Chapter 5, Problem 5.20P

To determine

(a)

If, $P_x=4,P_y=1$ then consumer's optimal basket is to be calculated.

Answer to Problem 5.20P

The consumer's optimal basketis 15 units of x and 60 units ofy.

Explanation of Solution

Budget constraint/line shows all possible combinations of two goods that can be purchased, given the level of income of the consumer and the market prices of both goods.

Indifference curve (IC) shows all possible combination of two goods that gives the same level of satisfaction to the consumer.

At the point of tangency of the budget line and the indifference curve, the consumption bundle is optimal. The slope of the IC and the budget line is equal at this point.

A consumer has utility function:

  $U\left(x,y\right)=xy$

X represents pizza and y represents other goods.

Given that −

Marginal utility of consumer

  $M{U}_x=y$

  $M{U}_y=x$

Income = $120

The budget line of the consumer is:

  $x.P_x+y.P_y=M$

  $4x+y=120$................Equ (1)

First find the values that satisfy the tangency condition, which is the requirement by putting $M{U}_x=y$ and $M{U}_y=x$ into tangency condition equation,

  $\begin{array}{l}\frac{MUx}{Px}=\frac{MUy}{P\text{y}}\\ \frac{y}{4}=\frac{x}{1}\end{array}$

  $y=4x$................Equ (2)

From Equ (1) and Equ (2), then −

  $\begin{array}{l}4x+y=120\\ 4x+4x=120\\ 8x=120\end{array}$

  $x=15$

Now the put the value of x form Equ (3) in Equ (1) to calculate the value of y −

  $\begin{array}{l}4x+y=120\\ 4\left(15\right)+y=120\\ y=120-4\left(15\right)\\ y=60\end{array}$

Hencethe consumer's optimal basket consist 15 units of x and 60 units of Y.

To determine

(b)

If the price of food reduced to $3, then income and substitution effect of consumer is to be calculated.

Answer to Problem 5.20P

The consumer's substitution effect is 2.7.

Explanation of Solution

Substitution effect: Substitution effect is the term used in economics to change in the relative prices of products on the basis of consumption pattern.

To calculate the consumer's income and substitution effect of a decreases in the price x to 3.

Use the new tangency conditions equation for this price.

It is given by the equation $\frac{y}{x}=3$.

The new budget constraints-

  $3x+y=120$.................Equ (1)

Put y =3x in Equ (1), to calculate the value of x.

  $\begin{array}{l}3x+y=120\\ 3x+3x=120\\ 6x=120\\ x=20\end{array}$

Put the value of x in Equ (1) to calculate the value of y −

  $\begin{array}{l}3x+y=120\\ 3\left(20\right)+3x=120\\ y=120-6\\ y=60\end{array}$

Since the consumer's utility is $15\times 60=900$ ,the decomposition basket for $y=3x$

is given:

  $\begin{array}{l}x\left(3x\right)=900\\ x^2=300\\ x=17.32\\ \text{And}\\ \text{17}\text{.32y=900}\\ \text{y = 51}\text{.9}\end{array}$

Thus, the consumer's optimal basket when $Px=3$ and $P_y=1$ is 20 units of x and 60 units of y.

The consumer's substitution effect is $17.3-15=\text{ 2}\text{.3}$ and the consumer's income effect is $20-17.3=\text{ 2}\text{.7}$.

To determine

(c)

The compensating variation of price change is to be calculated.

Answer to Problem 5.20P

The consumer's compensating variation of the price change is 16.2.

Explanation of Solution

Compensating variation is the quantity of earning of the consumer able to give after a price alteration to be just well of as before the price variation.

The consumers compensating variation of the amountalteration is equal to the difference between the consumers earning 120 the income requires to buy the decomposition basket.

The decomposition basket can be taken for hence the consumers compensating variation of the price change is $\left(x=17.32,y=51.9\right)$.

The decomposition basket can be calculated as −

  $3x+y=120$

Put the values of x and y in above equation −

Compensation variation = $17.32\times 3+51.9=120$

Compensation variation = 120 − 103.8 = 16.2

To determine

(d)

The equivalent variation of price change is to be calculated.

Answer to Problem 5.20P

The compensating variation of the price change $138.4-120=18.4$

Explanation of Solution

Equivalent variation is the volume of income the consumer would need before a price change to be just as well off after the price change.

The consumer's equivalent variation of the price change is equal to the amount needed to achieve the utility after the price change which is $xy=20\times \left(60\right)=1200$ and the consumer's income is 120.

  $\begin{array}{l}xy=1200\\ x\left(4x\right)=1200\\ x^2=300\\ x=17.3\end{array}$

If $x=17.3$ then $y=4\left(17.3\right)=69.2$

  $y=4x\text{ and }xy1200\text{ are }\left(x=17.3,y=69.2\right)$

This basket can be brought for $17.3\left(3\right)+69.2=138.4$

Hence, the consumer's equivalent variation of the price change is $138.4-120=18.4$