MICRO-Growth, Fluctuations and Innovation1. Explai

Growth, Fluctuations and Innovation 1. Explain what it means by steady state. 2. A production function is given as below where Y is output, K is capital and L is effective worker. a. Find steady state for capital per effective worker (k), output per effective worker (y) and consumption per effective worker (c) in term of parameters used in the model namely (saving rate), (population growth rate), (depreciation rate) and ( growth rate of efficient worker). b. Find the golden rule of k c. Find the level of saving rate in order to obtain the golden rule capital stock? Y = K  L 1−

Solution

a

Steady State: In the context of economic growth models, a steady state is a condition where key economic variables (such as capital per worker, output per worker, and consumption per worker) grow at a constant rate over time. This implies that the economy has reached a point where the growth rates of these variables are balanced by the rates of savings, population growth, depreciation, and technological progress

b

Steady State for Capital per Effective Worker (

k

): Given the production function

Y = K^\alpha L^{1-\alpha}

, we can express the steady state for capital per effective worker (

k

) as follows:

k^* = \left( \frac{s}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

where

s

is the saving rate,

n

is the population growth rate,

g

is the growth rate of effective workers, and

\delta

is the depreciation rate

c

Steady State for Output per Effective Worker (

y

): Using the production function and the steady state value of

k

, we can find the steady state for output per effective worker (

y

):

y^* = (k^*)^\alpha = \left( \frac{s}{n + g + \delta} \right)^{\frac{\alpha}{1-\alpha}}

d

Steady State for Consumption per Effective Worker (

c

): Consumption per effective worker in the steady state can be found by subtracting the steady state investment from the steady state output:

c^* = y^* - (n + g + \delta)k^*

c^* = \left( \frac{s}{n + g + \delta} \right)^{\frac{\alpha}{1-\alpha}} - (n + g + \delta) \left( \frac{s}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

e

Golden Rule Level of Capital (

k_{GR}

): The Golden Rule level of capital is the level of capital per effective worker that maximizes consumption per effective worker. It is found where the marginal product of capital equals the sum of the population growth rate, the growth rate of effective workers, and the depreciation rate:

MPK = \alpha k_{GR}^{\alpha-1} = n + g + \delta

k_{GR} = \left( \frac{\alpha}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

f

Saving Rate for Golden Rule Capital Stock: To find the saving rate that achieves the Golden Rule level of capital, we set the steady state capital per effective worker equal to the Golden Rule level of capital:

\left( \frac{s_{GR}}{n + g + \delta} \right)^{\frac{1}{1-\alpha}} = \left( \frac{\alpha}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

s_{GR} = \alpha

Answer

The steady state is a condition where key economic variables grow at a constant rate. The steady state values for capital per effective worker, output per effective worker, and consumption per effective worker are derived using the given parameters. The Golden Rule level of capital maximizes consumption per effective worker, and the saving rate to achieve this is equal to the output elasticity of capital (

\alpha

).

Key Concept

Steady State in Economic Growth Models

Explanation

The steady state is a balanced growth condition where the growth rates of key economic variables are constant, determined by the rates of savings, population growth, depreciation, and technological progress.

The recent health and economic crisis has reduced overall labor’s productivity. Based on WS-PS and PC-MR model, show the effect of this phenomenon on inflation and explain how the central bank responds to reach the medium-run equilibrium. Question 3 Using the 3-equation model of IS, PC and MR, differentiate the effects of temporary negative aggregate demand shock with those of permanent negative aggregate demand shock on the equilibrium interest rate and the central bank’s response to reaching the medium-run equilibrium. Question 4 COVID-19 pandemic has caused a drastic fall in autonomous investment in an economy. Based on the 3-equation model with the banking mark-up consideration, describe the impact of this phenomenon on the economy and explain how the central bank responds to reach the equilibrium. Question 5 A production function is given as below where Y is output, K is capital and L is effective worker. Suppose α is 0.7, growth rate of population (n) is 1%, depreciation rate of capital ( ) is 2% and growth rate of effective worker (g) is 3%. (a) If saving rate ( ) is 20%, find steady states for capital per effective worker (k), output per effective worker (y) and consumption per effective worker (c) in term of parameters used in the model namely α, , , and . [5 marks] (b) What is the golden rule for k? [3 marks] (c) What is the level of saving rate in order to obtain a golden rule capital stock? [2 marks] SELAMAT MAJU JAYA GOOD LUCK Y = K  L 1−

Sure, let's address each of your questions one by one. Question 1 Explain what it means by steady state.

Solution

a

Steady State: In the context of economic growth models, a steady state is a condition where key economic variables (such as capital per worker, output per worker, and consumption per worker) grow at a constant rate over time. This implies that the economy has reached a point where the growth rates of these variables are balanced by the rates of population growth, technological progress, and depreciation

Answer

A steady state is a condition where key economic variables grow at a constant rate over time, balanced by population growth, technological progress, and depreciation.

Key Concept

Steady State

Explanation

In a steady state, the economy's growth rates of capital, output, and consumption per worker are constant and balanced by other economic factors.

Question 2 A production function is given as below where Y is output, K is capital and L is effective worker. # (a) Find steady state for capital per effective worker (k), output per effective worker (y) and consumption per effective worker (c) in terms of parameters used in the model namely

\alpha

(saving rate),

n

(population growth rate),

\delta

(depreciation rate) and

g

(growth rate of efficient worker).

Solution

a

Steady State for Capital per Effective Worker (

k^*

): The steady state level of capital per effective worker is found by setting the change in capital per effective worker to zero. The equation is given by:

s \cdot f(k) = (n + g + \delta) \cdot k

Given the production function

Y = K^\alpha L^{1-\alpha}

, we have:

k^* = \left( \frac{s}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

Steady State for Output per Effective Worker (

y^*

):

y^* = f(k^*) = (k^*)^\alpha

Steady State for Consumption per Effective Worker (

c^*

):

c^* = y^* - s \cdot y^* = (1 - s) \cdot y^*

Answer

The steady states are: - Capital per effective worker:

k^* = \left( \frac{s}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

- Output per effective worker:

y^* = (k^*)^\alpha

- Consumption per effective worker:

c^* = (1 - s) \cdot y^*

Key Concept

Steady State in Solow Model

Explanation

The steady state levels of capital, output, and consumption per effective worker are derived by setting the change in capital per effective worker to zero and using the given production function.

# (b) Find the golden rule of

k

.

Solution

b

Golden Rule of Capital (

k_{GR}

): The golden rule level of capital is the level of capital per effective worker that maximizes consumption per effective worker. This is found by setting the marginal product of capital equal to the sum of the population growth rate, the growth rate of effective workers, and the depreciation rate:

MPK = n + g + \delta

Given the production function

Y = K^\alpha L^{1-\alpha}

, we have:

\alpha (k_{GR})^{\alpha-1} = n + g + \delta

Solving for

k_{GR}

:

k_{GR} = \left( \frac{\alpha}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

Answer

The golden rule level of capital per effective worker is

k_{GR} = \left( \frac{\alpha}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

.

Key Concept

Golden Rule of Capital

Explanation

The golden rule level of capital maximizes consumption per effective worker and is found by equating the marginal product of capital to the sum of the population growth rate, the growth rate of effective workers, and the depreciation rate.

# (c) Find the level of saving rate in order to obtain the golden rule capital stock.

Solution

c

Saving Rate for Golden Rule Capital Stock (

s_{GR}

): To find the saving rate that achieves the golden rule level of capital, we use the steady state condition for capital per effective worker:

s_{GR} \cdot f(k_{GR}) = (n + g + \delta) \cdot k_{GR}

Given

k_{GR} = \left( \frac{\alpha}{n + g + \delta} \right)^{\frac{1}{1-\alpha}}

and

f(k) = k^\alpha

, we have:

s_{GR} = \frac{(n + g + \delta) \cdot k_{GR}}{(k_{GR})^\alpha} = \alpha

Answer

The saving rate to obtain the golden rule capital stock is

s_{GR} = \alpha

.

Key Concept

Saving Rate for Golden Rule

Explanation

The saving rate that achieves the golden rule level of capital is equal to the output elasticity of capital (

\alpha

).

Question 3 Using the 3-equation model of IS, PC, and MR, differentiate the effects of temporary negative aggregate demand shock with those of permanent negative aggregate demand shock on the equilibrium interest rate and the central bank’s response to reaching the medium-run equilibrium.

Solution

a

Temporary Negative Aggregate Demand Shock: A temporary negative aggregate demand shock shifts the IS curve to the left, leading to a lower output and lower inflation in the short run. The central bank responds by lowering the interest rate to stimulate demand and return the economy to its potential output in the medium run

b

Permanent Negative Aggregate Demand Shock: A permanent negative aggregate demand shock also shifts the IS curve to the left, but the central bank's response is different. Since the shock is permanent, the central bank may not fully offset the shock with lower interest rates. Instead, the central bank may allow the economy to adjust to a new lower level of potential output, resulting in a lower equilibrium interest rate in the medium run

Answer

Temporary shocks lead to a short-term reduction in interest rates to stimulate demand, while permanent shocks result in a new lower equilibrium interest rate as the economy adjusts to a new potential output level.

Key Concept

Temporary vs. Permanent Aggregate Demand Shocks

Explanation

Temporary shocks are countered by lowering interest rates to restore demand, whereas permanent shocks lead to a new equilibrium with a lower interest rate and potential output.

Question 4 COVID-19 pandemic has caused a drastic fall in autonomous investment in an economy. Based on the 3-equation model with the banking mark-up consideration, describe the impact of this phenomenon on the economy and explain how the central bank responds to reach the equilibrium.

Solution

a

Impact of Fall in Autonomous Investment: A fall in autonomous investment shifts the IS curve to the left, leading to lower output and lower inflation. The banking mark-up consideration implies that the cost of borrowing increases, further reducing investment and aggregate demand

b

Central Bank Response: The central bank responds by lowering the policy interest rate to reduce the cost of borrowing and stimulate investment and consumption. This helps to shift the IS curve back to the right, aiming to restore the economy to its potential output and stabilize inflation

Answer

The fall in autonomous investment reduces output and inflation, and the central bank lowers interest rates to stimulate demand and restore equilibrium.

Key Concept

Impact of Investment Fall and Central Bank Response

Explanation

A fall in investment reduces demand, and the central bank lowers interest rates to counteract this and stabilize the economy.

Question 5 A production function is given as below where Y is output, K is capital and L is effective worker. Suppose $\alpha$ is 0.7, growth rate of population (n) is 1%, depreciation rate of capital ($\delta$) is 2% and growth rate of effective worker (g) is 3%. # (a) If saving rate (

s

) is 20%, find steady states for capital per effective worker (

k

), output per effective worker (

y

) and consumption per effective worker (

c

) in terms of parameters used in the model namely

\alpha

,

n

,

\delta

, and

g

.

Solution

a

Steady State for Capital per Effective Worker (

k^*

):

k^* = \left( \frac{s}{n + g + \delta} \right)^{\frac{1}{1-\alpha}} = \left( \frac{0.2}{0.01 + 0.03 + 0.02} \right)^{\frac{1}{1-0.7}} = \left( \frac{0.2}{0.06} \right)^{\frac{1}{0.3}} = \left( \frac{1}{0.3} \right)^{\frac{1}{0.3}} \approx 2.154

Steady State for Output per Effective Worker (

y^*

):

y^* = (k^*)^\alpha = (2.154)^{0.7} \approx 1.515

Steady State for Consumption per Effective Worker (

c^*

):

c^* = (1 - s) \cdot y^* = (1 - 0.2) \cdot 1.515 \approx 1.212

Answer

The steady states are: - Capital per effective worker:

k^* \approx 2.154

- Output per effective worker:

y^* \approx 1.515

- Consumption per effective worker:

c^* \approx 1.212

Key Concept

Steady State Calculations

Explanation

The steady state levels are calculated using the given parameters and the production function.

# (b) What is the golden rule for

k

?

Solution

b

Golden Rule of Capital (

k_{GR}

):

k_{GR} = \left( \frac{\alpha}{n + g + \delta} \right)^{\frac{1}{1-\alpha}} = \left( \frac{0.7}{0.01 + 0.03 + 0.02} \right)^{\frac{1}{1-0.7}} = \left( \frac{0.7}{0.06} \right)^{\frac{1}{0.3}} = \left( \frac{7}{0.6} \right)^{\frac{1}{0.3}} \approx 4.641

Answer

The golden rule level of capital per effective worker is

k_{GR} \approx 4.641

.

Key Concept

Golden Rule of Capital

Explanation

The golden rule level of capital maximizes consumption per effective worker and is derived using the marginal product of capital.

# (c) What is the level of saving rate in order to obtain the golden rule capital stock?

Solution

c

Saving Rate for Golden Rule Capital Stock (

s_{GR}

):

s_{GR} = \alpha = 0.7

Answer

The saving rate to obtain the golden rule capital stock is

s_{GR} = 0.7

.

Key Concept

Saving Rate for Golden Rule

Explanation

The saving rate that achieves the golden rule level of capital is equal to the output elasticity of capital (

\alpha

).

AskSia

Plus