Economics 164 assignment for each of the following

Question

For problems 1-4, use Excel to compute a times series consisting of 50 periods. Create a graph of your computed time series.

1. a. Suppose x_t+1 = 2x_t - 1 and x₀ = 1.

b. Same equation, with x₀ = 1.5.

c. Same equation, with x₀ = .5.

Something to think about: If this equation is interpreted as governing the behavior of a price of something over time, what value of x₀ makes economic sense? Is there more than one initial value that makes sense?

2. a. x_t+1 = .95x_t, x₀ = .01

b. x_t+1 = 1.2x_t, x₀ = .01

3. x_t+1 = 2 - x_t, try a couple of values for x₀ > 0.

4. a. x_t+1 = 3.6x_t(1 - x_t), x₀ = .5.

b. x_t+1 = 2.6x_t(1 - x_t), x₀ = .5.

5. For each of the following difference equations, compute the steady state value(s) for x_t. For each steady state, explain whether it is stable or unstable.

a. x_t+1 = .95x_t + 5

b. x_t+1 = 1.2x_t - .1

c. x_t+1 = 2x_t^.5

d. x_t+1 = x_t

e. x_t+1 = 4x_t

6. The following difference equation will turn out to be important for us later in the course.

s_t+1 = 1+ βθ - (βθ/s_t)

a. For now, let's not worry about what β and θ are, but we do need that 0 < βθ < 1. In fact, assume that βθ = 0.2. Find the steady state value(s) of s_t and explain in each case whether the steady state is stable or unstable.

b. It will turn out that s_t is the savings rate (that fraction of income that is not consumed) in a model with optimally chosen savings. What value of s₀ makes economic sense in this case? Is there more than one value that makes sense? Defend your answer.

7. The growth rate of a variable x can be expressed, (x_t+1 - x_t/x_t) = (x_t+1/x_t) - 1. Often times this will be approximated with log_t+1 - logx_t, where log is the natural logarithm.

a. Present the mathematical argument that justifies using this approximation. Under what conditions is this a good approximation and when is it a bad one?

b. Compute the true growth rate and the approximate one for the following examples:

i. x_t = .01 and x_t+1 = .02

ii. x_t = .01 and x_t+1 =  .011

iii. x_t = .01 and  x_t+1 = .0101

c. Suppose that x_t is a variable that grows 10 percent every period and that x₀ = 1.

i. How many periods does it take for x to double? Triple?

ii. Use Excel to plot x_t for 50 periods. On a separate graph, plot log x_t. Remember that in Excel, natural log is LN.

iii. How can you use the log graph to approximate how many periods it takes for x to double? Triple? How accurate is the result? Is this surprising? Explain.