L04 Practice Problems

The extra practice problems below (with answers) are for topics covered in this lesson. (Throughout the lesson, you'll find practice problems on many pages as well.) Completing these  problems will help you apply the topics covered in the lesson. WHILE THESE PROBLEMS WILL NOT BE GRADED, YOU ARE STRONGLY ENCOURAGED TO COMPLETE ALL THE PROBLEMS AS YOU PROCEED THROUGH THE LESSONS. There will be similar problems in   the graded homework and on the exams. If you have questions about a particular problem below, please use the "General Questions" discussion board in Canvas. Working together is encouraged!

A consumer has a current before-tax income of $100,000 and a future before-tax income or    $140,000. She has no current wealth. Her current taxes are $30,000 and her expected future taxes are $49,000. She wants her current consumption to be equal to her future consumption. The real interest rate is .1 (10%).

1. How much should her current consumption be? (Hint: Write an expression for the present value of her (after tax) lifetime resources. Let x be the value of her current and future consumption. Then the present value of her lifetime consumption is x + x/(1+.1) = 2.1x/1.1. Set this equal to the expression for the present value of lifetime consumption and solve for x.)

2. Given her current desired consumption, will the consumer be a current borrower or a current saver. Explain.

3. Suppose that her after tax income increases by $6300. How much will her current   consumption increase? How will this change affect her current saving or borrowing?

4. Suppose that instead of her current after tax income increasing by $6300, her future after tax income increases by $6300. How much will her current consumption increase? How will this affect her current saving or borrowing?

5. Suppose that everything is as in the original statement of the problem except that now the real interest rate is .25 (25%). What will her current consumption and current saving or borrowing  be now?

 Check the answer.

1. PVLR = 70,000 + 91,000/1.1 so we can write the consumer’s budget constraint as 2.1x/1.1 = 70,000 + 91,000/1.1 and solve for x.

Multiplying through by 1.1 we have 2.1x = 77,000 + 91,000 = 168,000, or

x = 168,000/2.1 = 80,000.

2. This consumer will be a current borrower, since her desired present consumption of 80,000 is greater than her current after tax income of 70,000. She will have to borrow $10,000.

3. If the consumer’s current after-tax income increases by 6300, then her PVLR will increase to 76,300 + 91,000/1.1

So her desired current consumption can be calculated by solving the equation 2.1x/1.1 = 76,300 + 91,000/1.1 for x.

Again, multiply through by 1.1 to get 2.1 x = (1.1)(76,300) + 91,000 =174,930, or x = 174,930/2.1 = 83,300.

Her current borrowing will decrease to 83,300 – 76,300 = 7000.

4. If the consumer’s future after-tax income increases by 6300, then her PVLR will be 70,000 + 97300/1.1

So her desired current consumption can be calculated by solving the equation 2.1x/1.1 = 70,000 + 97,300/1.1 for x.

Again, multiply through by 1.1 to get 2.1 x = (1.1)(70,000) + 97,300 =174,300, or x = 174,300/2.1 = 83,000.

Her current borrowing will increase to 83,000 – 70,000 = 13,000.

5. Then the present value of her lifetime consumption is x + x/(1+.25 = 2.25x/1.25.

PVLR = 70,000 + 91,000/1.25 so we can write the consumer’s budget constraint as 2.25x/1.25 = 70,000 + 91,000/1.25 and solve for x.

Multiplying through by 1.25 we have 2.25x = 87,500 + 91,000 = 178,500, or x = 178,500/2.25 = 79,333.33.

Her current borrowing will be 79,333.33 – 70,0000 = 9333.33.

1. PVLR = 70,000 + 91,000/1.1 so we can write the consumer’s budget constraint as 2.1x/1.1 = 70,000 + 91,000/1.1 and solve for x.

Multiplying through by 1.1 we have 2.1x = 77,000 + 91,000 = 168,000, or x = 168,000/2.1 = 80,000.

2. This consumer will be a current borrower, since her desired present consumption of 80,000 is greater than her current after tax income of 70,000. She will have to borrow $10,000.

3. If the consumer’s current after-tax income increases by 6300, then her PVLR will increase to 76,300 + 91,000/1.1

So her desired current consumption can be calculated by solving the equation 2.1x/1.1 = 76,300 + 91,000/1.1 for x.

Again, multiply through by 1.1 to get 2.1 x = (1.1)(76,300) + 91,000 =174,930, or x = 174,930/2.1 = 83,300.

Her current borrowing will decrease to 83,300 – 76,300 = 7000.

4. If the consumer’s future after-tax income increases by 6300, then her PVLR will be 70,000 + 97300/1.1

So her desired current consumption can be calculated by solving the equation 2.1x/1.1 = 70,000 + 97,300/1.1 for x.

Again, multiply through by 1.1 to get 2.1 x = (1.1)(70,000) + 97,300 =174,300, or x = 174,300/2.1 = 83,000.

Her current borrowing will increase to 83,000 – 70,000 = 13,000.

5. Then the present value of her lifetime consumption is x + x/(1+.25 = 2.25x/1.25.

PVLR = 70,000 + 91,000/1.25 so we can write the consumer’s budget constraint as 2.25x/1.25 = 70,000 + 91,000/1.25 and solve for x.

Multiplying through by 1.25 we have 2.25x = 87,500 + 91,000 = 178,500, or x = 178,500/2.25 = 79,333.33.

Her current borrowing will be 79,333.33 – 70,0000 = 9333.33.

Homer Simpson does not abide by the life cycle theory of consumption. Homer has a “let’s live life like it’s our last day” mentality and thus, he prefers to consume more today, relative to the future.  In particular, Homer prefers to consume exactly twice as much today (c), relative to consumption  next period (cf). Homer’s current income = $250K and his future expected income = $150K. He     has no wealth (neither current nor expected) since he lives like today is his last! Homer faces a     real interest rate of 0.10. Please answer the following questions.

1. State clearly what the slope of any budget line represents, in general, and then refer to this specific case (i.e., what is the slope of Homers budget constraint and what does this mean exactly? Be specific.

2. Solve for Homer’s optimal consumption basket today (C*) and his optimal consumption basket next period (Cf*). Please provide a completely labeled graph depicting these results and label this initial point as C*A . Be sure to label the no lending / no borrowing point = NL/NB).

3. Now Ben Bernanke and the Fed is not happy with the state of the economy (i.e., we are not at NAIRU) and therefore conduct massive amounts of open market purchases and some how     get the real rate of interest all the way down to 0.00 (that is correct, zero percent!). Resolve for Homer’s optimal basket, given the Fed’s expansionary policy and label as point C*B on your    diagram.

4. Did the Fed policy work as in stimulating the economy? That is, did the Fed policy successfully increase consumption, which represents about 70% of the economy (assume the economy is  made up with a bunch of Homers, just like this one!) Be sure to define what the income and substitution effects are and how they play a role in determining whether or not the Fed policy   is successful (as defined by increasing consumption). Also, comment on whether these

income and substitution effects work in the same or opposite direction (i.e., is it a tug of war or do they work in the same direction?) in this particular case.

5. Given the Fed’s behavior (i.e., r = 0), Homer becomes more cautious since he is thinking that  if interest rates are this low, something must be wrong. He grabs an economic textbook and reads about the ‘life cycle theory of consumption’ and along with discussions with his wife Marge, he decides to change his preferences to be consistent with perfect consumption smoothing, just like Dagwood was in our HW problem. Resolve for Homers optimal basket and label as point C*C on a NEW diagram. Be sure to label the no lending / no borrowing point =    NL/NB.

6. Now Ben Bernanke is monitoring the economic situation, given that he and the Fed lowered rates and sees that consumption has gone down. Why exactly has consumption gone down (use the terms saver and borrower in your answer)?

7. Given the fall in consumption, Ben and the Fed expand again and get real interest rates all the way down to -0.05 (negative 5%, but use decimals as we always do). Resolve for Homer’s new optimal consumption point and label as point C*D on the ‘new’ diagram above.

8. Did this latest expansionary monetary policy work in term of increasing consumption (assume there are a bunch of Homers that have changed preferences like this Homer did)? Why or why not, explain using the income and substitution effects. If you were advising Ben Bernanke and the Fed, what would you tell him to do to raise consumption and why?

 Check the answer.

1. Cf = [ (1 + r)(y + a) + yf + af] – (1 + r) C

Intercept

Slope

Slope represents trade – off or relative price of current consumption in terms of future consumption – in this case, if Homer consumes one more unit of current consumption, he is    giving up 1 + .1 units of future consumption- so 1 + r units of future consumption is the price of (one unit) current consumption.

2. C = 265.6 Cf = 132.8

3. C = 266.6 Cf = 133.33                   see diagram for question 2 above

4. It did increase consumption since the income and substitution effects work in the same         direction since Homer is a borrower (his income is 250K and his current consumption is        256.6K). Note that since real rates are falling, the price of current consumption, 1 plus r units of future consumption, falls and we (rationally) substitute away from future consumption        towards current consumption. We know this of course as the substitution effect. The income effect of a change in the real rate of interest depends critically on whether the consumer is a

borrower or a saver. Since Homer is a borrower, he likes the lower rates since he is financing current consumption with future resources. In other words, he really cares about the present value of future resources (since he is a borrower) and with lower rates, the present value of   future resources rises. Homer is wealthier causing him to consume more. Therefore, the       substitution effect and income effect work in the same direction.

5. C = 200 Cf = 200

6. Consumption has gone down since Homer has changed his preferences and went from being a borrower to being a saver. Of course this change in behavior, moving from being borrowers to being savers makes the Fed's job tougher if they are trying to stimulate the economy by     lowering rates. See below.

7. C = 198.7 Cf = 198.7

see diagram for question 5 above

8. Since Homer is now a saver, the lower rates cause him to spend less. The substitution effect, as we know, works to stimulate current consumption since lower rates lower the price of         current consumption in terms of future consumption (same as before). The income effect in    this case works in the opposite direction. Since Homer is a saver, the lower rates increase the cost of financing future consumption, as we know savers do. Put differently, since Homer is    now a saver, he cares about the future value of present resources. Lower rates lower the        future value of present resources and thus, Homer is poorer. The income effect dominates      here as current consumption falls. If Bernanke wanted to stimulate economy that consists of a bunch of savers, raise rates!!!!!

1. Cf = [ (1 + r)(y + a) + yf + af] – (1 + r) C

Intercept

Slope

Slope represents trade – off or relative price of current consumption in terms of future              consumption – in this case, if Homer consumes one more unit of current consumption, he is    giving up 1 + .1 units of future consumption- so 1 + r units of future consumption is the price of (one unit) current consumption.

2. C = 256.6 Cf = 132.8

3. C = 266.6 Cf = 133.33                   see diagram for question 2 above

4. It did increase consumption since the income and substitution effects work in the same         direction since Homer is a borrower (his income is 250K and his current consumption is        256.6K). Note that since real rates are falling, the price of current consumption, 1 plus r units of future consumption, falls and we (rationally) substitute away from future consumption        towards current consumption. We know this of course as the substitution effect. The income effect of a change in the real rate of interest depends critically on whether the consumer is a

borrower or a saver. Since Homer is a borrower, he likes the lower rates since he is financing current consumption with future resources. In other words, he really cares about the present value of future resources (since he is a borrower) and with lower rates, the present value of   future resources rises. Homer is wealthier causing him to consume more. Therefore, the       substitution effect and income effect work in the same direction.

5. C = 200 Cf = 200

6. Consumption has gone down since Homer has changed his preferences and went from being a borrower to being a saver. Of course this change in behavior, moving from being borrowers to being savers makes the Fed's job tougher if they are trying to stimulate the economy by     lowering rates. See below.

7. C = 198.7 Cf = 198.7

see diagram for question 5 above

8. Since Homer is now a saver, the lower rates cause him to spend less. The substitution effect, as we know, works to stimulate current consumption since lower rates lower the price of         current consumption in terms of future consumption (same as before). The income effect in    this case works in the opposite direction. Since Homer is a saver, the lower rates increase the cost of financing future consumption, as we know savers do. Put differently, since Homer is    now a saver, he cares about the future value of present resources. Lower rates lower the        future value of present resources and thus, Homer is poorer. The income effect dominates      here as current consumption falls. If Bernanke wanted to stimulate economy that consists of a bunch of savers, raise rates!!!!!

A certain economy has the following characteristics:

Cd = 130 + .6Y – 1000r

Id = 300 – 3000r

G = 250

and the full-employment level of output equals 1200.

1. Derive equation for desired national saving Sd as a function of Y and r.

2. Find the real interest rate that clears the goods market in two ways. Assume that the level of output is the full-employment level of output. Illustrate the equilibrium graphically.

3. Government purchases increase to 290. What is the new the equation describing desired    national saving? Illustrate the change graphically. What is the new equilibrium interest rate?

 Check the answer.

1. Sd = Y – Cd – G = Y – (130 +.6Y – 1000r) – 250 = .4Y – 380 + 1000r.

2. First, use Y = Cd + Id + G:

1200 = 130 +( .6)(1200) – 1000r + 300 – 3000r + 250

4000r = 130 + 720 + 300 + 250 – 1200 = 200, so

r = 200/4000 = .05.

Second, use Sd = Id:

(.4)(1200) -380 + 1000r = 300 – 3000r,

4000r = 380 – 480 + 300 = 200. so

= 200/4000 = .05.

3. Sd = Y – Cd – G = Y – (130 +.6Y – 1000r) – 290 = .4Y – 420 + 1000r = 60 + 1000r.

Using Sd = Id:

(.4)(1200) – 420 + 1000r = 300 – 3000r,

4000r = 420 – 480 + 300 = 240. so

r = 240/4000 = .06.

1. Sd = Y – Cd – G = Y – (130 +.6Y – 1000r) – 250 = .4Y – 380 + 1000r.

2. First, use Y = Cd + Id + G:

1200 = 130 +( .6)(1200) – 1000r + 300 – 3000r + 250

4000r = 130 + 720 + 300 + 250 – 1200 = 200, so

r = 200/4000 = .05.

Second, use Sd = Id:

(.4)(1200) -380 + 1000r = 300 – 3000r,

4000r = 380 – 480 + 300 = 200. so

= 200/4000 = .05.

3. Sd = Y – Cd – G = Y – (130 +.6Y – 1000r) – 290 = .4Y – 420 + 1000r = 60 + 1000r.

Using Sd = Id:

(.4)(1200) – 420 + 1000r = 300 – 3000r,

4000r = 420 – 480 + 300 = 240. so

r = 240/4000 = .06.

Analyze the effects of each of the following on national saving, investment, and the real interest rate. Explain your reasoning and illustrate it with an appropriate diagram.

1. Consumer confidence falls, so consumers decide to consume less and save more at every level of the real interest rate.

2. A new technology breakthrough increases the future marginal product of capital and expected future income.

 Check the answer.

1. If consumers decide to save more at every level of the real interest rate, the desired saving   curve will shift to the right. This shift causes an increase in the level of saving and investment and an decrease in the real interest rate.

2. The increase in the future marginal product of capital shifts the desired investment curve to     the right. The increase in expected future income causes households to increase their current consumption, thus reducing their desired saving, which shifts the desired saving schedule to   the left. Both shifts tend to increase the level of the expected real interest rate. The increase in desired investment causes an increase in investment and saving, but the decrease in desired saving causes a decrease in saving and investment, so it is not possible to say whether           saving and investment increase, decrease or stay the same.

1. If consumers decide to save more at every level of the real interest rate, the desired saving   curve will shift to the right. This shift causes an increase in the level of saving and investment and an decrease in the real interest rate.

2. The increase in the future marginal product of capital shifts the desired investment curve to     the right. The increase in expected future income causes households to increase their current consumption, thus reducing their desired saving, which shifts the desired saving schedule to   the left. Both shifts tend to increase the level of the expected real interest rate. The increase in desired investment causes an increase in investment and saving, but the decrease in desired saving causes a decrease in saving and investment, so it is not possible to say whether           saving and investment increase, decrease or stay the same.

A consumer has a current income of $80,000, an expected future income of $110,000, and no current wealth. The real interest rate is .1 (10%).

1. Calculate the present value of this consumer’s lifetime resources.

2. Carefully graph this consumer’s budget constraint, labeling all important quantities.

3. Suppose that this consumer wants to consume equal amounts in the present and in the future. Will this consumer be a current saver or a current borrower? Explain.

4. Suppose the real interest rate were to increase. Would this consumer’s current consumption  increase or decrease? Give a justification of your answer that includes a discussion of income and substitution effects

 Check the answer.

1. 80,000 + 110,000/1.1 = 180,000.

2.  

3. Clearly, this consumer will be a current borrower. It is easy to see that if this consumer were to consume $80,000 or less during the current period, he/she would be consuming $110,000 or  more in the future, so that his/her consumption could not be equal in the two periods. So the   consumer will want to consume more than $80,000 in the current period, which will make this  consumer a current borrower.

4. In this case, the consumer’s current consumption will decrease. An increase in the real         interest rate has two effects on current consumption, a substitution effect and an income       effect. The substitution effect arises because an increase in the real interest rate causes       current consumption to become more expensive relative to future consumption, so the          consumer has a tendency to substitute future consumption for current consumption, causing current consumption to decrease. The income effect depends upon whether the consumer is currently a saver or a borrower. In this case the consumer is a current borrower, so the         increase in the real interest rate is bad for the consumer. This can be seen in the diagram     above. The consumer’s current consumption of c* is greater than her current income of        $80,000, so she is a current borrower. When the real interest rate increases, she can no       longer afford (c*, cf*) so she must reduce both current and future consumption.

1. 80,000 + 110,000/1.1 = 180,000.

2.

3. Clearly, this consumer will be a current borrower. It is easy to see that if this consumer were to consume $80,000 or less during the current period, he/she would be consuming $110,000 or  more in the future, so that his/her consumption could not be equal in the two periods. So the   consumer will want to consume more than $80,000 in the current period, which will make this  consumer a current borrower.

4. In this case, the consumer’s current consumption will decrease. An increase in the real         interest rate has two effects on current consumption, a substitution effect and an income       effect. The substitution effect arises because an increase in the real interest rate causes       current consumption to become more expensive relative to future consumption, so the          consumer has a tendency to substitute future consumption for current consumption, causing current consumption to decrease. The income effect depends upon whether the consumer is currently a saver or a borrower. In this case the consumer is a current borrower, so the         increase in the real interest rate is bad for the consumer. This can be seen in the diagram     above. The consumer’s current consumption of c* is greater than her current income of        $80,000, so she is a current borrower. When the real interest rate increases, she can no       longer afford (c*, cf*) so she must reduce both current and future consumption.

Desired consumption for an economy is given by the equation

Cd = 1000 + .6Y – 4000r.

Government purchases are given by G = 1500.

1. Write an expression relating desired saving, Sd , to Y and r.

2. Suppose that the full-employment level of output is 10,000. Graph the relationship between desired saving, Sd , and the real interest rate r. (Your graph should include properly labeled axes and an indication of the scale on each axis.)

3. If desired investment for the economy is given by the equation Id = 2000 – 6000r, calculate the equilibrium real interest rate for the economy.

4. Using the equilibrium real interest rate that you calculated in part 3, calculate the equilibrium level of saving, investment, and consumption in the economy. Does Y = C + I + G in             equilibrium?

5. Add the relationship between desired investment and the real interest rate to your graph in part 2, and show the equilibrium values of r, Sd and Id from parts 3 and 4

 Check the answer.

1. Sd = Y – Cd – G = Y – (1000 +.6Y – 4000r) – 1500 = .4Y – 2500 + 4000r.

2. Sd = .4Y – 2500 + 4000r

Sd = (.4)(10,000) – 2500 + 2000r

Sd = 1500 + 4000r.

3. Set Sd = Id: 1500 + 4000r = 2000 – 6000r.

Then solve for r: 10,000r = 500 so

r = 500/10,000 = .05.

4. Sd = 1500 + 4000r = 1500 + (4000)(.05) = 1700. Id = 2000 – 6000r = 2000 – (6000)(.05) = 1700.

Cd = 1000 + .6Y – 4000r = 1000 + (.6)(10,000) – (4000)(.05) = 6800.

C + I + G = 6800 + 1700 +1500 = 10,000 = Y.

5. see above graph

1. Sd = Y – Cd – G = Y – (1000 +.6Y – 4000r) – 1500 = .4Y – 2500 + 4000r.

2. Sd = .4Y – 2500 + 4000r

Sd = (.4)(10,000) – 2500 + 2000r

Sd = 1500 + 4000r.

3. Set Sd = Id: 1500 + 4000r = 2000 – 6000r.

Then solve for r: 10,000r = 500 so

r = 500/10,000 = .05.

4. Sd = 1500 + 4000r = 1500 + (4000)(.05) = 1700. Id = 2000 – 6000r = 2000 – (6000)(.05) = 1700.

Cd = 1000 + .6Y – 4000r = 1000 + (.6)(10,000) – (4000)(.05) = 6800.

C + I + G = 6800 + 1700 +1500 = 10,000 = Y.

5. see above graph

Use a saving investment diagram (and an explanation) to show what happens to saving,         investment and the real interest rate in the following scenario. (Note: we are just looking at the goods market here.)

A practical technology that uses nuclear fusion to produce electricity from sea water is discovered. The effect of this innovation is expected to make energy cheap and abundant in the future.

 Check the answer.

The newly discovered abundance of cheap energy in the future will cause an increase in the    expected future marginal product of capital. The increase in the future marginal productivity of  capital will cause an increase in the level of desired investment at each level of the real interest rate. Graphically, this will be reflected in a rightward shift in the desired investment curve. In addition, there will be an increase in expected future income, which will cause an increase in both current and future consumption. This means that current desired national saving will decrease at every level of the real interest rate. (The substitution of current leisure for future leisure that we    found in question #1 will also contribute to a reduction in current desired national saving, since it  leads to a decline in current income.) In the new equilibrium, the real interest rate will increase,    but the effect on the level of saving and investment in the economy is ambiguous. (In the diagram below, S and I go down, but they could have gone up if the shift in desired investment had been   greater or the shift in desired saving had been smaller.)

The newly discovered abundance of cheap energy in the future will cause an increase in the        expected future marginal product of capital. The increase in the future marginal productivity of      capital will cause an increase in the level of desired investment at each level of the real interest    rate. Graphically, this will be reflected in a rightward shift in the desired investment curve. In         addition, there will be an increase in expected future income, which will cause an increase in both current and future consumption. This means that current desired national saving will decrease at every level of the real interest rate. (The substitution of current leisure for future leisure that we    found in question #1 will also contribute to a reduction in current desired national saving, since it  leads to a decline in current income.) In the new equilibrium, the real interest rate will increase,    but the effect on the level of saving and investment in the economy is ambiguous. (In the diagram below, S and I go down, but they could have gone up if the shift in desired investment had been   greater or the shift in desired saving had been smaller.)