ch8A Two-Period Model(中级宏观经济学,香港中文大学)
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– Consumer can put aside the work-leisure decision. – Incomes can differ across consumers.
• Zero wealth endowment in the current period. • Each consumer pays lump-sum taxes in both periods.
Consumer’s Lifetime Budget Constraint
• The slope of the lifetime budget constraint is -(1 + r).
• E is the endowment point ( I.e. where s = 0 ). • Points on BE s 0 consumer is a lender. • Points on EA s 0 consumer is a borrower.
2
• Given that U( ., .) is strictly quasiconcave, > 0.
Consumer’s Problem
• Similar to the one in Chapter 4, the consumer’s optimal consumption bundle is the one at which an indifference curve is tangent to the budget constraint. • This implies the following condition: 1 + r = MRS c,c’
2) The consumer likes diversity in consumption bundle.
– Corresponds to the consumer’s desire to smooth consumption over time. That is, consumer dislikes consuming a lot in a single period and very few in another.
c’
B D
A
c
Indifference Curves
• Slope of indifference curve = MRS c,c’ • The following are equivalent:
– – – – a preference for diversity diminishing MRS c,c’ convexity of indifference curve consumption smoothing
c y t s 1 r
• Substitute s into c + s = y - t, we have
c y t c y t 1 r 1 r
• RHS equals the present value of lifetime disposable income or lifetime wealth (we). • LHS equals the present value of lifetime consumption. • Note: we can think of (1 + r )-1 as relative price of c’ in terms of c, while the price of c is normalized to one.
Chapter 8
A Two-Period Model: Consumption-Savings Decision & Ricardian Equivalence
Two-Period Model of the Economy
• In Chapter 8, we will focus on the consumers’ and government’s behavior. Firms’ behavior is reintroduced in Chapter 9. • A consumer’s consumption-savings decision involves a tradeoff between current and future consumption. • By saving, a consumer gives up consumption in exchange for assets in the present, in order to consume more in the future. • The government’s decision concerning the financing of government expenditure, involves a trade-off between current and future taxes.
Comparative Statics
• To determine the effects of changes in y, y’ and r on c, c’ and s. • Totally differentiate the following system
U1 c, c 1 r. U 2 c, c
Consumer’s Preferences
• Consumer’s utility function is given by U( c, c’ ).
• Properties of Preferences:
1) More is preferred to less
– U(., .) is increasing in both arguments.
U1 c, c 0, U 2 c, c
y t
1 r
0,
பைடு நூலகம்
y t c c 0 1 r 1 r
• From the first two conditions, we obtain
MRSc ,c U1 c, c 1 r. U 2 c, c
• Future period budget constraint: c’ = y’ - t’ + ( 1 + r )s. • Apart from the after-tax income, the consumer receives the interest and principal on savings. • Since there are only two periods, there is no incentive for any savings in the second period. So the consumer will consume everything he has. • Next, we want to combine the two constraints into one lifetime budget constraint. • From c’ = y’ - t’ + ( 1 + r )s, we have
Consumer’s Problem ( Appendix pp.640-643 )
• The consumer chooses c and c’ to maximize U(c, c’) subject to the lifetime budget constraint. Max U(c, c’)
• Notations:
– y and t denote real income and tax in 1st period. – y’ and t’ denote real income and tax in 2nd period.
• Current period budget constraint: c + s = y - t. • Consumers’ after-tax income can either be saved ( s ) or consumed ( c ).
• Consider the bordered Hessian matrix
U11 1 r U12 U12 1 r U 22 A 1 r 1
• The determinant of A is
U11 21 r U12 1 r U 22
Consumers
• Assume that there are m consumers ( m is a large number ). • Each consumer lives for two periods, current and future period. • Each consumer receive exogenous income in both periods.
• Saving > 0 ( < 0 ) Buys ( sells ) a bond with part of his/her income The consumer is a lender ( borrower ) on the credit market.
• What is a bond ? – A bond issued ( by the government ) in the current period is a promise to pay a certain amount, say 1 + r units, of consumption good in the future period. 1 unit of c can be exchanged for 1 + r units of c’ in the credit market. – r is thus the real interest rate on each bond. • Remark: we assume that a consumer can lend and borrow at the same real interest rate, i.e. the credit market is perfect.
c, c’
subject to
c
c y t y t 1 r 1 r
• Lagrangian
y t c L U c, c y t c 1 r 1 r 1 r
• First-order conditions:
3) Current and future consumption are normal goods. – Let A be the original optimal choice. – Suppose there is a parallel upward shift in the constraint (a pure income increase). Then the new optimal choice must lie on BD. – This is also related to the consumer’s desire to smooth consumption over time.
and
y t c y t c 0 1 r 1 r
• Assuming dt = dt’ = 0, then in matrix form, we have
U11 1 r U12 U12 1 r U 22 dc 1 r dc 1 U2 0 0 dr 1 r dy dy y t c 1 1
• Zero wealth endowment in the current period. • Each consumer pays lump-sum taxes in both periods.
Consumer’s Lifetime Budget Constraint
• The slope of the lifetime budget constraint is -(1 + r).
• E is the endowment point ( I.e. where s = 0 ). • Points on BE s 0 consumer is a lender. • Points on EA s 0 consumer is a borrower.
2
• Given that U( ., .) is strictly quasiconcave, > 0.
Consumer’s Problem
• Similar to the one in Chapter 4, the consumer’s optimal consumption bundle is the one at which an indifference curve is tangent to the budget constraint. • This implies the following condition: 1 + r = MRS c,c’
2) The consumer likes diversity in consumption bundle.
– Corresponds to the consumer’s desire to smooth consumption over time. That is, consumer dislikes consuming a lot in a single period and very few in another.
c’
B D
A
c
Indifference Curves
• Slope of indifference curve = MRS c,c’ • The following are equivalent:
– – – – a preference for diversity diminishing MRS c,c’ convexity of indifference curve consumption smoothing
c y t s 1 r
• Substitute s into c + s = y - t, we have
c y t c y t 1 r 1 r
• RHS equals the present value of lifetime disposable income or lifetime wealth (we). • LHS equals the present value of lifetime consumption. • Note: we can think of (1 + r )-1 as relative price of c’ in terms of c, while the price of c is normalized to one.
Chapter 8
A Two-Period Model: Consumption-Savings Decision & Ricardian Equivalence
Two-Period Model of the Economy
• In Chapter 8, we will focus on the consumers’ and government’s behavior. Firms’ behavior is reintroduced in Chapter 9. • A consumer’s consumption-savings decision involves a tradeoff between current and future consumption. • By saving, a consumer gives up consumption in exchange for assets in the present, in order to consume more in the future. • The government’s decision concerning the financing of government expenditure, involves a trade-off between current and future taxes.
Comparative Statics
• To determine the effects of changes in y, y’ and r on c, c’ and s. • Totally differentiate the following system
U1 c, c 1 r. U 2 c, c
Consumer’s Preferences
• Consumer’s utility function is given by U( c, c’ ).
• Properties of Preferences:
1) More is preferred to less
– U(., .) is increasing in both arguments.
U1 c, c 0, U 2 c, c
y t
1 r
0,
பைடு நூலகம்
y t c c 0 1 r 1 r
• From the first two conditions, we obtain
MRSc ,c U1 c, c 1 r. U 2 c, c
• Future period budget constraint: c’ = y’ - t’ + ( 1 + r )s. • Apart from the after-tax income, the consumer receives the interest and principal on savings. • Since there are only two periods, there is no incentive for any savings in the second period. So the consumer will consume everything he has. • Next, we want to combine the two constraints into one lifetime budget constraint. • From c’ = y’ - t’ + ( 1 + r )s, we have
Consumer’s Problem ( Appendix pp.640-643 )
• The consumer chooses c and c’ to maximize U(c, c’) subject to the lifetime budget constraint. Max U(c, c’)
• Notations:
– y and t denote real income and tax in 1st period. – y’ and t’ denote real income and tax in 2nd period.
• Current period budget constraint: c + s = y - t. • Consumers’ after-tax income can either be saved ( s ) or consumed ( c ).
• Consider the bordered Hessian matrix
U11 1 r U12 U12 1 r U 22 A 1 r 1
• The determinant of A is
U11 21 r U12 1 r U 22
Consumers
• Assume that there are m consumers ( m is a large number ). • Each consumer lives for two periods, current and future period. • Each consumer receive exogenous income in both periods.
• Saving > 0 ( < 0 ) Buys ( sells ) a bond with part of his/her income The consumer is a lender ( borrower ) on the credit market.
• What is a bond ? – A bond issued ( by the government ) in the current period is a promise to pay a certain amount, say 1 + r units, of consumption good in the future period. 1 unit of c can be exchanged for 1 + r units of c’ in the credit market. – r is thus the real interest rate on each bond. • Remark: we assume that a consumer can lend and borrow at the same real interest rate, i.e. the credit market is perfect.
c, c’
subject to
c
c y t y t 1 r 1 r
• Lagrangian
y t c L U c, c y t c 1 r 1 r 1 r
• First-order conditions:
3) Current and future consumption are normal goods. – Let A be the original optimal choice. – Suppose there is a parallel upward shift in the constraint (a pure income increase). Then the new optimal choice must lie on BD. – This is also related to the consumer’s desire to smooth consumption over time.
and
y t c y t c 0 1 r 1 r
• Assuming dt = dt’ = 0, then in matrix form, we have
U11 1 r U12 U12 1 r U 22 dc 1 r dc 1 U2 0 0 dr 1 r dy dy y t c 1 1