STATS-Consider a two-period, endowment, and small

Solution by Steps

step 1

The budget constraint for period 1 can be written as:

C_{1}^{T} + p_{1}C_{1}^{N} = Q_{1}^{T} + p_{1}Q_{1}^{N} + B_{1}

where

B_{1}

is the amount of bonds purchased in period 1

step 2

The budget constraint for period 2 can be written as:

C_{2}^{T} + p_{2}C_{2}^{N} = Q_{2}^{T} + p_{2}Q_{2}^{N} - (1 + r^{\star})B_{1}

where

(1 + r^{\star})B_{1}

is the repayment of bonds with interest

Answer

The budget constraints for periods 1 and 2 are:

C_{1}^{T} + p_{1}C_{1}^{N} = Q_{1}^{T} + p_{1}Q_{1}^{N} + B_{1}

C_{2}^{T} + p_{2}C_{2}^{N} = Q_{2}^{T} + p_{2}Q_{2}^{N} - (1 + r^{\star})B_{1}

Key Concept

Budget constraints in each period

Explanation

The budget constraints represent the balance between consumption, endowment, and bond transactions in each period.

Question 2: Derive the household's intertemporal budget constraint.

step 1

Substitute

B_{1}

from the period 1 budget constraint into the period 2 budget constraint:

B_{1} = C_{1}^{T} + p_{1}C_{1}^{N} - Q_{1}^{T} - p_{1}Q_{1}^{N}

step 2

Substitute this expression for

B_{1}

into the period 2 budget constraint:

C_{2}^{T} + p_{2}C_{2}^{N} = Q_{2}^{T} + p_{2}Q_{2}^{N} - (1 + r^{\star})(C_{1}^{T} + p_{1}C_{1}^{N} - Q_{1}^{T} - p_{1}Q_{1}^{N})

step 3

Simplify the equation to get the intertemporal budget constraint:

C_{1}^{T} + p_{1}C_{1}^{N} + \frac{C_{2}^{T}}{1 + r^{\star}} + \frac{p_{2}C_{2}^{N}}{1 + r^{\star}} = Q_{1}^{T} + p_{1}Q_{1}^{N} + \frac{Q_{2}^{T}}{1 + r^{\star}} + \frac{p_{2}Q_{2}^{N}}{1 + r^{\star}}

Answer

The intertemporal budget constraint is:

C_{1}^{T} + p_{1}C_{1}^{N} + \frac{C_{2}^{T}}{1 + r^{\star}} + \frac{p_{2}C_{2}^{N}}{1 + r^{\star}} = Q_{1}^{T} + p_{1}Q_{1}^{N} + \frac{Q_{2}^{T}}{1 + r^{\star}} + \frac{p_{2}Q_{2}^{N}}{1 + r^{\star}}

Key Concept

Intertemporal budget constraint

Explanation

The intertemporal budget constraint combines the budget constraints of both periods, accounting for the time value of money.

Question 3: Derive the optimality conditions associated with this problem.

step 1

Solve the intertemporal budget constraint for

C_{1}^{T}

:

C_{1}^{T} = Q_{1}^{T} + p_{1}Q_{1}^{N} + \frac{Q_{2}^{T}}{1 + r^{\star}} + \frac{p_{2}Q_{2}^{N}}{1 + r^{\star}} - p_{1}C_{1}^{N} - \frac{C_{2}^{T}}{1 + r^{\star}} - \frac{p_{2}C_{2}^{N}}{1 + r^{\star}}

step 2

Substitute this expression for

C_{1}^{T}

into the utility function:

U = \log \left( Q_{1}^{T} + p_{1}Q_{1}^{N} + \frac{Q_{2}^{T}}{1 + r^{\star}} + \frac{p_{2}Q_{2}^{N}}{1 + r^{\star}} - p_{1}C_{1}^{N} - \frac{C_{2}^{T}}{1 + r^{\star}} - \frac{p_{2}C_{2}^{N}}{1 + r^{\star}} \right) + \log C_{1}^{N} + \log C_{2}^{T} + \log C_{2}^{N}

step 3

Take the derivatives of the resulting utility function with respect to

C_{1}^{N}

,

C_{2}^{T}

, and

C_{2}^{N}

and set them equal to zero:

\frac{\partial U}{\partial C_{1}^{N}} = 0

\frac{\partial U}{\partial C_{2}^{T}} = 0

\frac{\partial U}{\partial C_{2}^{N}} = 0

Answer

The optimality conditions are derived by setting the partial derivatives of the utility function with respect to

C_{1}^{N}

,

C_{2}^{T}

, and

C_{2}^{N}

to zero.

Key Concept

Optimality conditions for utility maximization

Explanation

The optimality conditions are found by taking the partial derivatives of the utility function with respect to the consumption variables and setting them to zero.

Question 4: Write down the market-clearing conditions in the nontradable goods market in periods 1 and 2.

step 1

The market-clearing condition for nontradable goods in period 1 is:

C_{1}^{N} = Q_{1}^{N}

step 2

The market-clearing condition for nontradable goods in period 2 is:

C_{2}^{N} = Q_{2}^{N}

Answer

The market-clearing conditions are:

C_{1}^{N} = Q_{1}^{N}

C_{2}^{N} = Q_{2}^{N}

Key Concept

Market-clearing conditions

Explanation

The market-clearing conditions ensure that the consumption of nontradable goods equals the endowment of nontradable goods in each period.

Question 5: What is the value of

r_{1}

under perfect capital mobility?

step 1

Under perfect capital mobility, the domestic interest rate

r_{1}

equals the world interest rate

r^{\star}

:

r_{1} = r^{\star}

Answer

The value of

r_{1}

under perfect capital mobility is:

r_{1} = r^{\star}

Key Concept

Perfect capital mobility

Explanation

Under perfect capital mobility, the domestic interest rate aligns with the world interest rate.

Solution by Steps

step 1

The profit maximization condition for the firm is given by the marginal product of investment being equal to the cost of investment. The production function is

Y_{t} = A_{t} I_{t-1}^{\alpha}

step 2

The firm's profit in period 2 is

\Pi_{2} = Y_{2} - q_{1} I_{1}

step 3

To maximize profit, set the marginal product of investment equal to the cost of investment:

\frac{\partial \Pi_{2}}{\partial I_{1}} = 0

step 4

This gives us

A_{2} \alpha I_{1}^{\alpha-1} = q_{1}

. Solving for

I_{1}

, we get

I_{1} = \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

step 5

The derivative of

I_{1}

with respect to

r_{1}

is zero since

r_{1}

does not appear in the expression. The derivative of

I_{1}

with respect to

q_{1}

is negative

step 6

Graphically, in the

(I_{1}, r_{1})

plane, the investment schedule is a vertical line. An increase in

q_{1}

shifts the investment schedule to the left

Answer

I_{1} = \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

Question 2: Optimal

\Pi_{2}

step 1

Using the production function

Y_{2} = A_{2} I_{1}^{\alpha}

and the expression for

I_{1}

from Question 1, we have

Y_{2} = A_{2} \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{\alpha}{1-\alpha}}

step 2

The profit

\Pi_{2}

is then

Y_{2} - q_{1} I_{1}

step 3

Substituting

I_{1}

, we get

\Pi_{2} = A_{2} \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{\alpha}{1-\alpha}} - q_{1} \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

step 4

Simplifying,

\Pi_{2} = A_{2} \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{\alpha}{1-\alpha}} - A_{2} \alpha \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

Answer

\Pi_{2} = A_{2} \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{\alpha}{1-\alpha}} - A_{2} \alpha \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

Question 3: Household Maximization Problem

step 1

The household's budget constraints are

C_{1} + B_{1}^{h} = \Pi_{1} + (1 + r_{0}) B_{0}^{h}

and

C_{2} + B_{2} = \Pi_{2} + (1 + r_{1}) B_{1}^{h}

step 2

The intertemporal budget constraint is

C_{1} + \frac{C_{2}}{1 + r_{1}} = \Pi_{1} + (1 + r_{0}) B_{0}^{h} + \frac{\Pi_{2}}{1 + r_{1}}

step 3

The household maximizes

U = \log C_{1} + \beta \log C_{2}

subject to the intertemporal budget constraint

step 4

The first-order condition is

\frac{1}{C_{1}} = \beta \frac{1}{C_{2}} (1 + r_{1})

step 5

Solving for

C_{1}

, we get

C_{1} = \frac{1}{1 + \beta (1 + r_{1})} \left(\Pi_{1} + (1 + r_{0}) B_{0}^{h} + \frac{\Pi_{2}}{1 + r_{1}}\right)

step 6

The derivative of

C_{1}

with respect to

r_{1}

is negative, and the derivative with respect to

q_{1}

is also negative

step 7

Graphically, in the

(C_{1}, r_{1})

plane, the consumption schedule is downward sloping. An increase in

q_{1}

shifts the consumption schedule downward

Answer

C_{1} = \frac{1}{1 + \beta (1 + r_{1})} \left(\Pi_{1} + (1 + r_{0}) B_{0}^{h} + \frac{\Pi_{2}}{1 + r_{1}}\right)

Question 4: National Savings

S_{1}

step 1

National savings

S_{1}

is defined as

S_{1} = Y_{1} - C_{1} - I_{1}

step 2

Using the production function

Y_{1} = A_{1} I_{0}^{\alpha}

and the expressions for

C_{1}

and

I_{1}

, we get

S_{1} = A_{1} I_{0}^{\alpha} - \frac{1}{1 + \beta (1 + r_{1})} \left(\Pi_{1} + (1 + r_{0}) B_{0}^{h} + \frac{\Pi_{2}}{1 + r_{1}}\right) - \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

step 3

The derivative of

S_{1}

with respect to

r_{1}

is negative, and the derivative with respect to

q_{1}

is also negative

step 4

Graphically, in the

(S_{1}, r_{1})

plane, the saving schedule is downward sloping. An increase in

q_{1}

shifts the saving schedule downward

Answer

S_{1} = A_{1} I_{0}^{\alpha} - \frac{1}{1 + \beta (1 + r_{1})} \left(\Pi_{1} + (1 + r_{0}) B_{0}^{h} + \frac{\Pi_{2}}{1 + r_{1}}\right) - \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

Question 5: Current Account

CA_{1}

step 1

The current account in period 1 is

CA_{1} = S_{1} - I_{1}

step 2

Using the expression for

S_{1}

and

I_{1}

, we get

CA_{1} = A_{1} I_{0}^{\alpha} - \frac{1}{1 + \beta (1 + r_{1})} \left(\Pi_{1} + (1 + r_{0}) B_{0}^{h} + \frac{\Pi_{2}}{1 + r_{1}}\right) - 2 \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

step 3

The derivative of

CA_{1}

with respect to

r_{1}

is negative, and the derivative with respect to

q_{1}

is also negative

step 4

Graphically, in the

(CA_{1}, r_{1})

plane, the current account schedule is downward sloping. An increase in

q_{1}

shifts the current account schedule downward

Answer

CA_{1} = A_{1} I_{0}^{\alpha} - \frac{1}{1 + \beta (1 + r_{1})} \left(\Pi_{1} + (1 + r_{0}) B_{0}^{h} + \frac{\Pi_{2}}{1 + r_{1}}\right) - 2 \left(\frac{A_{2} \alpha}{q_{1}}\right)^{\frac{1}{1-\alpha}}

Question 6: Effect of Increase in

q_{1}

step 1

In a small open economy with perfect capital mobility, an increase in

q_{1}

increases the cost of investment goods

step 2

This leads to a decrease in

I_{1}

, shifting the investment schedule to the left

step 3

The decrease in

I_{1}

reduces

Y_{2}

and

\Pi_{2}

, leading to a decrease in

C_{1}

and

S_{1}

step 4

The current account

CA_{1}

also decreases as

S_{1}

decreases more than

I_{1}

step 5

Graphically, the investment, saving, and current account schedules all shift downward

Answer

An increase in

q_{1}

decreases

I_{1}

,

C_{1}

,

S_{1}

, and

CA_{1}

.

Question 7: Closed Economy

step 1

In a closed economy, the effect of an increase in

q_{1}

on

I_{1}

,

C_{1}

, and

S_{1}

remains the same

step 2

However, the current account

CA_{1}

is zero by definition in a closed economy

step 3

The decrease in

I_{1}

leads to a decrease in

Y_{2}

and

\Pi_{2}

, reducing

C_{1}

and

S_{1}

step 4

Graphically, the investment and saving schedules shift downward, but the current account remains zero

Answer

In a closed economy, an increase in

q_{1}

decreases

I_{1}

,

C_{1}

, and

S_{1}

, but

CA_{1}

remains zero.

Key Concept

The relationship between investment, consumption, savings, and the current account in response to changes in the cost of investment goods.

Explanation

An increase in the cost of investment goods (

q_{1}

) leads to a decrease in investment, which in turn affects production, profits, consumption, savings, and the current account. In a closed economy, the current account remains zero.

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