Econ 100B: Economic Analysis { MacroeconomicsProblem Set #2 { SolutionsDue Date: July 6, 20171. Comparing the output and unemployment gaps shown on slide 26 of lecture 8:(a) Would you characterize the direction of these fluctuations as generally procyclical,countercyclical, or acyclical? Why?The direction of these fluctuations is generally countercyclical because almost all of themovements in the
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**Econ 100B: Economic Analysis { Macroeconomics**

Problem Set #2 { Solutions

Due Date: July 6, 2017

1. Comparing the output and unemployment gaps shown on slide 26 of lecture 8:

(a) Would you characterize the direction of these fluctuations as generally procyclical,

countercyclical, or acyclical? Why?

The direction of these fluctuations is generally countercyclical because almost all of the

movements in the output gap have a corresponding opposite direction of movement

by the unemployment gap.

(b) Many of the relationships in our macroeconomic models are linear; two variables

are related by the equation for a straight line *y *= *mx*+*b*. What is the relationship

between (i) the sign of the slope *m *and (ii) the cyclicality (pro or counter) seen

in a time series? Briefly explain.

If variables *y *and *x *are *procyclical *then the direction of change of one is the same as

the other: if one increases so does the other and if one decreases so does the other.

This would imply that *m > *0. Similarly, if variables *y *and *x *are *countercyclical *then

the direction of change in one is the opposite of that of the other: if one variable

increases the other decreases and if one variable decreases the other increases. This

would imply that *m < *0.

2. Compare and contrast the timing and direction of the fluctuations of *Y *and CPI

inflation (shown on slide 29 of lecture 5) during (i) the recession associated with the

financial crisis just before 2010 and (ii) the recession in the mid 1970s. Read the

final paragraph on page 296 (312) of the first (second) edition of Mishkin to find the

name of the phenomenon depicted in mid-1970 recession: what is the name of this

phenomenon?

In the recent financial crisis the fluctuations of *Y *and CPI inflation show the fluctuation

in CPI inflation to be procyclical with a lag. By contrast, the mid-70s recession shows the

fluctuation in CPI inflation to be countercyclical and coincident. This phenomenon of a

contemporaneous increase in inflation and decrease in aggregate output is referred to as

*stagflation*.

3. Who cares more about short-run models: Classicals or Keynesians? Why?

The Keynesians care more about short-term models. The Classicals see return to equilibrium

as a rapid phenomenon that needs no intervention, and they consequently focus on longterm models. The Keynesians see return to equilibrium as a potentially long phenomenon

and that intervention can help shorten the time it takes to return to equilibrium, and they

consequently focus on short-term models.

1

4. Beginning with the second equation for *Y *in the middle of slide 24 of lecture 6, carry

out the algebra needed to obtain the equation for the IS curve.

Beginning with the IS curve

*Y *= *C *+ *mpc × Y d *+ *I *+ *G *+ *NX - *(*ζC *+ *ζI *+ *ζNX*) *r ; *

we see that the substitution *Y d *= *Y - T *= *Y - T *yields

*Y *= *C *+ *mpc × Y - T* + *I *+ *G *+ *NX - *(*ζC *+ *ζI *+ *ζNX*) *r : *

Collecting terms in *Y *we find that | (1)

(2)*Y *(1 + *mpc*) = *C - mpc × T *+ *I *+ *G *+ *NX - *(*ζC *+ *ζI *+ *ζNX*) *r ; *(3)

which, after dividing both sides by (1 + *mpc*) and moving *-mpc × T *to the end of the

sum of autonomous components, yields the IS curve:

*Y *= | *-* | *r * | (4)

*C *+ *I *+ *G *+ *NX - mpc × T* | (*ζC *+ *ζI *+ *ζNX*)

(1 *- mpc*) | (1 *- mpc*) 5. If the IS curve does not shift when autonomous investment increases by $250 MM and

autonomous taxes increase by $325 MM,

(a) what is the value of the expenditure multiplier?

The expenditure multiplier is given by

1

1 *- mpc*

(5)

where *mpc *is the marginal propensity to consume. Thus, if we know the *mpc *we

know the expenditure multiplier. Given that the IS curve did not shift following the

changes in these autonomous components it follows that these changes have offset

(or cancelled) one another. Referring to Equation (4) we see that this implies that

∆*I - mpc × *∆*T *= 0 *; *(6)

or

$250 MM *- mpc × *$325 MM = 0 *; *(7)

or

*mpc *=

$250 MM

$325 MM = 0*:*769 *: *(8)

and it follows that the expenditure multiplier is

1

1 *- mpc*

=

1

1 *- *0*:*769

= 4*:*33 (9)

2

(b) what is the value of the tax multiplier?

The tax multiplier is also related to the marginal propensity to consume by

*-*

mpc

1 *- mpc*

= *-*3*:*33 (10)

6. In our derivation of the IS curve we made the simple assumption that taxes are completely autonomous: *T *= *T *. A more realistic tax function would be one where taxes

increase with income as described by *T *= *T *+ *tY *. Follow the steps you undertook in

question (4) with this more realistic tax function to derive a new IS-curve equation.

Returning to the IS curve

*Y *= *C *+ *mpc × Y d *+ *I *+ *G *+ *NX - *(*ζC *+ *ζI *+ *ζNX*) *r ; *

but now substituting with *Y d *= *Y - T *= *Y - T - tY *yields | (11)*Y *= *C *+ *mpc × Y - T - tY * + *I *+ *G *+ *NX - *(*ζC *+ *ζI *+ *ζNX*) *r :*

(12)

Collecting terms in *Y *we find that

*Y *[1 *- mpc *(1 *- t*)] = *C - mpc × T *+ *I *+ *G *+ *NX - *(*ζC *+ *ζI *+ *ζNX*) *r ; *(13)

which, after dividing both sides by [1 *- mpc *(1 *- t*)] and moving *-mpc × T *to the end of

the sum of autonomous components, yields the IS curve:

*Y *=

*C *+ *I *+ *G *+ *NX - mpc × T *

[1 *- mpc *(1 *- t*)] *-*

(*ζC *+ *ζI *+ *ζNX*)

[1 *- mpc *(1 *- t*)]*r *(14)

7. Referring to this article on Bloomberg:

(a) What \short-term tool" did the People’s Bank of China use to inject money into

the Chinese economy?

The reverse-repurchase agreement.

(b) Briefly explain how this short-term tool works (i.e. how does using this tool result

in more money in the Chinese economy? (*Some online research may need to be*

done to answer this question.).

In a repurchase agreement one party sells a security to another party and agrees to

repurchase that security at an agreed time in the future and at a agreed price. Typical

times involves are short, such as overnight or a week. A summary of this approach to

managing liquidity can be found on Investopia

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