ECON 445
Spring 2007
Professor Paul Rothstein
Problem Set 1
Due February 6 at the start of class
Worth 32 points (2 points per question)
Show all of your work!
Problem 1 (2 points)
Consider a firm which plans to maximize its profit. It faces the following (inverse)
demand function:
P(Q)=100 −Q
2
Its total cost function is:
TC(Q)=2Q2
1. What is its profit maximizing quantity of output, and for what price does it sell
the output?
Problem 2 (10 points)
Suppose that you derive utility from consuming two commodities Xand Y.Your
utility function is:
U(X, Y )=X0.5+Y
Suppose the price of Xis 1 dollar, the price of Yis 4 dollars, and your income is 24
dollars. So your budget constraint is:
X+4Y=24
Assume you need to allocate your income to maximize your utility.
2. Write down the LaGrangean and the three first order conditions. Denote the
Lagrange multiplier for the budget constraint as λ.
3. Solve explicitly (meaning, in this case, numbers) for the solution to the first
order conditions: X∗,Y∗,andλ∗.
4. Compute the value of the objective function at the solution.
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