MACRO-explain the answer in more details on questi

explain the answer in more details on question a to e

Solution

a

To solve for

p_t

in terms of

m_t

and

E_t p_{t+1}

, start with the given money demand equation:

m_t - p_t = c - b(E_t p_{t+1} - p_t)

Rearrange to isolate

p_t

:

m_t - c = p_t - b(E_t p_{t+1} - p_t)

m_t - c = p_t(1 + b) - bE_t p_{t+1}

p_t(1 + b) = m_t - c + bE_t p_{t+1}

p_t = \frac{m_t - c + bE_t p_{t+1}}{1 + b}

b

Using the law of iterated projections, express

E_t p_{t+1}

in terms of

E_t m_{t+1}

and

E_t p_{t+2}

:

E_t p_{t+1} = \frac{E_t m_{t+1} - c + bE_t p_{t+2}}{1 + b}

c

Iterate this process forward to express

p_t

in terms of

m_t, E_t m_{t+1}, E_t m_{t+2}, \ldots

:

p_t = \frac{m_t - c}{1 + b} + \frac{b}{1 + b} E_t p_{t+1}

Substitute

E_t p_{t+1}

iteratively:

p_t = \frac{m_t - c}{1 + b} + \frac{b}{1 + b} \left( \frac{E_t m_{t+1} - c}{1 + b} + \frac{b}{1 + b} E_t p_{t+2} \right)

Continue this process:

p_t = \sum_{i=0}^{\infty} \left( \frac{b}{1 + b} \right)^i \frac{E_t m_{t+i} - c}{1 + b}

Given the no-bubbles condition:

\lim_{i \to \infty} E_t \left[ \left( \frac{b}{1 + b} \right)^i p_{t+i} \right] = 0

Thus:

p_t = \sum_{i=0}^{\infty} \left( \frac{b}{1 + b} \right)^i \frac{E_t m_{t+i} - c}{1 + b}

d

Intuitively, an increase in

E_t m_{t+i}

for any i > 0 raises

p_t

because higher expected future money supply increases the expected future price level, which in turn increases the current price level due to the forward-looking nature of price setting

e

Suppose expected money growth is constant, so

E_t m_{t+i} = m_t + g i

. Solve for

p_t

in terms of

m_t

and

g

:

p_t = \sum_{i=0}^{\infty} \left( \frac{b}{1 + b} \right)^i \frac{m_t + g i - c}{1 + b}

p_t = \frac{m_t - c}{1 + b} \sum_{i=0}^{\infty} \left( \frac{b}{1 + b} \right)^i + \frac{g}{1 + b} \sum_{i=0}^{\infty} i \left( \frac{b}{1 + b} \right)^i

Using the geometric series sum formula:

\sum_{i=0}^{\infty} \left( \frac{b}{1 + b} \right)^i = \frac{1}{1 - \frac{b}{1 + b}} = 1 + b

\sum_{i=0}^{\infty} i \left( \frac{b}{1 + b} \right)^i = \frac{\frac{b}{1 + b}}{\left( 1 - \frac{b}{1 + b} \right)^2} = \frac{b}{(1 + b)^2}

Thus:

p_t = \frac{m_t - c}{1 + b} (1 + b) + \frac{g}{1 + b} \frac{b}{(1 + b)^2}

p_t = m_t - c + \frac{gb}{(1 + b)^2}

An increase in

g

raises

p_t

because it increases the expected future money supply, leading to higher current price levels

Answer

The final answers are as follows: -

p_t = \frac{m_t - c + bE_t p_{t+1}}{1 + b}

-

E_t p_{t+1} = \frac{E_t m_{t+1} - c + bE_t p_{t+2}}{1 + b}

-

p_t = \sum_{i=0}^{\infty} \left( \frac{b}{1 + b} \right)^i \frac{E_t m_{t+i} - c}{1 + b}

- An increase in

E_t m_{t+i}

for any i > 0 raises

p_t

because it increases the expected future price level. -

p_t = m_t - c + \frac{gb}{(1 + b)^2}

, and an increase in

g

raises

p_t

.

Key Concept

The relationship between money supply, expected future money supply, and price levels.

Explanation

The price level

p_t

is influenced by the current money supply

m_t

and the expected future money supply

E_t m_{t+i}

. An increase in expected future money supply raises the current price level due to the forward-looking nature of price setting.

explain in details the question a

Solution

a

To solve for

p_{t}

in terms of

m_{t}

and

E_{t} p_{t+1}

, we start with the given money demand equation:

m_{t} - p_{t} = c - b \left( E_{t} p_{t+1} - p_{t} \right)

First, we rearrange the terms to collect

p_{t}

on one side:

m_{t} - p_{t} = c - b E_{t} p_{t+1} + b p_{t}

Next, we combine the

p_{t}

terms:

m_{t} - c = p_{t} (1 + b) - b E_{t} p_{t+1}

Solving for

p_{t}

, we get:

p_{t} (1 + b) = m_{t} - c + b E_{t} p_{t+1}

Finally, we isolate

p_{t}

:

p_{t} = \left( \frac{b}{1 + b} \right) E_{t} p_{t+1} + \left( \frac{1}{1 + b} \right) (m_{t} - c)

Thus, the solution for

p_{t}

in terms of

m_{t}

and

E_{t} p_{t+1}

is:

p_{t} = \left( \frac{b}{1 + b} \right) E_{t} p_{t+1} + \left( \frac{1}{1 + b} \right) (m_{t} - c)

Answer

The solution for

p_{t}

in terms of

m_{t}

and

E_{t} p_{t+1}

is:

p_{t} = \left( \frac{b}{1 + b} \right) E_{t} p_{t+1} + \left( \frac{1}{1 + b} \right) (m_{t} - c)

Key Concept

Solving for the price level

p_{t}

in terms of the money stock

m_{t}

and the expected future price level

E_{t} p_{t+1}

.

Explanation

By rearranging the given money demand equation and isolating

p_{t}

, we can express the current price level as a function of the current money stock and the expected future price level.

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