Question

Let the production function be given by
$ Y_t = A_t K_t ^{\alpha} H_t ^{\beta} 𝐿_t^{1−\alpha−\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

• A. 2 percent
• B. 3 percent
• C. 5 percent
• D. 10 percent

Answer 1

The Correct Option is B

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

Step 1: Cobb-Douglas Growth Rate Formula

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

g_Yt = g_At + αg_Kt + βg_Ht + (1 − α − β)g_Lt

Step 2: Substituting Given Values

We are given:

• g_Yt = 10%
• α = 1/5, β = 2/5
• g_Kt = 5%, g_Ht = 5%, g_Lt = 10%

Substituting into the formula:

10% = g_At + (1/5 × 5%) + (2/5 × 5%) + (2/5 × 10%)

Step 3: Simplifying

• 1/5 × 5% = 1%
• 2/5 × 5% = 2%
• 2/5 × 10% = 4%

Thus, we get:

10% = g_At + 1% + 2% + 4%

g_At = 10% − 7% = 3%

Final Answer:

The growth rate of A_t is 3%.