MACRO-1. Consider an overlapping generations model

Solution

a

Steady State Equilibrium: The steady state equilibrium in an overlapping generations model with production is characterized by the condition where capital per unit of effective labor,

k

, remains constant over time. This occurs when the net investment in capital equals the depreciation of capital. Mathematically, this can be expressed as:

k_{t+1} = \frac{(1-\delta)k_t + s f(k_t)}{n a}

where

s

is the savings rate,

f(k_t)

is the production function per unit of effective labor,

n

is the population growth rate, and

a

is the productivity growth rate. In the steady state,

k_{t+1} = k_t = k^{ss}

, leading to:

k^{ss} = \frac{(1-\delta)k^{ss} + s f(k^{ss})}{n a}

Aggregate consumption per unit of effective labor,

c_t

, is given by:

c_t = f(k_t) - (n a - 1 + \delta)k_t

If k_1 = k^{gr} < k^{ss}, then

k_t

will converge to

k^{ss}

over time. Since

k^{gr}

is less than

k^{ss}

, aggregate consumption per unit of effective labor will initially rise as

k_t

increases towards

k^{ss}

b

Immediate Effect of Earthquake: If an earthquake destroys half of the capital stock at

t=2

, the new capital stock

k_2

will be:

k_2 = \frac{1}{2} k^{ss}

Time Path of Economy: The economy will experience a sudden drop in capital per unit of effective labor. The production function will shift downwards, leading to a decrease in output and consumption. Over time, the economy will gradually rebuild its capital stock, moving back towards the steady state equilibrium

k^{ss}

. The time path can be illustrated with a diagram showing the initial drop in

k_t

and its gradual recovery. Long Term Consequences: In the long term, the economy will return to its steady state equilibrium

k^{ss}

. However, the temporary loss of capital will result in lower output and consumption in the short run. The speed of recovery will depend on the savings rate and the productivity growth rate

Answer

The steady state equilibrium is characterized by constant capital per unit of effective labor, and aggregate consumption per unit of effective labor will rise if k_1 = k^{gr} < k^{ss}. An earthquake that destroys half of the capital stock will cause an immediate drop in

k_2

, followed by a gradual recovery towards the steady state equilibrium.

Key Concept

Steady State Equilibrium

Explanation

The steady state equilibrium occurs when capital per unit of effective labor remains constant, and aggregate consumption per unit of effective labor will rise if the initial capital is below the steady state level. An earthquake causing a sudden loss of capital will lead to a temporary drop in output and consumption, followed by a recovery towards the steady state.

Solution

a

Definition of Elasticity: Elasticity of output per unit of effective labor,

y^*

, with respect to the rate of population growth,

n

, measures the responsiveness of

y^*

to changes in

n

b

Steady State Condition: In the steady state, the capital per unit of effective labor,

k^*

, is determined by the equation

s f(k^*) = (n + g + \delta) k^*

, where

s

is the savings rate,

f(k^*)

is the production function,

g

is the growth rate of technology, and

\delta

is the depreciation rate

c

Production Function: Given

\alpha_K(k^*) = \frac{1}{3}

, we assume a Cobb-Douglas production function

f(k) = k^{\alpha}

, where

\alpha = \frac{1}{3}

d

Output per Unit of Effective Labor: The output per unit of effective labor,

y^*

, is given by

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

e

Capital per Unit of Effective Labor: From the steady state condition,

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

f

Elasticity Calculation: The elasticity of

y^*

with respect to

n

is given by

\frac{\partial y^*}{\partial n} \cdot \frac{n}{y^*}

. Using the chain rule and the expressions for

y^*

and

k^*

, we get

\frac{\partial y^*}{\partial n} = \alpha (k^*)^{\alpha - 1} \cdot \frac{\partial k^*}{\partial n}

g

Derivative of

k^*

:

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

h

Simplifying: Substituting

\alpha = \frac{1}{3}

,

g = 0.02

, and

\delta = 0.03

, we find the elasticity

\frac{\partial y^*}{\partial n} \cdot \frac{n}{y^*} = -\frac{1}{3} \cdot \frac{n}{n + g + \delta}

i

Impact of Change in

n

: For

n

falling from 0.02 to 0.01, the change in

y^*

can be approximated by

\Delta y^* \approx -\frac{1}{3} \cdot \frac{\Delta n}{n + g + \delta} \cdot y^*

Answer

A fall in

n

from 2% to 1% would raise

y^*

by approximately 0.5%.

Key Concept

Elasticity of output per unit of effective labor with respect to the rate of population growth

Explanation

The elasticity measures how sensitive the output per unit of effective labor is to changes in the population growth rate. In this case, a decrease in the population growth rate leads to an increase in the output per unit of effective labor.

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