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## ECON 100C University of California Irvine Numerical Economics Worksheet

### Question Description

Instructions

For this assignment, you will have to use spreadsheet software (such as Excel). To get full credit, please submit your all of your answers in one single spreadsheet file. In order to write text in a spreadsheet, you can use “Insert Text.”

Make sure to read carefully all of the instructions.

Goals

• Being able to plot the IS curve in Excel
• Being able to use the spreadsheet to study the impact of different changes to parameters (“comparative statics”) or changes to the model assumptions

Description of the economy

Consider an economy identical to that described in the lecture. Potential output

$\stackrel{}{Y}$

¯is given by the production model, thus

$\stackrel{}{Y}$

¯ = A ¯ K ¯ α L ¯ 1 α, where

$\stackrel{}{K}$

¯ is the supply of capital stock and

$\stackrel{}{L}$

¯ is the labor force. Short-run output is given by

$\stackrel{}{Y}$

~ = Y Y ¯ Y ¯, where

$Y$

is actual output. Consumption is given by

$C$

= a ¯ c Y ¯ + x ¯ ( Y Y ¯ ). Investment is given by

$I$

= a ¯ i Y ¯ b ¯ ( R r ¯ ) Y ¯, where

$R$

is the real short-term interest rate and

$\stackrel{}{r}$

¯ is the real long-run interest rate (which is equal to the marginal product of capital, MPK). Government purchases are given by

$G$

= a ¯ g Y ¯. The economy is closed, so there are no imports and no exports. The national accounting identity tells us that

$Y$

= C + I + G.

For now, assume that the parameters are as follows:

• $\stackrel{}{A}$

¯ = 1

• $\stackrel{}{L}$

¯ = 1

• $\stackrel{}{K}$

¯ = 1

• $\alpha$

= 0.3

• ${\stackrel{}{a}}_{}$

¯ c = 0.2

• ${\stackrel{}{a}}_{}$

¯ g = 0.5

• ${\stackrel{}{a}}_{}$

¯ i = 0.3

• $\stackrel{}{x}$

¯ = 0.3

• $\stackrel{}{b}$

¯ = 10

Questions

(a) 2 points. Input the parameter values in Excel (we will call this set of parameters “Baseline”). Add a row for the aggregate demand shock,

$a$

¯ = a ¯ c + a ¯ i + a ¯ g 1 (use a formula, so that it will be automatically updated if you change any of the parameters later on).

(b) 2 points. Compute the equilibrium long-run real interest rate,

$r$

¯, using the formula

$r$

¯ = M P K (also have Excel make the computation, so that it will be automatically updated if we change parameters). Your spreadsheet should look like

(c) 4 points. In the lecture notes, we derived the formula for the IS curve, which gives the relation between the short-run real interest rate

$R$

and the short-run output

$Y$

~ :

$Y$

~ = a ¯ b ¯ ( R r ¯ ) 1 x ¯ ( 1 )

We are now going to plot it. To do so we will need two columns: one column with values for the interest rate (in increments of 0.01 from 0 to 0.5), and another column with the corresponding values for the short-run output. Then we will be able to create a scatter plot with the column for short-run output on the x-axis and the column for short-run interest rate on the y-axis.

If you need help, you can follow these steps:

• Create a column for R. Input the first two values, R=0 and R=0.01, by hand. Select the two cells you just filled in and use the dropdown technique to fill in the cells below up to 0.5. You should get a column that looks like {0, 0.01, 0.02, 0.03,.., 0.5}.
• In the next column, compute
$Y$

~, using formula (1). Note that when you enter a formula, if you would like one of the cells that the formula refers to to remain the same when you use the dropdown method (for example, you always want the formula to contain the value from B6, you don’t want the formula to be updated to B7, B8, etc.), you can use \$ signs around the letter (using the previous example, you would write \$B\$6 instead of B6).

• To create the scatter plot, use the same method as you used in the Week 3 numerical assignment when you had to plot the Beveridge curve. Recall that we want R on the y-axis and
$Y$

~ on the x-axis. Name this plot “Baseline IS curve.” Add axes titles and a legend to your graph.

By now your graph should look like this:

(d) 2 points. Let’s do a few checks to make sure that the graph is correct. First, you expect the curve to be downwards-sloping (a higher interest rate should correspond to a lower short-run output, because it discourages investment and investment increases output). Is that the case? Second, because the aggregate shock so far is equal to

$a$

¯ = 0, we expect actual output to be equal to potential output (so short-run output equals 0) when the short-run interest rate,

$R$

, is exactly equal to the long-run interest rate,

$r$

¯. In your graph, what is the value of

$R$

when

$Y$

~ = 0? Does it correspond to what we expect? (if not, you should fix your spreadsheet because something is wrong).

(e) 4 points. In the column next to the one you used to input your original parameter values, let’s input a new set of parameter values. Increase “>

$a$

¯ g by 0.8, so that your new government demand shock is

$a$

¯ g = 1.3. This means that the government is spending 0.8 additional units of government purchases per unit of potential output. Let’s leave all of the other values identical to the original ones. Repeat the steps described in (c) to plot the new IS curve. Plot it on the same graph as the baseline IS curve, so that we can see the two curves together. Name the new one “IS curve with more government spending.”

(f) 2 points. How does the new IS curve compare to the original one? According to the baseline IS curve, a short-run interest rate of 0.4 would correspond to a short-run output of _____? According to the new IS curve, a short-run interest rate of 0.4 would correspond to a short-run output of ____? What does it mean for short-run output to be negative?

(g) 2 points. We learned in the lecture that when the government spends 1 extra unit (that is, when

$a$

¯ g, the government demand shocks, goes up by one), it increases short-run output by

$1$

1 x ¯, which is greater than 1. This is because the unit spent directly increases output by 1 through

$G$

, but then also indirectly increases consumption. Indeed, the increase in

$G$

leads to an increase in income by 1. This increase in income is going to be partially spent by households(a fraction

$x$

¯ of the extra income is spent). This extra spending in turn increases income, and again a fraction

$x$

¯ is spent by households, etc… Overall the total increase in output is

$1$

+ x ¯ + x ¯ 2 + . . . . . . + x ¯ = 1 1 + x ¯. If the government spends 0.8 extra units originally (instead of 1), then the total increase in output will be

$0.8$

+ 0.8 x ¯ + 0.8 x ¯ 2 + . . . + 0.8 x ¯ = 0.8 1 x ¯. This phenomenon is called the government-purchase multiplier. Using this formula, compute the total increase in output that must have followed the increase in government spending from

$a$

¯ g = 0.5to

$a$

¯ g = 1.3.

Note: You can check your answer using the numbers calculated fro questions (c) and (e). If you pick any column (any R), and compute the difference between the

$Y$

~ computed with the baseline and the

$Y$

~ computed with after the government spending increase, it should be exactly equal to what you computed using the formula in (g).

(h) 4 points. Now assume that there is a natural disaster and half of the original capital stock,

$K$

¯, is destroyed. Create a new set of parameter values to take this change into account. All of the other parameters are equal to their baseline values. Plot the new IS curve on the same graph as the previous two. Name this curve “IS curve after capital destruction.”

(i) 3 points. How does the new IS curve compare to the baseline IS curve? If you look at the equation for the IS curve, (1), you can see that it does not include

$K$

¯. However, we know that a curve will shift if and only if one of the parameters present in the equation (but not present on the axes), changes. Then, why did the IS curve shift?

Hint: Check which of the parameters present in the IS curve formula changed between the baseline and the new parameter values.

(j) 5 points. Now we will change the equation for consumption, so that

$C$

= a ¯ c Y ¯ + x ¯ ( Y Y ¯ ) + b ¯ c ( R r ¯ ) Y ¯. The extra term implies that consumption now depends on the difference between the short-run interest rate and the long-run interest rate, and it does so positively (households consume more when the short-run interest rate is higher, for example because they are lenders so higher interest rates is a good thing for them). The resulting IS curve is

$Y$

~ = a ¯ ( b ¯ b ¯ c ) ( R r ¯ ) 1 x ¯(I suggest you practice and check that you can get this equation by yourself).

Using the same steps as described before, plot this new IS curve, assuming

$b$

¯ c = 20, and using the baseline parameter values for the rest. What can you say about this new IS curve? How can you explain its shape?

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