Alternative to Ricardian equivalence and calculus

Consider the intertemporal consumption � labor model. Suppose that the lifetime utility function is given by

v(B1c1, l1,B2c2, l2) = u(B1c1, l1) + u(B2c2, l2)

= 2 √ B1c1 + 2

√ l1 + 2

√ B2c2 + 2

√ l2

v = 2 √ B1c

1 2 1 + 2l

1 2 1 + 2

√ B2c

1 2 2 + 2l

1 2 2

which is a slight modi�cation of the utility function presented in chapter 5. The modi�cation is that preference shifters B1 and B2 enter the lifetime utility function, with B1 the preference shifter in period 1 and B2 is the preference shifter in period 2. In each of the two periods the function u takes the form

u(Btct, lt) = 2 √ Btct + 2

√ lt

Also t1 = 0.15 , t2 = 0.2 , w1 = 0.2 , w2 = 0.25 , r = 0.15, B1 = 1 , and B2 = 1.2.

1. Construct the marginal rate of substitution functions between consumption and leisure in each of period one and period two. (Hint: These expressions will be functions of consumption and leisure � you are not being asked to solve for any numerical values yet.) How does the preference shifter a�ect this intratemporal margin?

− ∂u ∂l1 ∂u ∂c1

= − l − 1

2 1

B 1 2 1 (c1)

− 1 2

= − √ c1√ B1l1

= dc1 dl1

An increase in B1 �attens the indi�erence curve, thus increasing the preference l1.

2. Construct the marginal rate of substitution function between period-one consumption and period-two consumption. (Hint: Again, you are not being asked to solve for any numerical values yet.) How do the preference shifters a�ect this intertemporal margin?

− ∂u ∂c1 ∂u ∂c2

= − B

1 2 1 (c1)

− 1 2

B 1 2 2 (c2)

− 1 2

= − √ B1c2√ B2c1

= dc2 dc1

If B1/B2 rises, the preference for c2 rises relative to c1.

3. Using the expressions you developed in parts a and b along with the lifetime budget constraint (ex- pressed in real terms, etc.) and the given numerical values, solve numerically for the optimal choices of consumption in each of the two periods and of leisure in the two periods. (Hint: You need to set up and solve the appropriate Lagrangian. Note that the computations here are messy and the �nal answers do not necessarily work out �nicely�. To preserve some numerical accuracy, carry out your computations to at least four decimal places.)

L = 2 √ B1c1 + 2

√ l1 + 2

√ B2c2 + 2

√ l2

+ λ

( (1 + r)a0 + (1− t1)w1 +

(1− t2)w2 1 + r

) −λ

( c1 +

c2 1 + r

+ (1− t1)w1l1 + (1− t2)w2l2

1 + r

)

Lc1 = √ B1c

− 1 2

1 −λ = 0

Lc2 = √ B2c

− 1 2

2 − λ

1 + r = 0

Ll1 = l − 1

2 1 −λ(1− t1)w1 = 0

Ll2 = l − 1

2 2 −

λ(1− t2)w2 1 + r

= 0

1

√ B1√ B2

√ c2√ c1

= 1 + r

√ c2 = (1 + r)

√ c1

√ B2√ B1

c2 = (1 + r) 2 c1 B2 B1√

l2√ l1

= (1− t1)w1 (1− t2)w2

(1 + r)

√ l2 =

√ l1 (1− t1)w1 (1− t2)w2

(1 + r)

l2 = l1

( (1− t1)w1 (1− t2)w2

)2 (1 + r)

2

√ l1B1√ c1

= 1

(1− t1)w1√ l1 =

√ c1√

B1 (1− t1)w1 l1 =

c1

B1 ((1− t1)w1) 2 1

A = c1 + c2

1 + r + (1− t1)w1l1 +

(1− t2)w2l2 1 + r

A = c1 + (1 + r)

2 c1

B2 B1

1 + r + (1− t1)w1l1 +

(1− t2)w2 1 + r

l1

( (1− t1)w1 (1− t2)w2

)2 (1 + r)

2

A = c1 + (1 + r)c1 B2 B1

+ (1− t1)w1l1 + (1 + r) ((1− t1)w1)

2

(1− t2)w2 l1

= c1 + (1 + r)c1 B2 B1

+ (1− t1)w1 c1

B1 ((1− t1)w1) 2 + (1 + r)

((1− t1)w1) 2

(1− t2)w2 c1

B1 ((1− t1)w1) 2

= c1 + (1 + r)c1 B2 B1

+ c1

B1 (1− t1)w1 + (1 + r)

c1 B1 (1− t2)w2

A = c1

( 1 + (1 + r)

B2 B1

+ 1

B1 (1− t1)w1 +

1 + r

B1 (1− t2)w2

) Once we have c1, we can, from the equations above, calculate c2

c2 = (1 + r) 2 c1 B2 B1

and l1 l1 =

c1

B1 ((1− t1)w1) 2

And with l1 we can calculate l2.

l2 = l1

( (1− t1)w1 (1− t2)w2

)2 (1 + r)

2

4. Based on your answer in part c, how much (in real terms) does the consumer save in period 1? What is the asset position that the consumer begins period 2 with?

y1 = (1− l1) (1− t1)w1 s1 = y1 − c1 a1 = (1 + r) ·s1

2

5. Suppose that B2 were instead higher at 1.6. How are your solutions in parts c and d a�ected? Provide brief interpretation in terms of � consumer con�dence. �

6. Suppose r rises. How does this a�ect consumption and leisure in each period? Let's �nd the budget constraint for c2 in terms of c1. In other words, we put c2 on the vertical axis and c1 on the horizontal axis.

(1 + r)a0 + (1− t1)w1 + (1− t2)w2

1 + r = c1 +

c2 1 + r

+ (1− t1)w1l1 + (1− t2)w2l2

1 + r

As above, let's call the left-hand-side of the above equation A.

A = (1 + r)a0 + (1− t1)w1 + (1− t2)w2

1 + r

Then the budget constraint is

A = c1 + c2

1 + r + (1− t1)w1l1 +

(1− t2)w2l2 1 + r

Solve for c2.

A− c1 − (1− t1)w1l1 − (1− t2)w2l2

1 + r =

c2 1 + r

(1 + r)A− (1 + r) (1− t1)w1l1 − (1− t2)w2l2 − (1 + r)c1 = c2 Thus an increase in r steepens the slope on c1. This means a unit increase in c1 is associated with a larger decrease in c2 than before the increase in r. In other words, c1 is more expensive in terms of c2, so the consumer chooses less c1 and more c2, assuming no income e�ect. If the consumer is a saver, the rise in r will increase consumption of both c1 and c2, so the net e�ect on c1 is ambiguous, though c2 will de�nitely increase due to both the substitution and income e�ects. Now consider the e�ect of an increase in r on leisure. For this we need the budget constraint in which l2 is on the vertical axis and l1 on the horizontal axis.

(1 + r)A− (1 + r) (1− t1)w1l1 − (1 + r)c1 − c2 = (1− t2)w2l2 (1 + r)A

(1− t2)w2 −

(1 + r)c1 (1− t2)w2

− c2

(1− t2)w2 −

(1 + r) (1− t1)w1 (1− t2)w2

l1 = l2

Similar to above, an increase in r also steepens the slope on l1, meaning that a unit increase in l1 is associated with a larger decline in l2 than before the rise in r. That is, l1 has become more expensive in terms of l2, leading the consumer to choose less l1 and more l2. Also similar to above, if the consumer is a saver, the increase in r will raise her income, increasing her demand for both l1 and l2. Thus the net e�ect on l1 is ambiguous, though l2 will de�nitely increase due to both the substitution and income e�ects. By the way the budget constraint can be usefully rewritten.

(1 + r)a0 + (1− t1)w1 + (1− t2)w2

1 + r = c1 +

c2 1 + r

+ (1− t1)w1l1 + (1− t2)w2l2

1 + r

Note that (1− t1)w1 − (1− t1)w1l1 = (1− t1)w1 (1− l1)

and (1− t2)w2

1 + r −

(1− t2)w2l2 1 + r

= (1− t2)w2 (1− l2)

1 + r

Therefore the budget constraint can be condensed by combining terms with (1− t1)w1 and with (1− t2)w2/(1 + r).

(1 + r)a0 + (1− t1)w1 (1− l1) + (1− t2)w2 (1− l2)

1 + r = c1 +

c2 1 + r

The left-hand-side is the present value of lifetime income (both interest income and labor income); the right-hand-side is the present value of consumption.

7. How would an increase in w1 a�ect consumption and leisure in both periods?

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