Quiz 1

1.  An econometrician estimates that the daily supply of airline seats flying one-way between New York and Washington is:
\(Q_S = 4P^{1.5}\)
What is the elasticity of supply for these seats?
2.  Someone wants to take out a 30-yr. mortgage worth $400,000, and the annual interest rate on the mortgage is going to be 6%. After 30 years, the mortgage will be entirely paid off. What will the annual payments be closest to?
3.  \(f(x) = \ln (x^2)\) What is \(\frac{df(x)}{dx}\)?
4.  In Milton Friedman's article on the quantity theory of money, he notes that \(M_t V_t = P_t Y_t\). Assume V (velocity of money) is fixed and real output (Y) is always falling by 1% each year. If the central bank wants prices to fall by 2% per year every year (a concept known as the Friedman Rule) what annual rate should the central bank grow the money supply at?

Quiz 2

1.  In the very long run, what value will this first order differential equation head toward?
\(\frac{dx}{dt} = 3 - 2x(t)\)
\(x(0) = 0\)
2.  Solve this equation for x(t) if:
\(\frac{dx}{dt} = 0.05x(t)\)
\(x(0) = 10\)
3.  Consider this quite basic optimal control problem:
Maximize \(V = \int\limits _0 ^1 [x(t) - \frac{10}{2} u(t)^2] dt\)
Subject to: \(x(1) = 5\) and \(\lambda (1) = 0\)
4. I suggest treating this as a continuous-time problem:
You start with $1,000 right now in your investment account, and every year you have to pay a $60 fee from it. Every year, your remaining balance grows by 4%. How much will your account approximately be worth in 10 years? In the very long run?

Quiz 3

1.  What's the optimal amount of good 1 and good 1 for a consumer with this utility function and budget constraint:
\(U(c_1, c_2) = (c_1 ^{0.5})(c_2^{0.5})\)
\(100 = c_1 + 3c_2\)
2.  Consider a firm with this Leontief production process: 
\(Y = min(K_1, K_2)\)
The prices of the two types of capital goods are 5 and 10 respectively. What is the cost function of the firm?
3.  Consider this CES (constant elasticity of substitution) production function:
\(Y = (K^{0.5} + L^{0.5})^2\)
Does this function have a higher elasticity of substitution than a Cobb-Douglas?
4.  Consider a competitive, price-taking firm that is trying to maximize its profits and trying to figure out how many workers to hire:
\(TR = 2zL^{0.5}\)
\(TC = wL\)
What is the optimal number of workers to hire?
______________________________________
Answers:
  1. 1.5
  2. $30,000
  3. 2/x
  4. -3%
  5. not sure
  6. not sure
  7. not sure
  8. not sure
  9. c1= 50 and c2= 50/3
  10. TC = 15
  11. No
  12. not sure