Chapter 19, Problem 6PA

(a)

To determine

Explain how much Ms. N consumes at each period.

Explanation of Solution

Given information:

Utility function of Ms. N is $\text{U}=\text{ln}\left(\text{C1}\right)+\text{ln}\left(\text{C2}\right)+\text{ln}\left(\text{C3}\right)$ (1)

Budget constraint is $120,000.

Calculation:

Generally, consumers allocate their income for each of consumption. Therefore, the MRS (Marginal Rate of Substitution) between any two different periods is one plus rate of interest. The given up amount in order to obtain another commodity to maintain same level of utility or marginal rate of substitution between period 1 and 2 is ${\text{MRS}}_{1,2}=\frac{{\text{C}}_2}{{\text{C}}_1}$. Likewise, the marginal rate of substitution between period 2 and 3 is ${\text{MRS}}_{2,3}=\frac{{\text{C}}_3}{{\text{C}}_2}$. Thus, with the limited budget constraint ($120,000), her optimal consumption bundle is the sum of all consumption bundles in each period. Symbolically, it is shown below:

${\text{C}}_1+{\text{C}}_2+{\text{C}}_3=120,000$

The consumption bundle purchased in each period is equal, which is ${\text{C}}_1={\text{C}}_2={\text{C}}_3$ . Thus, by using the $120,000 budget constraint, she can use $40,000$\left(\frac{120,000}{3}\right)$ for each period.

Economics Concept Introduction

Marginal Rate of Substitution (MRS): Marginal Rate of Substitution represents the rate at which an individual can give up some amount of a commodity in order to obtain another commodity while maintaining the same level of utility.

(b)

To determine

Explain how much Mr. D consumes in each period.

Explanation of Solution

Given information:

Utility function of Mr. D is $\text{U}=2\text{ln}\left(\text{C1}\right)+\text{ln}\left(\text{C2}\right)+\text{ln}\left(\text{C3}\right)$ (2)

Budget constraint is $120,000.

Calculation:

The marginal rate of substitution between period 1 and 2 is ${\text{MRS}}_{1,2}=\frac{2{\text{C}}_2}{{\text{C}}_1}$. Likewise, the marginal rate of substitution between period 2 and 3 is ${\text{MRS}}_{2,3}=\frac{{\text{C}}_3}{{\text{C}}_2}$. The optimal bundle, which he can satisfy with his limited budget constraint and the marginal condition is shown below:

${\text{C}}_1+{\text{C}}_2+{\text{C}}_3=120,000$ (2.A)

The person considers that the value of present consumption is twice as the future consumption and valued future consumption equally. This written as follows:

${\text{C}}_1=2{\text{C}}_2$ (2.B)

${\text{C}}_2={\text{C}}_3$ (2.C)

Substitute Equation (2.B) and (2.C) in Equation (2.A) for C₁ and C_3, respectively to calculate the value of C₂.

$\begin{array}{c}{\text{2C}}_2+{\text{C}}_2+{\text{C}}_2=120,000\\ {\text{4C}}_2=120,000\\ {\text{C}}_2=\frac{120,000}{4}\\ {\text{C}}_2=30,000\end{array}$

Mr. D has $30,000 as wealth in period C₂.

Equation (2.C) reveals that the value of time period C₃ is equal to C₂. Thus, the wealth in period C₃ is $30,000.

Equation (2.B) reveals that the value of time period C₁ is equal to 2C₂. Thus, the wealth in period C₁ is $60,000$\left(2\times 30,000\right)$.

Economics Concept Introduction

Marginal Rate of Substitution (MRS): Marginal Rate of Substitution represents the rate at which an individual can give up some amount of a commodity in order to obtain another commodity while maintaining the same level of utility.

(c)

To determine

Explain how much Mr. D consumes in period 2 and period 3.

Explanation of Solution

Given information:

Utility function of Mr. D in period 2 is $\text{U}=\text{ln}\left(\text{C1}\right)+2\text{ln}\left(\text{C2}\right)+\text{ln}\left(\text{C3}\right)$ (3)

Calculation:

The person wealth in period 1 (C₁) is $60,000. Thus, the total wealth for period 2 and 3 is $60,000. This can be written as follows:

${\text{C}}_{\text{2}}+{\text{C}}_{\text{3}}=\text{60,000}$ (4)

The person considers that the value of present consumption is twice as the future consumption. This written as follows:

${\text{C}}_2=2{\text{C}}_3$ (5)

Substitute Equation (5) in Equation (4) for C₂ to calculate the value of C₃. This can be written as follows:

$\begin{array}{c}{\text{2C}}_{\text{3}}+{\text{C}}_{\text{3}}=\text{60,000}\\ {\text{3C}}_{\text{3}}=\text{60,000}\\ {\text{C}}_{\text{3}}=\frac{60,000}{3}\\ {\text{C}}_{\text{3}}=20,000\end{array}$

The value of C₃ is $20,000.

Substitute the value of C₃ in Equation (4) to calculate the value of C₂.

$\begin{array}{c}{\text{C}}_{\text{2}}+\text{20,000}=\text{60,000}\\ {\text{C}}_{\text{2}}=\text{60,000}-\text{20,000}\\ {\text{C}}_{\text{3}}\text{=40,000}\end{array}$

The value of C₂ is $40,000. Thus, the person changed his plan from period 1 to period 2 which indicates the time inconsistent.

(d)

To determine

Calculate the value of different time periods.

Explanation of Solution

The person wealth in period 1 (C₁) is $60,000. Thus, the total wealth for period 2 and 3 is $60,000. This can be shown in Equation (4) as ${\text{C}}_{\text{2}}+{\text{C}}_{\text{3}}=\text{60,000}$. In period 1, the person valued ${\text{C}}_2={\text{C}}_3$ is denoted in Equation (2.C)

Substitute Equation (2.C) in Equation (4) for C₂ to calculate the value of C₃. This can be written as follows:

$\begin{array}{c}{\text{C}}_{\text{3}}+{\text{C}}_{\text{3}}=\text{60,000}\\ {\text{2C}}_{\text{3}}=\text{60,000}\\ {\text{C}}_{\text{3}}=\frac{60,000}{2}\\ {\text{C}}_{\text{3}}=30,000\end{array}$

The value of C₃ is $30,000. Since C₂ is equal to C₃, the value of C₂ is $30,000.

However, in period 2, the value of C₂ is $40,000 and the value of C₃ is $20,000. The person should follow the original consumption plan in order to avoid the consumption constraint. This selection of consumption plan is an example of Laibson’s pull-of-instant-gratification model. Thus, it shows that the person face time inconsistent preference and constraining the person’s future choice made him better off.