Question

Larry’s utility is a function of nuts (X) and berries (Y), given by U = 20ln X + 4Y .

a. Derive equations for Larry’s demand functions for X and Y, assuming an interior optimum.

b. If the price of berries (PY) gets too high, Larry will stop consuming them altogether. That price is his reservation price for berries. Derive a formula for Larry’s reservation price for berries as a function of the price of nuts and his income (PX and M).

c. Graph out Larry’s demand curve for berries if M=50 and Px =3. Graph his demand curve if M increases to 100 and Px is unchanged

Answer 1

U = 20 ln X + 4Y

Budget line: M = X.Px + Y.Py

(a) Utility is maximized when MUx/MUy = Px/Py

MUx = U/X = 20/X

MUy = U/Y = 4

MUx/MUy = (20/X) / 4 = 5/X = Px/Py

X.Px = 5Py

X = 5Py/Px [Demand function for Py]

Substituting in budget line,

M = 5Py + Y.Py

Y.Py = M - 5Py

Y = (M - 5Py)/Py [Demand function for Py]

(b) When reservation price is reached, Y = 0.

(M - 5Py)/Py = 0

M - 5Py = 0

5Py = M

Py = M/5

(c) Y = (M - 5Py)/Py

When M = 50 & Px = 3, Y = (50 - 5Py)/Py

When M = 100 & Px = 3, Y = (100 - 5Py)/Py

Data table used:

Py	Y (given M = 50)	Y (given M = 100)
1	45	95
2	20	45
5	5	15
10	0	5

In following graph, D1 & D2 are the demand curves for Y when M = 50 and M = 100 respectively.