Please Do not answer the question 1,2,3, 4,5, and 6. Need only question 7 to 9.
ID: 1139890 • Letter: P
Question
Please Do not answer the question 1,2,3, 4,5, and 6. Need only question 7 to 9.
Exercise 1 - Consumer choice problems (duality) I
Preferences over goods 1 and 2 are given by u(q1, q2) = (q1 a)(q2 b)
the expenditure function is e(p1, p2, v¯) = p1 a + p2 b + 2 vp¯ 1p2.
7. State Shephard’s Lemma and apply it to derive the Hicksian demands.
8. State Roy’s identity and apply it to derive the Marshallian demands.
9. Verify that the Slutsky equation holds.
5. Show that inverting the expenditure function with respect to its third argument gives the indirect utility function, and vice-versa.
6. Show that evaluating the Marshallian demands when y = e(p1, p2, v¯) gives the Hicksian demands. Then show that evaluating the Hicksian demands when ¯v = v(p1, p2, y) gives the Marshallian demands.
1. Set up the consumer’s utility maximisation problem when prices are p1, p2 and available income is y. Assuming there exists an interior solution, set up the Lagrangian and solve for the Marshallian demands. (You may wish to substitute them back into the budget constraint to check that your answers are correct).
2. Compute the indirect utility function v(p1, p2, y).
3. Now set up the consumer’s expenditure minimisation problem when prices are p1, p2 and the minimum required utility level is ¯v. Assuming there exists an interior solution, set up the Lagrangian and solve for the Hicksian demands. (You may wish to substitute them back into the utility constraint to check that your answers are correct).
4. Show that the expenditure function is e(p1, p2, v¯) = p1 a + p2 b + 2 vp¯ 1p2.
Explanation / Answer
7.Stephard lemma is a huge part of microeconomics in theory of the firm and consumer choice. This theory mainly states thet the indifference curves of expenditure and cost function are convex in nature.This clearly implies that the consumer will pick the ideal amount to buy the product where he will get certain level of utility. Later lemma is named after Ronald stephard who is the proffessor in the university gave the distance formula
It derives the relationship between Cost fuctions and hicksian demand or it can be re expressed in Roys identity which explains a relation between the indirect utility fuction and Marshallian deamnd function.
8. Now this can be explained through the derivatives
The indirect utility function v(p_{1},p_{2},w)} v(p1,p2 w) is the maximand of the constrained optimization problem characterized by the following Lagrangian:
L = U(x1,x2)+ Y(w-p1x1-p2x2)
9.Slusky Equation: This thoery is mainly into checking the price change in Uncompensated demand with the Price change and compensated demand with the price change with the income change in certain cases
This can also be explained in elasticities.