Simplifying

  • 1/5 Γ— 5% = 1%
  • 2/5 Γ— 5% = 2%
  • 2/5 Γ— 10% = 4%

Thus, we get:

10% = gAt + 1% + 2% + 4%

gAt = 10% βˆ’ 7% = 3%

Final Answer:

The growth rate of At is 3%.

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