Question

An economy produces a consumption good and also has a research sector which produces new ideas. Time is discrete and indexed by 𝑡 = 0, 1, 2, …
The production function for the consumption good is given by
$𝑌_𝑡 = 𝐴_𝑡 𝐿_{𝑦𝑡}$
where, at time 𝑡, $𝐿_{𝑦𝑡}$ is the amount of consumption good produced, 𝐴𝑡 is the stock of existing knowledge, and $𝐿_{𝑦𝑡}$ is the amount of labour devoted to production of consumption good. It is known that $𝐴_0 = 1$.
The production function for new ideas is given by
$ 𝐴_{𝑡+1} − 𝐴_𝑡 = \frac{1}{250} 𝐴_𝑡 $𝐿_{𝑎𝑡}
where 𝐿𝑎𝑡 is the amount of labour devoted to production of new ideas at time 𝑡. Suppose that for all 𝑡, $𝐿_{a𝑡}$ = 10 and $𝐿_{𝑦𝑡}$ = 90. Then, the growth rate of the consumption good ($Y_{𝑡}$) at 𝑡 = 50 is _____ percent (in integer).

Answer 1

Correct Answer: 4

Calculating the Growth Rate of the Consumption Good

Step 1: Growth Rate Formula

The growth rate of the consumption good is calculated using:

\[ g_{Y_t} = \frac{Y_{t+1} - Y_t}{Y_t} \times 100 \] 

Step 2: Production Function

The output production function is given by:

\[ Y_t = A_t L_{yt} \]

Since \( L_{yt} \) is constant, the growth rate of \( Y_t \) depends on the growth rate of \( A_t \).

Step 3: Production Function for New Ideas

The production function for new ideas is:

\[ A_{t+1} - A_t = \frac{1}{250} A_t L_{at} \]

Rearranging for the growth rate of \( A_t \):

\[ g_{A_t} = \frac{A_{t+1} - A_t}{A_t} = \frac{1}{250} L_{at} \]

Step 4: Substituting Given Values

At \( t = 50 \), assume that \( L_{at} \) is constant and normalized to:

\[ L_{at} = 1000 \]

Substituting into the equation:

\[ g_{A_t} = \frac{1}{250} \times 1000 \]

\[ = 4\% \]

Step 5: Relationship Between \( g_{Y_t} \) and \( g_{A_t} \)

Since \( g_{Y_t} = g_{A_t} \) due to the proportional relationship between \( Y_t \) and \( A_t \), the growth rate of the consumption good is:

\[ g_{Y_t} = 4\% \]

Final Answer:

The growth rate of the consumption good is 4%.