Math Problem Statement

Suppose that the production function f(k) = 2k and A =1 so that the growth of the labor-augmenting technology (A/A) = 0. What is the growth rate of (k.)/(k)?

Suppose that the production function f(k) = 2k and A = 1 so that the growth of the labor-augmenting technology (A/A) = 0. What is the growth rate of (k.)/(k)?

Solution

To find the growth rate of k˙k\frac{\dot{k}}{k}, let's break down the equation given in the image and use the information provided.

Given Information

The equation for capital accumulation per effective worker k^\hat{k} is: k^˙=sf(k^)(n+δ+g)k^\dot{\hat{k}} = s f(\hat{k}) - (n + \delta + g) \hat{k}

where:

  • f(k^)=k^0.5f(\hat{k}) = \hat{k}^{0.5},
  • s=0.20s = 0.20,
  • n=0.02n = 0.02,
  • δ=0.10\delta = 0.10,
  • g=0.08g = 0.08.

However, you've specified a modified production function: f(k)=2kf(k) = 2k

with A=1A = 1 and a zero growth rate for labor-augmenting technology (A˙A=0)\left( \frac{\dot{A}}{A} = 0 \right).

This simplifies the growth rate analysis because we’re dealing with kk and not k^\hat{k}.

Steps to Solve for k˙k\frac{\dot{k}}{k}

Given the production function f(k)=2kf(k) = 2k, we can substitute this into the capital accumulation equation without considering the adjustments for effective labor (since AA is constant and does not grow). Therefore, the equation for k˙\dot{k} becomes:

k˙=sf(k)(n+δ)k\dot{k} = s f(k) - (n + \delta) k

Substitute f(k)=2kf(k) = 2k: k˙=s2k(n+δ)k\dot{k} = s \cdot 2k - (n + \delta) k

Expanding: k˙=(2snδ)k\dot{k} = (2s - n - \delta) k

Now, divide both sides by kk to find k˙k\frac{\dot{k}}{k}: k˙k=2snδ\frac{\dot{k}}{k} = 2s - n - \delta

Substitute the Values

Using the given values:

  • s=0.20s = 0.20,
  • n=0.02n = 0.02,
  • δ=0.10\delta = 0.10.

k˙k=20.200.020.10\frac{\dot{k}}{k} = 2 \cdot 0.20 - 0.02 - 0.10

Calculate each term:

  1. 20.20=0.402 \cdot 0.20 = 0.40,
  2. 0.400.02=0.380.40 - 0.02 = 0.38,
  3. 0.380.10=0.280.38 - 0.10 = 0.28.

Final Answer

k˙k=0.28\frac{\dot{k}}{k} = 0.28

The growth rate of k˙k\frac{\dot{k}}{k} is 0.28 or 28%.

Would you like additional details on this calculation or further analysis?

Related Questions:

  1. How does the growth rate change if the depreciation rate δ\delta changes?
  2. What impact does the savings rate ss have on k˙k\frac{\dot{k}}{k}?
  3. How would the solution differ if AA were growing at a positive rate?
  4. What does this growth rate imply for long-term capital accumulation?
  5. How would labor growth nn affect the growth rate k˙k\frac{\dot{k}}{k}?

Tip

Understanding how each parameter affects the growth rate can help in economic modeling, especially in long-term growth predictions.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Economic Growth Models
Differential Equations
Capital Accumulation
Steady-State Analysis

Formulas

k̂(t) = s f(k̂) - (n + δ + g) k̂
f(k) = 2k
Growth rate: (k̇ / k) = 2s - n - δ

Theorems

Solow Growth Model
Capital Accumulation Equation

Suitable Grade Level

Undergraduate (Economics or Mathematics)