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Econ 400
Winter 2014
Midterm Exam
Name:
65 points total. Please answer all four questions. For full credit, remember to
write legibly and show your work.
REMINDER: It is Department of Economics policy that no graphing calculators or phones are to be used during the exam.
Academic integrity is the cornerstone of the Department’s rules for student conduct and evaluation of student learning. Students accused of academic misconduct will be referred directly to the Office of Community Standards and Student Conduct for disciplinary action pursuant to the Student Conduct Code and, if found guilty, will be subject to sanctions. Sanctions range from a disciplinary warning, to academic probation, to immediate dismissal for the Department and the University, depending on the seriousness of the misconduct. Dismissal can be, and has been, applied even for first offenses. Moreover, a grade of zero can be assigned by the instructor for the course.
1. Preferences and Utility (14 points) a. (6 points) Bob enjoys cookies (x) according to the utility function U(x)=20x-tx^2 , where t is a parameter that reflects how hungry he is. Cookies are costless in Bob’s world and so there is no income constraint. Using the envelope theorem, calculate how Bob’s maximum utility from eating cookies varies with t.
b. (8 points) Are the following utility functions quasi-concave? Show why. i) 𝑈𝑈(𝑋𝑋, 𝑌𝑌) = ln(𝑋𝑋) + ln(𝑌𝑌)
ii) 𝑈𝑈(𝑋𝑋, 𝑌𝑌) = min(𝑋𝑋, 𝑌𝑌) (Hint: You can use a diagram or sample values here)
c. (3 points) Calculate the consumer’s income elasticity of demand.
3. Slutsky Decomposition (21 points)
The utility function for an individual is given by 𝑈𝑈(𝑋𝑋, 𝑌𝑌) = 𝑋𝑋.^75 𝑌𝑌.^25. Prices for the two goods are PX and PY respectively, and income is I. The uncompensated demand functions for
the two goods X and Y are:
a. (6 points) Derive the compensated demand function for good X, XC(PX,PY,I).
b. (3 points) For a small increase in the price of X , what is the total change in the quantity demanded of X? Express your answer in terms of price(s) and income.
c. (6 points) Using the Slutsky decomposition, calculate the substitution and income effects for the price change in part (d). Again, express your answer in terms of price(s) and income.
b. (10 points) If the individual has utility over wealth (𝑊𝑊) = −𝑒𝑒 −𝐴𝐴𝐴𝐴^ , show that the premium that this person would pay to avoid a fair gamble of h is independent of initial wealth.
