(5.3) PCM intertemporal optimisation

Question 1

2

Assume a budget constraint 1.2C₀ + C₁ = 120 with all earnings in period 0. What is the interest rate?

Answer 1

• (1+r) = 1.2
• r = 0.2 = 20%

Question 2

4

Assume a budget constraint 1.2C₀ + C₁ = 120 with all earnings in period 0. What are the earnings in period 0?

Answer 2

• sub in C₁ = 0
• 1.2C₀ + 0 =120
• C₀ = 100
• y₀ = 100

Question 3

3

Assume a budget constraint 1.2C₀ + C₁ = 120 with all earnings in period 0. What is the MRS?

Answer 3

• MRS = (1+r)
• r = 0.2
• MRS = 1.2

Question 4

2

Assume a budget constraint 1.2C₀ + C₁ = 120 with all earnings in period 0. What is the slope of the budget line?

Answer 4

• -(MRS)
• -1.2

Question 5

1

Assume a budget constraint 1.2C₀ + C₁ = 120 with all earnings in period 0. If they save 20, consume 80 in period 0, what will they consume in period 1?

Answer 5

• 20(1.2) = 24 consumption in period 1

Question 6

4

Draw an indifference curve such that consumption in period 1 is 0. Interpret this result

Answer 6

• Corner solution at C₁ = 0
• highest attainable indifference curve touches the budget constraint at this corner point
• MRS = 0
• consumer is unwilling to substitute any consumption in period 0 in exchange for consumption in period 1

Question 7

3

For an interior solution, what does the condition MRS = 1+ r indicate?

Answer 7

• tangency condition
• MRS between present and future consumption equals the market exchange rate
• so maximises utility

Question 8

3

What is necessary for consumer preferences to be represented by u(c0, c1)?

Answer 8

• completeness
• transivity
• continuity

Question 9

2

How do we guarantee that intertemporal IDCs are downward sloping and convex to origin?

Answer 9

• non-satiation
• convexity

Question 10

2

What 2 constraints are consumers subject to in deciding intertemporal consumption?

Answer 10

• Intertemporal budget constraint: C1 = [y1 + (1+r)y0] - (1+r)c0
• Non-negativity constraints: C0, C1 ≥ 0

Question 11

3 + photo

Describe borrowers

Answer 11

• Consume relatively more in period 0 than 1
• Consume more than income in period 0
• C₀ > y₀

Question 12

3 + photo

Describe savers

Answer 12

• Consume relatively more in period 1 than 0
• Consume less than income in period 0
• C₀ < y₀

Question 13

3 + photo

Describe someone who neither borrows nor saves

Answer 13

• Consume exactly their endowment
• C₀ = y₀, C₁ = y₁
• IDC tangent to budget line at endowment pt

Question 14

2

What direction does an interest rate increase change a borrower?

Answer 14

• Generally worse off
• Generally reduces C0 consumption i.e. borrow less

Question 15

3

Explain a decomposition for a borrower after an interest rate rise (normal goods)

Answer 15

• Sub effect: Makes borrowing more expensive → reduces C₀, increases C₁
• Income effect: Makes the borrower poorer → reduces both C₀ and C₁ (if normal goods)
• Net effect: Both effects reduce borrowing (reduce C₀)

Question 16

3

Explain a decomposition for a borrower after an interest rate decrease (normal goods)

Answer 16

• Sub effect: Makes borrowing cheaper → increases C₀, decreases C₁
• Income effect: Makes the borrower richer → increases both C₀ and C₁ (if normal goods)
• Net effect: Both effects increase borrowing (increase C₀)

Question 17

3

Explain a decomposition for a saver after an interest rate rise (normal goods)

Answer 17

• Sub effect: Makes saving more rewarding → reduces C₀, increases
  saving
• Income effect: Makes the saver richer → increases both C₀ and C₁ (if normal
  goods)
• Net effect: Ambiguous - substitution effect encourages more saving, income
  effect encourages less saving

Question 18

3

Explain a decomposition for a saver after an interest rate decrease (normal goods)

Answer 18

• Sub effect: Makes saving less rewarding → increases C₀, decreases saving
• Income effect: Makes the saver poorer → decreases both C₀ and C₁ (if normal
  goods)
• Net effect: Ambiguous - sub effect encourages less saving, income
  effect encourages more saving

Question 19

2

How does the net effect of interest rate changes for borrowers and savers differ (both types of goods)?

Answer 19

• Borrowers: unambigious
• Savers: ambiguous

Question 20

2 + photo

Draw a graph decomposing an interest rate increase for a borrower (normal goods)

Answer 20

• A -> D: sub effect
• D -> B: income effect

Question 21

3 + photo

Draw a graph decomposing an interest rate increase for a saver (normal goods)

Answer 21

• A -> D: sub effect
• D -> B: income effect
• Note: overall effect unclear

Question 22

1

What is the special case where an interest rate change can make the borrower better off?

Answer 22

• Interest rate large enough to make them switch from borrowing to saving

Question 23

2

Compare the IDC steepness of savers/borrowers?

Answer 23

• Saver - flat IDC
• Borrower - steep IDC

Question 24

2

Compare the model of optimal intertemporal consumption and optimal consumption between 2 goods

Answer 24

• assumptions we make about the shape of preferences are exactly the same
• analysis we do (sub and income effect) is very similar

Question 25

# 1 if c0 is an inferior good, what happens when income rises

Answer 25

* income effect causes c0 to fall, c1 to rise

Question 26

# 1 if c0 is an inferior good, what happens when income falls

Answer 26

* income effect causes c0 to rise, c1 to fall

Question 27

# 2 If consumption begins at endowment pt and interest rate changes, how does this affect the consumer?

Answer 27

* Cannot be worse off (OG bundle can still be consumed) * but may be better off (can reoptimise to higher IDC)

Question 28

# 1 When moving to reoptimising pt, what should we state to justify that we can reoptimise?

Answer 28

* assume that standard preferences satisfy our assumptions