Practice 1 | Môn Kinh tế kinh doanh - Đại học Ngoại Thương

lOMoAR cPSD| 58540065 Business Economics Revision and Practice 1 Tran Vu Trung1
School of Economics and International Business
Foreign Trade University May 2025
Question 1: Preferences and Utility
Suppose that preferences of Antony (denoted by the subscript A) over the consumption
of two goods, denoted good 1 and good 2, take the form:
with α + β = 1,α,β > 0.
1. Derive the Marshallian demands for Antony
2. Derive the indirect utility function vA(p1,p2,I). Show that it is homogeneous of
degree zero in prices and income. Verify the Roy’s Identity.
3. Derive the Hicksian demands hAi (p1,p2,U¯), i = 1,2.
4. Derive the expenditure function eA(p1,p2,U¯). Show that it is homogeneous of
degree 1 in prices. Verify the Shephard’s Lemma.
5. Using your results from previous parts, verify the Slutsky equation.
6. (BONUS) This type of exercise is not examined but it’s worth trying at home. If the
utility form is changed as follows:
uA(x1,x2) = min{αx1,βx2} with α,β ≥ 1.
Derive the Marshallian demands for Andy xAi (p1,p2,I), i = 1,2. Are the goods
Marshallian complements or substitutes?
Derive the indirect utility function vA(p1,p2,I). Show that it is homogeneous of
degree zero in prices and income.
Derive the expenditure function e(p1,p2,U¯). Show that it is homogeneous of degree 1 in prices.
Derive the Hicksian demands hAi (p1,p2,U¯), i = 1,2. Are the goods Hicksian complements or substitutes? 1
1 Instructor’s email: trungvt@ftu.edu.vn lOMoAR cPSD| 58540065
Question 2: Production and Profit Maximization
The production function is f(x) = 20x − x2 and the price of output is normalized to 1. Let
w be the price of the x-input. We must have x ≥ 0.
1. What is the first-order condition for profit maximization if x > 0?
2. For what values of w will the optimal x be zero?
3. For what values of w will the optimal x be 10?
4. What is the factor demand function?
5. What is the profit function?
6. What is the derivative of the profit function with respect to w?