Problem Set 2(Suggested Solution): Spring 1998

**1.** a) Z = C + I + G = 100 +0.6(Y-100) + 50 + 250= 340 +0.6Y

In equilibrium, Y=Z, so Y= (1/(1-0.6))*(340)= 2.5*340= 850.

In equilibrium, Y=850, C=550, Private Saving = Y_d-C = 200,
Public Saving = (T-G) = 100-250 = -150

Multiplier= 1/(1-0.6)= 2.5

b) Change in Y= Multiplier* Change in autonomous spending.

So Y would go down by 2.5*100= 250, so Y = 600.

Consumption = 400, Private Saving = 100, Public Saving = -50

The multiplier remains the same

c) The marginal propensity to consume(MPC) is higher, 0.8. This implies a larger multiplier 1/(1-0.8)= 5, therefore the equilibrium level of output will be higher ( in (a) Y=1600, in (b) Y= 1100). With a larger MPC the demand feedback effects are larger, therefore aggregate production ends up being larger too.

d) Demand = C + I + G = 550 + 50 + 250 = 850 = Production

Savings= Pr_S + Pub_S = 200 - 150 = 50 = Investment

e) Change in GDP = Multiplier * Change in autonomous spending.

Given a multiplier of 2.5, then
change in government spending = 200/2.5 = 80, so G=330.

If the MPC is 0.8 as in c) then

change in government spending = 200/5 = 40.

If G cannot change, the required change in autonomous spending would
still be the same, that change translated into tax changes would be:

Change autonomous spending= -c_1 *Change in taxes.

Change in taxes= - 80/0.6 = -133.33.

Taxes would have to be reduced by
133.33. Since T=100, this implies a positive transfer of 33.33.
If MPC=0.8, then taxes should be cut by 50, then T=50.

**2.** The main implication from proportional taxation is a change in the
multiplier.

C= c_o+c_1(Y-tY)= c_o+c_1(1-t)*Y; 0<t<1

multiplier = 1/(1-c_1(1-t)) < 1/(1-c_1)

The multiplier would be smaller now. Given a certain increase in
income, the increase in consumption, and therefore in demand, will be
smaller, because for every increase of 1 unit in income, disposable
income only increase (1-t). Before an increase in income translated
into a one-for-one increase in disposable income.

**3.** BS=tY-G

a) An increase in business confidence would more than likely increase the level of investment by firms, it might also increase consumer confidence. Autonomous spending would go up, therefore given the level of government spending, the budget surplus would increase. Output increases, income increases by the same amount, and tax revenues increase by t*increase in income.

b) An increase in government spending has a direct negative effect on the BS, but then it has a positive effect through tax revenues (Y goes up when G go up). What is the final answer?

dBS/dG = t*(dY/dG) - 1 = t*multiplier - 1

----- = t/[1-c_1(1-t)] - 1

----- = [ t(1-c_1) - (1-c_1)] / [(1-c_1(1-t)]

----- < 0

So an increase in G has a negative on the budget surplus, that is, the government deficit goes up.

c) A decrease in consumer confidence would translate into a reduction of c_o and possibly c_1, both effects would imply a reduction in output and income and therefore in the budget surplus

d) The value of the multiplier in this case is 1. Let's see why: A balance budge t means G=T, therefore dG=dT.

Y = C+ I + G = c_o + c_1*(Y-T) + I + G

dY = c_1*(dY-dT) + dG

dY = (1/1-c_1)[dG - c_1*dT]

-- = [dG - c_1*dG] / (1-c_1)

-- = dG

Therefore an increase in government purchases accompanied by an increase in taxe such that in the new equilibrium the Budget Surplus is exactly the same an in the original equilibrium will result in an increase in output equal to the increase in government purchases. There is no induced increase in consumption spending, higher taxes to balance the budget offset the income expansion.

**4.**
a) Answer will vary by individuals.

b) The two natural restrictions in the text are that the marginal propensity to consume is positive, and less than one.

**5.**
a) False. We have seen that there is a multiplier, and it has the
same meaning and same mathematical formula as in the simpler models
of Chapter 3.

b) False. In the models of Chapter 4,firms take a "wait and see" attitude before changing production. Thus, they respond to changes in demand by changing their inventories. See footnote 4 in the chapter.

c) False. A simulation is one way to calculate the new equilibrium GDP, but it is unnecessarily tedious. A better way is to use the multiplier for the model.

**6.**
a) In equilibrium, with constant Y, Y_t+1 is the same as Y_t, so we
can ignore the time subscripts and solve this model just as in chapter
3.

Thus, Y = 402.

b) The table below summarizes the change in C, Z, and Y as a result of reduction in G by 100.

Year | t | t+1 | t+2 |
---|---|---|---|

Change in C | 0 | -75 | -56.25 |

Change in Z | -100 | -75 | -56.25 |

Change in Y | 0 | -100 | -75 |

c) Graph.

d) The multiplier is 1/(1-.75) = 4, so the 100 fall in G should cause
a 400 drop in equilibrium Y, to 2.

In equilibrium Y_t= Z_t, so demand will also be 2.

C = .5+.75(Y_t-Y) = 50+ .75(200-100) = -73

[Please note that there is a typo in the question. The autonomous consumption (c_o) is meant to be 50 and not 0.5. As a result of this mistake, we thus ended up with a negative number for consumption which is not realistic. In any case, we assume that each of you use 0.5 as the value foe c_0, and thus, the answer here is -73.]

e) The change in output during the first five periods (T+1 to t+5) is -100-75-56.25-42.19-31.64 = -305. This is slightly more than 75% of the total change in output (-400). thus, the answer is at least five periods.