Consider a consumer with M = $120 to spend (income) and faces prices PX = $15 and PY = $5 for goods X and Y. Her utility function is U(X, Y) = X1/2 + Y1/2. a Carefully. express the consumer’s choice problem, using the given information (this is where you write out the max operator, the choice variables, the objective function, and the budget constraint). b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it to determine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape. To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether |MRS(X, Y)| is diminishing along an indifference curve. c. If the indifference curves are C-shaped write out the budget line and the equal slopes condition that characterize an interior solution to the consumer’s choice problem. Use the particulars for the given consumer. Solve these conditions to find the interior solution. On the other hand, if the indifference curves are linear or )-shaped, look for a boundary solution where the consumer either buys all X or all Y. Determine which boundary is the solution by comparing the consumer’s utility at (M/PX, 0) and (0, M/PY), which are the corners of the budget line. d. Now leave the prices and incomes as variables and solve for the demand functions for goods X and Y: X*(PX, PY, M) and Y*(PX, PY, M). Neatly show your work.
Consider a consumer with M = $120 to spend (income) and faces
$5 for goods X and Y. Her utility function is U(X, Y) = X1/2 + Y1/2.
a Carefully. express the consumer’s choice problem, using the given information (this is where
you write out the max operator, the choice variables, the objective function, and the budget constraint).
b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it to
determine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape.
To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether
|MRS(X, Y)| is diminishing along an indifference curve.
c. If the indifference curves are C-shaped write out the budget line and the equal slopes condition
that characterize an interior solution to the consumer’s choice problem. Use the particulars for the given
consumer. Solve these conditions to find the interior solution. On the other hand, if the indifference curves
are linear or )-shaped, look for a boundary solution where the consumer either buys all X or all Y. Determine
which boundary is the solution by comparing the consumer’s utility at (M/PX, 0) and (0, M/PY), which are
the corners of the budget line.
d. Now leave the prices and incomes as variables and solve for the demand functions for goods
X and Y: X*(PX, PY, M) and Y*(PX, PY, M). Neatly show your work.
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