Chapter 4, Problem 1NP

a)

To determine

The consumer’s saving and consumption.

Explanation of Solution

A consumer is making saving plans for this year and next year.

Let her income is Y, saving is S, and consumption is C, which is same in both the years.

Income in year 1 is Y₁.

Income in year 2 is Y₂.

The consumer’s saving in the 1^st year is Y₁-C

The consumer’s saving in second year is Y₂-C.

Saving the 1^st year earn interest rate R%.

Let consumer pay X amount as tuition fee.

Then, the mathematical expression for tuition fee would be:

  $X=\left(Y_1-C\right)\left(1+r\right)+\left(Y_2-C\right)$

Given,

Y₁ = Y₂ = $50,000,

X = $16,800,

R = 10%,

  $\begin{array}{l}\left(Y_1-C\right)\left(1+r\right)+\left(Y_2-C\right)=\$16,800\hfill \\ \left(50,000-C\right)\left(1.1\right)+\left(50,000-C\right)=\$16,800\hfill \\ \left(50,000-C\right)\left(1.1+1\right)=\$16,800\hfill \\ \left(50,000-C\right)\left(2.1\right)=\$16,800\hfill \\ 50,000-C=8,000\hfill \end{array}$

After rearranging this equation, the value of consumption and saving is derived:

C = $42,000

S = $8,000 (= $50,000-$42000)

b)

To determine

The change in saving and consumption when consumer’s current income rises from $50,000 to $54,200.

Explanation of Solution

New income Y₁ = $54,200,

Y₂ = $50,000

X = $16,800,

R = 10%,

  $\begin{array}{l}\begin{array}{l}\left(Y_1-C\right)\left(1+r\right)+\left(Y_2-C\right)=\$16,800\hfill \\ \left(54,200-C\right)\left(1.1\right)+\left(50,000-C\right)=\$16,800\hfill \end{array}\\ 109,620–2.1C=\$16,800\end{array}$

After rearranging this equation, the value of consumption and saving is derived:

C = $44,200

S = $10,000 (= $54,200-$44200).

Therefore, a rise in current income decreases the saving and increases the consumption.

c)

To determine

The change in saving and consumption when consumer’s expected income rises from $50,000 to $54,200 next year.

Explanation of Solution

Y₁ = $50,000

New expected income, Y₂ = $54,200,

X = $16800,

R = 10%,

  $\begin{array}{l}\left(Y_1-C\right)\left(1+r\right)+\left(Y_2-C\right)=\$16,800\hfill \\ \left(50,000-C\right)\left(1.1\right)+\left(54,200-C\right)=\$16,800\hfill \\ 109,200–2.1C=\$16,800\hfill \end{array}$

After rearranging this equation, the value of consumption and saving is derived:

C = $44,000

S = $6,000 (= $50,000-$44000).

Therefore, a rise in future income decreases the saving and increases the consumption.

d)

To determine

The change in saving and consumption when during the current year consumer receives an inheritance of $1050 (an increase in wealth, not income).

Explanation of Solution

Y₁ = Y₂ = $50,000,

X = $16,800,

R = 10%,

The amount of inheritance I = $1,050.

  $\begin{array}{l}\begin{array}{l}\left(Y_1+I-C\right)\left(1+r\right)+\left(Y_2-C\right)=\$16,800\hfill \\ \left(50,000+1,050-C\right)\left(1.1\right)+\left(50,000-C\right)=\$16,800\hfill \end{array}\\ 106,155–2.1C=\$16,800\end{array}$

After rearranging this equation, the value of consumption and saving is derived:

C = $42,550

S = $7450 (= $50,000-$42550).

Therefore, a rise in wealth leads to decrease the saving and increase the consumption.

e)

To determine

The change in saving and consumption when the expected tuition payment for next year rises from $16,800 to $18,900.

Explanation of Solution

Y₁ = Y₂ = $50,000,

New expected tuition payment X = $18,900,

R = 10%,

  $\begin{array}{l}\left(Y_1-C\right)\left(1+r\right)+\left(Y_2-C\right)=\$18,900\hfill \\ \left(50,000-C\right)\left(1.1\right)+\left(50,000-C\right)=\$18,900\hfill \\ 105,000–2.1C=\$18,900\hfill \end{array}$

After rearranging this equation, the value of consumption and saving is derived:

C = $41000

S = $9000 (= $50,000-$41000).

Therefore, a rise in expected tuition fee for next year leads to increase the saving and decrease the consumption.

f)

To determine

The change in saving and consumption when rate of interest rises from 10% to 24%.

Explanation of Solution

The real interest rate rises from 10% to 24%.

Y₁ = Y₂= $50,000,

X = $16800,

New rate R = 24%,

  $\begin{array}{l}\left(Y_1-C\right)\left(1+r\right)+\left(Y_2-C\right)=\$16800\hfill \\ \left(50,000-C\right)\left(1.24\right)+\left(50,000-C\right)=\$16,800\hfill \\ 112,000–2.24C=\$16,800\hfill \end{array}$

After rearranging this equation, the value of consumption and saving is derived:

C = $42500

S = $7500 (= $50,000-$42500).

Therefore, a rise in wealth leads to decrease the saving and increase the consumption.