Question

In Problem 3.7 of Chapter 3, we considered Julie’s preferences for food F and clothing C. Her utility function was U(F, C) = FC. Her marginal utilities were MUF = C and MUC = F. You were asked to draw the indifference curves U = 12, U = 18, and U = 24, and to show that she had a diminishing marginal rate of substitution of food for clothing. Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on food and clothing. a) Using a graph (and no algebra), find the optimal (utility-maximizing) choice of food and clothing. Let the amount of food be on the horizontal axis and the amount of clothing be on the vertical axis. b) Using algebra (the tangency condition and the budget line), find the optimal choice of food and clothing. c) What is the marginal rate of substitution of food for clothing at her optimal basket? Show this graphically and algebraically. d) Suppose Julie decides to buy 4 units of food and 4 units of clothing with her $12 budget (instead of the optimal basket). Would her marginal utility per dollar spent on food be greater than or less than her marginal utility per dollar spent on clothing? What does this tell you about how she should substitute food for clothing if she wanted to increase her utility without spending any more money?

Answer 1

(a)

(b) Now, by algebra , tangency condition of IC and budget line implies : MRS = MU_F/ MU_C = P_F/ P_C.

MU_F = C and MU_C = F

P_F = 1 and P_C =2

Put these values, we get,

C/F = 1/2

2 C = F.

Budget line equation is P_F F + P_C C =12

Now, put 2C = F we get ,

2C + 2 C = 12

4C = 12

C = 3.

And F = 2C = 2(3) =6 .

Hence, at the optimum Julie choose to consume 6 units of Food and 3 units of clothing.

(c) MRS = C/F . So, at optimum MRS = 3/6 = 1/2 . And this is equal to the price ratio.

(d) If Julie decides to buy 4 units of food and 4 units of clothing, then

MU_F/ P_F = 4/1 = 4

and MU_C/ P_C =4/2 = 2.

So, MU_F/ P_F > MU_C/ P_C.

It implies that the consumer could reallocate spending by purchasing more food and less clothing to increase utility . At the bundle (4,4) total utility is 16 and the consumer income is $12 . By giving up one of clothing , she saves $2 which can than be used to buy 2 units of food.

Hence, by reallocating spending toward the good with the highest amount of good from the minimum amount to spend , the consumer increased total utility while remaining within the same spending.