Question

3.10. The utility that Julie receives by consuming food F and clothing C is given by U(F, C) = FC. For this utility function, the marginal utilities are MU_F = C and MU_C = F.

a) On a graph with F on the horizontal axis and C on the vertical axis, draw indifference curves for U = 12, U = 18, and U= 24.

b) Do the shapes of these indifference curves suggest that Julie has a diminishing marginal rate of substitution of food for clothing? Explain.

c) Using the marginal utilities, show that the MRS_F,C = C/F. What is the slope of the indifference curve U = 12 at the basket with 2 units of food and 6 units of clothing? What is the slope at the basket with 4 units of food and 3 units of clothing? Do the slopes of the indifference curves indicate that Julie has a diminishing marginal rate of substitution of food for clothing? (Make sure your answers to parts (b) and (c) are consistent!)

Answer 1

a) Draw a graph with axes as instructed. Calculate three points on each indifference curve by: 1) selecting an indifference curve (utility) 2) picking a quantity of one good and 3) solving for the quantity of the other good. For example, on the indifference curve U=5, when F=1, C=5; when F=5, C=1, and when F=2.5, C=2 (in each case U(F,C)=F*C=5). Then connect the dots with a line that extends to the axes to draw the indifference curves.
b) If the indifference curves are convex to the origin, Julie has a decreasing marginal rate of substitution (the more food she has, the less clothing she will give up to get more food). If the indifference curves are linear, Julie has a constant marginal rate of substitution (she will give up food for clothing at the same rate regardless of how much food she has).
c) The shape (curved or linear) of the indifference curves should make the slopes and therefore the answer to this question fairly obvious. MRS = -slope. You can calculate the slope using the two points (2,5) and (5,2) given. If the slope is the same at both points, Julie has a constant marginal rate of substitution.

(a)

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(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.