Reference no: EM131230862
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 xt+1 = 2xt - 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. xt+1 = .95xt, x0 = .01
b. xt+1 = 1.2xt, x0 = .01
3. xt+1 = 2 - xt, try a couple of values for x0 > 0.
4. a. xt+1 = 3.6xt(1 - xt), x0 = .5.
b. xt+1 = 2.6xt(1 - xt), x0 = .5.
5. For each of the following difference equations, compute the steady state value(s) for xt. For each steady state, explain whether it is stable or unstable.
a. xt+1 = .95xt + 5
b. xt+1 = 1.2xt - .1
c. xt+1 = 2xt.5
d. xt+1 = xt
e. xt+1 = 4xt
6. The following difference equation will turn out to be important for us later in the course.
st+1 = 1+ βθ - (βθ/st)
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 st and explain in each case whether the steady state is stable or unstable.
b. It will turn out that st 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, (xt+1 - xt/xt) = (xt+1/xt) - 1. Often times this will be approximated with logt+1 - logxt, 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. xt = .01 and xt+1 = .02
ii. xt = .01 and xt+1 = .011
iii. xt = .01 and xt+1 = .0101
c. Suppose that xt 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 xt for 50 periods. On a separate graph, plot log xt. 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.
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