1. First we need to calculate the present value of the offer in Method 2. This takes a little time but is not hard. It is:
68,081.2 = 100,000/(1.05)t
which is one equation in one unknown, t. To solve it, isolate the (1.05)t on the left hand side. This yields
(1.05)t = 100,000/68,081.2 = 1.468834
We can solve for t by increasing t until we find the answer. Consider
the following table:
| t | (1.05)t | 100,000/(1.05)t |
| 1 | 1.05 | 95,238.10 |
| 2 | 1.1025 | 90,702.95 |
| 3 | 1.157625 | 86,383.76 |
| 4 | 1.21550625 | 82,270.25 |
| 5 | 1,276281563 | 78,352.62 |
| 6 | 1.340095641 | 74,621.54 |
| 7 | 1.407100423 | 71,068.13 |
| 8 | 1.477455444 | 67,683.94 |
So the answer is: Unless you expect your parents to die within the next 7 years, you are better off to take Method 2 (it's present value is 68,081.2 which is bigger than the expected value of the 100,000 in 8 years or later.)
2. a. With an interest rate of 0%, the expected value is $300,000
b. With an interest rate of 5%, the expected value is a bit harder.
It is given by:
100,000 + 100,000/(1.05) + 100,000/(1.05)2
100,000 + 95,238.10 + 90,702.95 = $285,941.05
c. With an interest rate of 10%, the expected value is given by:
100,000 + 100,000/(1.10) + 100,000/(1.10)2
100,000 + 90,909.09 + 82,644.63 = $273,553.72
[Note: the money in question 1 comes at the end of the year, and the money in question 2 comes now. This is the reason why the first term is discounted in problem 1 and not discounted in problem 2.]
3. The first thing we need to do is to determine what happens to the LM curve in this case. Consider Figure 1 which shows what happens to the LM curve if p e moves from 5% to 2.5%. Point A is the point on the p e = 5% LM curve which goes with a nominal interest rate of 10% and a real interest rate of 5%. Income YA is the income at which money supply equals money demand when the nominal interest rate is 10%. Now if the expectations of inflation are reduced to p e = 2.5%, the income which goes with money supply and money demand equilibrium and a 10% nominal interest rate does not change, but the real interest rate which goes with that nominal interest rate does change, to 7.5%. This means that the LM curve in the r,Y plane shifts up. Both point A and point A' are, given p e, associated with i = 10%.
Now we need to include the IS curve. The IS curve in the r,Y plane is
unaffected by changes in p e. Figure
2 gives the results. The LM curve shifts up and the equilibrium moves from
point A to point B. The results now are mostly quite easy:
C - declines because Y declines and C = C(Y-T);
I - declines because Y declines and r increases and I = I(Y,r);
Y - declines as the graph indicates;
r - increases as the graph indicates, and
i - decreases. To convince yourself of this consider what would happen
if the IS curve were vertical. Then, assuming that point A in Figure 1
to be the original equilibrium, point A' in Figure 1 would be a representation
of the new equilibrium. In that case the increase in r would be sufficient
to keep the nominal interest rate at 10%. Before the change i = 10% (made
up of r = 5% and p e = 5%) and after
the equilibrium, i = 10% (r = 7.5% and p e
= 2.5%). But the IS curve is not vertical, and the real interest
rate does not go up by 2.5% it goes up less. Hence the nominal interest
rate goes down.
4. The consumption level would go up if there are any people who wanted to finance consumption but had been unable to obtain loans. There clearly would be a short run boost to consumption (basically because of an increase in loans to the relatively young.) The marginal propensity to consume would decrease, i.e. the sensitivity of consumption to changes in current income would decrease. Peoples' consumption would no longer be be as severely affected by changes in current income.
5. The machine should be purchased if the present value of profits is greater than the cost of the machine. The formula for the present value, given the depreciation rate of 10 percent and an interest rate of r is given by:
10,000 { 1/[1-.9/(1+r)] }
a. 10,000 { 1/[1-.9/(1.05)]}
10,000 { 1/(1-.8571428570 }
10,000 { 7 } = 70,000 which exceeds the purchase price of 50,000, so
BUY.
b. 10,000 { 1/[1-[.9/1.10] }
10,000 { 1/(1-.81818181)}
10,000 [ 5.5 ] = 55,000 which exceeds the purchase price of 50,000,
so BUY
c. 10,000 { 1/[1-[.9/1.15] }
10,000 { 1/(1-.782608696)}
10,000 [ 4.6 ] = 46,000 which is below the purchase price of 50,000,
so DO NOT BUY
6. With r= 4 percent, the present discounted value of $20,000 in 30
years is 20,000/(1.04)30= 20,000/3.43 = 6,167.13.
The investor should sell the wine now, since its current price of 7,000
is greater than the present discounted value of its expected future price.
7. A discount bond offers just one payment at maturity. The general formula relating a discount bond's price ($P), face value ($F), and yield (i), given maturity in n periods is:
$P = $F/(1+i)n, so the formula for determining interest is
i = ($F/$P)1/n - 1
a. i = (10,000/8,000)1/2 - 1
1.118 -1 = .118 or 11.8%
b. i = (10,000/900)1/2 - 1
1.054 - 1 = .054 or 5.4%
8. The general formula for yield to maturity of an n-year bond is:
int
. (1/n)(i1t + ie1t+1 + ie
1t+2 + ... )
i.e. it is the average of the expected one year rates in the future.
a. i1t = 4%
b. i2t = (1/2)(.04 + .05) = .045 or 4.5%
c. i3t = (1/3)(.04 + .05 + .06) = .05 or 5%
9. Mathematically, this problem is very similar to number 5 (about the pretzel machine). The price of a share of stock is the present value of the stream of dividends it is expected to pay into the future, i.e., assuming a constant interest rate, r,
Qt = Dt/(1+r) + Dt+1/(1+r)2 + Dt+2/(1+r)3 + Dt+3/(1+r)4+ ...
a. If r=.05 and dividends start out at 100 and grow by 3 percent a year then
Q = 100/1.05 + 100(1.03)/(1.05)2 + 100(1.03)2/(1.05)3
+ 100(1.03)3/(1.05)4 + ...
= 100/1.05 { 1 + [1.03/1.05] + [1.03/1.05]2
+ [1.03/1.05]3 + ... }
Now we have another infinite geometric series, and we know the sum.
Q = 100/1.05 [1/(1 - 1.03/1.05)] = $5,000
b. We can simply substitute .10 for the .05 in the formula above, yielding
Q = 100/1.10 [1/(1 - 1.03/1.10)] = $1,429
10. a. The election of this president will cause future taxes to decrease. This will cause an increase in private spending (consumption), and a rightward shift in the current IS curve. This, in turn, will cause an increase in current output and - because the Fed does nothing - an increase in future output as well, tending to increase private spending still further.
The rightward shift in the IS curve, however will raise the current and expected future interest rate. This will tend to decrease private spending somewhat, through its effect on human and nonhuman wealth and also on investment, but it cannot reverse the increase in private spending. (If it did, the IS curve would not shift rightward in the first place, and thus, the interest rate would not rise.)
b. The decrease in future taxes will initially cause private spending to increase and shift the current IS curve rightward. Ordinarily, this would raise current and future output, but here the Fed is assumed to prevent this, with a contractionary monetary policy that shifts the LM curve upward. As a result, current output does not change, and current and future interest rates rise by even more than in a. above.
The effect on private spending might seem, at first glance, to be ambiguous: Future taxes have decreased, tending to increase private spending, but the interest rate has increased, tending to decrease private spending. However, we know that the Fed is acting to keep output unchanged, and that in equilibrium, output and total spending (private plus government) are equal. Since there was no change in government spending, and no change in output, we know there can be no change in private spending either.
c. Once again, The decrease in future taxes will increase private spending and shift the current IS curve rightward. Ordinarily, this would raise current and future interest rates. but his time the Fed is assumed to prevent any change in interest rates - by shifting the LM curve downward. This will cause current output to increase even more, and future output as well causing a further rightward shift in the IS curve. Since future taxes have decreased, and current and future output have increased, with no change in the current or future interest rate, private spending must increase.