Question

There are two goods in the world, pumpkins (x1) and apple cider (x2). Pumpkins are $2 each. Cider is $7 per gallon for the first two gallons. After the second gallon, the price of cider drops to $4 per gallon. a) Peter's income is $54. Draw his budget line. Solve for the intercepts on the x1 and x2 axes, and the kink in the budget line. b) Peter's utility function is u(x1,x2)= x1 + 3(x2). Find Peter's optimal consumption bundle. c) Paul's income is $22. Draw his budget line. Solve for the intercepts on the x1 and x2 axes, and the kink in the budget line. d) Paul's utility function is u(x1,x2)= min (3x1,2x2). Find Paul's optimal consumption bundle (x1*,x2*).

Answer 1

a) The budget line intercepts are
`\( (27, 0) \)` on the
`\(x_1\)` axis and
`\( (0, 6) \)` on the \
`(x_2\)` axis. The kink occurs at
`\( (20, 2) \).`

b) Peter's optimal consumption bundle is
`\( (13, 6) \).`

c) The budget line intercepts are.
`\( (11, 0) \)`on the
`\(x_1\)`axis and \( (0, 3) \) on the \
`(x_2\)`axis. The kink occurs at
`\( (8, 2) \).`

d) Paul's optimal consumption bundle is
`\( (8, 2) \).`

a) Peter's budget line can be expressed as:
`\(2x_1 + 7x_2 = 54\)`for the first two gallons of cider and
`\(2x_1 + 4x_2 = 54\)` for any additional gallons. To find the intercepts, set
`\(x_2 = 0\)` to find the intercept on the
`\(x_1\)` axis and vice versa. The kink in the budget line occurs when
`\(2x_1 + 7x_2 = 54\)` transitions to
`\(2x_1 + 4x_2 = 54\).`

b) To find Peter's optimal consumption bundle, maximize
`\(u(x_1, x_2) = x_1 + 3x_2\)` subject to the budget constraint found in part a.

c) Paul's budget line can be expressed as:
`\(2x_1 + 7x_2 = 22\)` for the first two gallons of cider and
`\(2x_1 + 4x_2 = 22\)`for any additional gallons. Find intercepts and the kink as in part a.

d) Paul's utility function is
`\(u(x_1, x_2) = \min(3x_1, 2x_2)\)`. To find Paul's optimal consumption bundle, maximize this utility subject to the budget constraint found in part c.