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------------------------------------------------ Suppose Shallan consumes stew (S) and flatbread (F), and her utility function is [tex]U(S,F) = S^3F[/tex]. The price of stew is 1, and the price of flatbread is 3. Shallan's income is [tex]M[/tex]. In this problem, you will find Shallan's utility-maximizing combination of stew and flatbread. (Note: Some of your answers in this problem may be functions of [tex]M[/tex].)

a. Write down Shallan's budget constraint. (3 points)

b. Find Shallan's marginal utility of stew. (2 points)

c. Find Shallan's marginal utility of flatbread. (2 points)

d. Find Shallan's marginal rate of substitution. (2 points)

e. What is the slope of Shallan's budget constraint? (2 points)

Answer :

a. Shallan's budget constraint can be written as:1S + 3F = M

b. To find Shallan's marginal utility of stew (MUS, we differentiate the utility function U(S,F) with respect to stew (S):

MUS= ∂U/∂S = 3S²F

c. marginal utility of flatbread (MUF, we differentiate the utility function U(S,F) with respect to flatbread (F):MUF= ∂U/∂F = S³

d. The marginal rate of substitution (MRS) represents the rate at which Shallan is willing to substitute stew for flatbread while keeping her utility constant. It is given by the ratio of the marginal utilities:

MRS = MUS/ MUF= (3S²F) / (S³) = 3F/S

e. The slope of Shallan's budget constraint represents the rate at which she can trade stew for flatbread in the market. It is given by the ratio of the prices of stew and flatbread:Slope = Price of stew / Price of flatbread = 1/3

Note: The s to parts b, c, d, and e are not functions of M since they depend on the variables S and F only.

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