Chapter 4, Problem 4.22P

To determine

(a)

Units of food, consumer would consume to maximizing utility is to be calculated.

Explanation of Solution

The consumer purchases food (F) and other goods (Y). The per unit price of F and Y is equal to $2 and $1 respectively. And the income of the consumer is equal to $12. The utility function of the consumer is given as follows-

  $\text{U}\text{= }\text{FY}$

Marginal utility of the two goods can be calculated as follows-

  $\begin{array}{c}\frac{\partial U}{\partial F}=Y\\ ={\text{MU}}_F\\ \text{and, }\frac{\partial U}{\partial Y}=C\\ ={\text{MU}}_Y\end{array}$

And the budget constraint of the consumer can be calculated as follows-

  $\text{2F + Y = 12 }\left(\text{1}\right)$

The rate at which consumer is willing to sacrifice some units good F to get an additional unit of good Y is known as marginal rate of substitution (MRS). It measures the slope of the indifference curve. The ratio of price of good F to price of good Y measures slope of the budget constraint.

At the optimum level of consumption, slope of the indifference curve is equal to the slope of the budget constraint. Mathematically, it can be expressed as follows,

  $\frac{{\text{MU}}_{\text{F}}}{{\text{MU}}_{\text{Y}}}\text{=}\frac{{\text{P}}_{\text{F}}}{\text{P}}\left(\text{2}\right)$

Plug the values of prices and marginal utilities in (2).

  $\begin{array}{c}\frac{{\text{MU}}_{\text{F}}}{{\text{MU}}_{\text{Y}}}\text{=}\frac{{\text{P}}_{\text{F}}}{\text{P}}\\ \frac{\text{Y}}{\text{F}}\text{ =}\frac{\text{2}}{\text{1}}\\ \text{Y = 2F }\left(3\right)\end{array}$

Plug (3) in (1).

  $\begin{array}{c}\text{2F + Y = 12 }\\ \text{2F +}\text{2F = 12}\\ \text{4F = 12}\\ \text{F = 3}\end{array}$

Plug value of F equal to 3 in (3).

  $\begin{array}{l}\text{Y = 2}\left(\text{3}\right)\hfill \\ \text{= 6}\hfill \end{array}$

Thus, optimal level of food and other goods is equal to 3 units and 6 units respectively. This implies that the consumer will consume 3 units of food at the maximum utility level.

To determine

(b)

To calculate cash subsidy and present optimal consumption basket on an optimal choice diagram after cash subsidy.

Explanation of Solution

The cash subsidy provided by government should be enough so that her income increases as much that she can now consume 8 units of F.

Due to cash subsidy, the budget line changes as follows:

  $\begin{array}{l}\begin{array}{l}\left(\text{12 + S}\right)\text{ = 2F + Y}\\ \text{Plug}\text{F=8}\text{and}\text{Y}\text{=}\text{2F,}\end{array}\hfill \\ \begin{array}{l}\left(\text{12 + S}\right)\text{ = 2}\left(\text{ 8}\right)\text{ + 2}\left(F\right)\\ \left(\text{12 + S}\right)\text{ = 16 + 2}\left(8\right)\\ \left(\text{12 + S}\right)\text{ = 16 + 16}\end{array}\hfill \\ \begin{array}{c}\text{12 + S = 32}\\ \Rightarrow \text{S}\text{= \$20}\end{array}\hfill \end{array}$

Therefore, the government should provide cash subsidy of $20 to the consumer so that he can consume 8 units of F. Plug F equal to 8 in (3) to calculate optimal unit of Y.

  $\begin{array}{l}\text{Y = 2}\left(8\right)\hfill \\ \text{= 16}\hfill \end{array}$

Thus, optimal consumption basket after cash subsidy includes 8 units of F and 16 units of Y.

Graphically, the point at which budget line is tangent to indifference curve gives the optimal level of consumption.

The figure (1) below plots the budget constraint along with the indifference curves of the consumer.

  

At point A (8,16), budget constraint is tangent to indifference curve showing optimal consumption basket.

To determine

(c)

Analysis of the effect of food stamp vouchers on consumption pattern and present optimal consumption basket on an optimal choice diagram after provision of food stamp scheme.

Explanation of Solution

Food stamp scheme:

It is a scheme under which government provides vouchers that can only be used to buy a good and cannot be used for meeting other desires.These vouchers are generally provided to ensure food safety of low-income or poor people.

As an alternative to the cash subsidy in part (b), the government is also considering giving consumers food stamps, that is, vouchers with a cash value that can only be redeemed to purchase food.

If the government is giving her a voucher of $16 to purchase food, she has $16 voucher and $12 as previous income to purchase good.

Now, she will consume 8 units of food with $16 voucher and 12 units of other goods with her income of $12. So, her new optimal consumption bundle is (8, 12).

The figure (2) below shows the optimal consumption basket after provision of food stamp scheme.

  

At point B (8,12), budget constraint is tangent to indifference curve showing optimal consumption basket.