# In public economics

In public economics, we are often interested in transfer programs that maximize some social
welfare function. For the current exercise, we will be interested in maximizing a utilitarian
social welfare function, which is equivalent to maximizing the utility of a randomly chosen
individual.
Suppose a random process assigns individuals different characteristics that affect their
ability to earn income. If these random characteristics were observable, the government could
use these characteristics as a basis for lump sum transfers and achieve the first best outcome
(see the discussion in the course textbook at the beginning of section 17.4). In practice,
however, we can only observe rough indicators of these characteristics, such as income. The
challenge is to design the best transfer program based only on these indicators, yielding what
we call the “second best” solution.
In this problem, we will consider a world where people choose to spend some of their
time in leisure / and the remainder of their time (1 -/) to earn an income Y (the maximum
available leisure has been normalized to be 1). We will assume that all individuals in our
society have a common utility function U(C, /) with consumption and leisure as arguments.
If a person receives a wage w, income and leisure are related by Y = (1-[) – w. Consumption
is equal to earned income Y plus any government transfers , so that C = Y + T. Thus,
the utility function can be written simply
U(C, D) = U(Y +T, 1-Y/w)
The particular form of the utility function has intentionally been left unspecified, but in
the following problems you should draw your plots for a well-behaved utility function that
features a believable trade-off between consumption and leisure. It is sufficient to solve
everything drawing graphs only-no need for a mathematical representation of the utility
function.
1. Plotting the utility function for a single individual.
(a) Suppose there are no government transfers (7 = 0). Plot utility vs. income
for a given individual (i.e. U on the y-axis, and Y on the x-axis). Discuss how
the shape of the utility curve relates to decreasing returns and to the resulting
trade-off between consumption and leisure. Mark the individual’s optimal choice
of earned income. Problem 7.22.
The one-year LIBOR rate is 10% with annual compounding. A bank trades swaps
where a fixed rate of interest is exchanged for 12-month LIBOR with payments being
exchanged annually. Two- and three-year swap rates (expressed with annual
compounding) are 11% and 12% per annum. Estimate the two- and three-year LIBOR
zero rates when LIBOR discounting is used. 38.
The current price of a medical company’s stock is 75. The expected value of the stock
price in three years is 90 per share. The stock pays no dividends.
You are also given
The risk-free interest rate is positive.
ii)
There are no transaction costs.
ifi)
Investors require compensation for risk.
The price of a three-year forward on a share of this stock is X, and at this price an
investor is willing to enter into the forward.
Determine what can be concluded about X.
(A) X <75 (B) X = 75 (C) 75 < X <90 (D) X = 90 (E) 90 < X 14) You are given the following information about Stock X, Stock Y, and the market: (i) The annual effective risk-free rate is 4%. (ii) The expected return and volatility for Stock X, Stock Y, and the market are shown in the table below: Expected Return Volatility Stock X 5.5% 40% Stock Y 4.5% 35% Market 6.0% 25% (iii) The correlation between the returns of stock X and the market is -0.25. (iv) The correlation between the returns of stock Y and the market is 0.30. Assume the Capital Asset Pricing Model holds. Calculate the required returns for Stock X and Stock Y, and determine which of the two stocks an investor should choose. (A) The required return for Stock X is 3.20%, the required return for Stock Y is 4.84%, and the investor should choose Stock X. (B) The required return for Stock X is 3.20%, the required return for Stock Y is 4.84%, and the investor should choose Stock Y. (C) The required return for Stock X is 4.80%, the required return for Stock Y is 4.84%, and the investor should choose Stock X. (D) The required return for Stock X is 6.40%, the required return for Stock Y is 3.16%, and the investor should choose Stock Y. (E) The required return for Stock X is 3.50%, the required return for Stock Y is 3.16%, and the investor should choose both Stock X and Stock Y. 38) An insurance company has a variable annuity linked to the S&P 500 index. A guaranteed minimum death benefit (GMDB) specifies the beneficiary will receive the greater of the account value and the original amount invested, if the policyholder dies within the first three years of the annuity contract. If the policyholder dies after three years, the beneficiary will receive the account value. Out of every 1000 policies sold, the company expects 10 deaths in each of years one, two, and three. Thus they also expect that 970 will survive the first three years. Assume the deaths occur at the end of the year. You are given the following at-the-money European call and put option prices, expressed as a percentage of the current value of the S&P 500 index. Duration Call Price Put Price (years) 18.7% 15.8% 2 26.2% 20.6% 3 31.6% 23.4% Calculate the expected value of the guarantee when the annuity is sold, expressed as a percentage of the original amount invested. (A) 0.23% (B) 0.32% (C) 0.52% (D) 0.60% (E) 0.76% 3. Solve for the new equilibrium allocation of labor, /*, the equilibrium before-tax wage w*, the equilibrium after-tax wage w*, and the competitive price of capital r*. How do these relate to your solution in part 1? Why?