Write down the households utility maximization problem

Question

Econ 3102 Section 003 Problem Set

Problem 1: Elasticity and the Consumption/Savings Decision

Suppose that you have a function f(x). The elasticity of f is defined as

?(x) = d ln f(x)/d ln x

Notice that for any x₁ close to x

?(x) = d ln f(x)/d ln x ≈ ln f(x₁) - ln f(x)/ln x₁ - ln x ≈ % change in f/% change in x

so the definition of elasticity that you are taught in Econ 1101 is actually just an approximation of the definition of ?(x) above.

i) Show that

?(x) = (f′(x)/f(x))x

(HINT: Notice that ln f(x) = ln f(e^{ln x}). Take the derivative of this expression with respect to ln x.)

Now, consider a version of our simple model of the household's savings/consumption decision. The household derives utility from consumption today c and consumption tomorrow c′. Its utility function is given by

u(c, c′) = c^1-σ - 1/1 - σ + β(c′^1-σ - 1/1 - σ)

where σ > 0 and 0 < β < 1. The household earns income today y and income tomorrow y′. The interest rate is 1 + r. The household does not pay taxes.

ii) Show that for any x > 0,

lim_σ→1(x^1-σ - 1/1 - σ) = ln x

(HINT: Note that d/dy x^y = x^y ln x)

iii) Write down the household's utility maximization problem subject to the present value budget constraint. Show that it is optimal for the household to choose c and c′ such that

c′/c = [β(1 + r)]^1/σ                              (1)

Solve for the household's optimal choice of c and c′ in terms of the preference parameters β and σ, income y and y′, and the interest rate 1 + r.

iv) The intertemporal elasticity of substitution is defined as

?_IT = d ln (c′/c)/d ln(1 + r)

This is a measure of how much the household wants to substitute from consumption today into consumption tomorrow in response to changes in the real interest rate (which, recall, determines the relative price of consumption today and consumption tomorrow). Use (1) to show that

?_IT = 1/σ

v) Notice that it follows from (1) that

ln c′/c = 1/σ ln(1 + r) + 1/σ ln β

Now, the left-hand-side of this equation is, approximately, the growth rate of consumption g_c, and ln(1 + r) ≈ r. So this can be rewritten as

g_c = 1/σr + constant

So, if we wanted to measure σ, we could do so by regressing the growth rate of real consumption on the interest rate. I have detrended quarterly real per capita consumption growth and the real interest rate for the U.S. economy from 2003 to 2015 and put the results in the file data.xls on the course Moodle site. Open this file, create a scatter plot with consumption growth on the y-axis and the real interest rate on the xaxis. Be sure to label the axes. Add a trend-line to the graph, and display the equation for the trend line. The x coefficient of the trendline is equal to 1/σ. What value of σ does this exercise imply? Include a copy of your graph with the problem set.

The value of σ from v.) is one half of what is called the Equity Premium Puzzle. What is the Equity Premium Puzzle? Well, the return from holding stocks (equity) is higher on average than the risk-free return of holding U.S. government debt. This is called the equity premium. The reason that the premium exists is that there is some risk to holding stocks (if the price of stocks falls, which happens sometimes, you get a lower return on your investment). The higher return compensates investors for the risk they take by putting their money in stocks. How much higher should this return be? That depends on how much people dislike risk and how much risk there is in investing in stocks. You can measure the amount of risk associated with investing in stocks by computing the variance of stock returns. Now, how much do people dislike risk? It turns out that the parameter σ also determines how much households in the model dislike risk, with higher σ implying people dislike risk more. The estimate of σ you get from v.) is too low to justify the size of the equity premium we observe, given the measured risk of holding stocks. In fact, it is way too low. It would have to be about ten times higher. So, to recap, there is a large premium for holding stocks instead of U.S. government debt, and the risk of investing in stocks is fairly low. This implies that people should hate risk. But if you try to estimate how much people dislike risk, you find that they just don't care about risk that much. This is the Equity Premium Puzzle, one of the great unsolved problems of modern economics.

Aggregate Supply and Aggregate Demand Shifts

Carefully explain, using graphs, the effects of the following shocks on first period aggregate supply and aggregate demand in our two period macro model. For each shock, what happens to first period consumption, first period investment, first period employment, the first wage, and the first period real interest rate (NOTE: Some shocks may have ambiguous effects).

i) An increase in government spending in the first period, G

ii) An increase in government spending in the second period, G′

iii) An increase in productivity in the first period, z

iv) An increase in productivity in the second period, z

v) An increase in the capital stock, K

vi) An increase in current taxes T.

Attachment:- Data.rar