Consumption and Savings Master en Econom´ ıa Internacional Universidad Auton´ oma de Madrid Fall 2014 Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 1 / 75
Objectives There are important differences between the short- and the long-run relationship between aggregate consumption and income. Over long time horizons, (aggregate) consumption and income are almost perfectly correlated. At business cycle frequencies this one-to-one relationship is broekn. In particular, consumption tends to fluctuate less than income as the agents use their savings to smooth consumption. The main objective of this lecture is to consider standard models of inter-temporal consumption and savings decisions that help to understand these differences. Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 2 / 75
Outline 1 Partial equilibrium in two periods 2 General equilibrium in two periods with competitive credit markets 3 Extensions ◮ N periods ◮ Uncertainty and risk aversion ◮ Borrowing constraints 4 Testable predictions ◮ Permanent income hypothesis ◮ Random walk hypothesis ◮ Life-cycle hypothesis Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 3 / 75
Savings motives Why do people save? 1 Consumption smoothing - agents dislike fluctuations in consumption spending. 2 Precautionary motives — fear of unemployment etc. 3 To finance consumption during retirement 4 Purchase of real estate or durable consumption goods. For the moment we focus on 1. Saving for retirement is treated at the end of the course, while precautionary savings is left for later. Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 4 / 75
Basic assumptions We consider an economy that lasts for two periods, t = 1, 2. In both periods, each agent j ∈ J receives a known endowment y j , t ≥ 0. Agents have access to a perfectly competitive bond markets Individual agents take the real interest rate R as given We assume complete information and we impose the condition that agents have to honor their debts. Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 5 / 75
Bonds A bond is a piece of paper with a promise of a future payment of a certain amount of goods to the holder of the bond. Bonds are issued by borrowers and handed over to lenders. All bonds are supposed to expire in one period. We denote the (real) interest rate by R . A household that buys b j , t in t will receive ( 1 + R t ) b j , t units of the good in period t + 1. We assume complete information. All loans are repaid. Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 6 / 75
Budget constraints The one-period budget constraint of a representative household can be written as c t + b t = y t + ( 1 + R ) b t − 1 When b t − 1 > 0, the resources of the household exceed the value of income y t + ( 1 + R ) b t − 1 > y t By contrast, when b t − 1 < 0, the household has to devote part of this period’s resources to pay back the loan plus interest y t + ( 1 + R ) b t − 1 < y t Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 7 / 75
Budget constraints in a two-period setting In a tw-period setting, a representative household faces the following pair of budget constraints: c 1 + b 1 = y 1 , c 2 + b 2 = y 2 + ( 1 + R ) b 1 . Note that savings in the second period are necessarily equal to zero. The household would like to borrow an infinite amount, but there are no lenders! Hence, c 2 = y 2 + ( 1 + R ) b 1 Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 8 / 75
Inter-temporal budget constraint The two budget constraints can be consolidated into a single lifetime budget constraint. Note that: b 1 = y 1 − c 1 , Using this expression we can write the second-period budget constraint as c 2 = y 2 + ( 1 + R )( y 1 − c 1 ) . Dividing both sides by ( 1 + R ) yields the following expression: c 2 y 2 c 1 + = y 1 + = x . 1 + R 1 + R � �� � � �� � Present Value of Lifetime Consumption Present Value of Lifetime Income Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 9 / 75
The inter-temporal budget constraint Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 10 / 75
Preferences We use the following standard specification of time-separable preferences: U ( c 1 , c 2 ) = u ( c 1 ) + β u ( c 2 ) , The same function u ( . ) defines utility in both periods, but the utility from future consumption is discounted at rate β ≤ 1. 1 The discount factor is commonly expressed as β = 1 + ρ , where ρ represents the rate of time preference. Finally, u ′ ( . ) = ∂ u ( . ) / ∂ c t > 0 and u ′′ ( . ) = ∂ u ′ ( . ) / ∂ c t < 0 In many examples we will consider u ( c t ) = log ( c t ) with lim c t → 0 u ′ ( c t ) = ∞ . This avoids corner solutions. Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 11 / 75
Inter-temporal marginal rate of rate of substitution (IMRS) The indifference curves in this two-period setting are pairs of current and future consumption that offer the same level of utility u 0 : U ( c 1 , c 2 ) = u ( c 1 ) + β u ( c 2 ) = u 0 Along the indifference curve u ′ ( c 1 ) dc 1 + β u ′ ( c 2 ) dc 2 = 0 And the IMRS is defined as: = − u ′ ( c 1 ) dc 2 β u ′ ( c 2 ) dc 1 Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 12 / 75
Indifference curves Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 13 / 75
Optimization problem The most general statement of the agent’s optimization problem is: max { c 1 , b 1 , c 2 } U ( c 1 , c 2 ) = u ( c 1 ) + β u ( c 2 ) s.t. c 1 + b 1 = y 1 c 2 + b 2 = y 2 + ( 1 + R ) b 1 ≥ 0 c 1 ≥ c 2 0 Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 14 / 75
The Optimization Problem In compact form the optimization problem can be written as max c 1 , c 2 [ u ( c 1 ) + β u ( c 2 )] (1) s.t. c 2 1 + R + c 1 = x . (2) Substituting the budget constraint into the objective function we obtain: c 1 [ u ( c 1 ) + β u (( x − c 1 )( 1 + R ))] max (3) Note: we rule out corner solutions and we impose equality in (2). Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 15 / 75
The solution The FOC associated with our optimization problem max c 1 [ u ( c 1 ) + β u (( x − c 1 )( 1 + R ))] is given by: u ′ ( c 1 ) + β u ′ (( x − c 1 )( 1 + R ) )( − 1 )( 1 + R ) = 0. � �� � c 2 Or alternatively, u ′ ( c 1 ) = u ′ ( c 2 ) β ( 1 + R ) � �� � � �� � MC MB u ′ ( c 1 ) = 1 + R . β u ′ ( c 2 ) � �� � IMRS Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 16 / 75
Graphical representation of solution Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 17 / 75
Consumption Euler equation The optimality condition — known as the Consumption Euler equation — u ′ ( c 1 ) β u ′ ( c 2 ) = ( 1 + R ) is nothing else than the intertemporal variant of a well-known result in consumption theory. IMRS = price ratio of consumption in both periods Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 18 / 75
Slope of the optimal consumption profile According to the Euler equation u ′ ( c 1 ) = β ( 1 + R ) u ′ ( c 2 ) there are two opposing forces that affect the inter-temporal choices of the agent Time preference The real interest rate Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 19 / 75
Perfect consumption smoothing When β ( 1 + R ) = 1, the Euler eqn. reduces to: u ′ ( c 1 ) = u ′ ( c 2 ) . Given the strict concavity of u ( . ) , this implies c 1 = c 2 . Intuition : Recall that β = 1 / ( 1 + ρ ) . Hence, the necessary condition for perfect consumption smoothing can be written as: β ( 1 + R ) = 1 + R 1 + ρ = 1, which is only satisfied if R = ρ . Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 20 / 75
Slope of consumption profiles Using the same line of argument, one can easily demonstrate that the consumption Euler eqn u ′ ( c 1 ) β u ′ ( c 2 ) = ( 1 + R ) implies the following three results β ( 1 + R ) < 1 = ⇒ c 1 > c 2 β ( 1 + R ) = 1 = ⇒ c 1 = c 2 β ( 1 + R ) > 1 = ⇒ c 1 < c 2 The above conditions play a central role in any dynamic macro model with endogenous consumption choices. Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 21 / 75
Example: logarithmic utility Suppose that the agent’s preferences satisfy u ( c 1 ) + β u ( c 2 ) = log ( c 1 ) + β log ( c 2 ) . In this case, we can write the maximand as max c 1 [ log ( c 1 ) + β log (( x − c 1 )( 1 + R ))] and the associated FOC is: 1 + β 1 ( − 1 )( 1 + R ) = 0 c 1 c 2 1 = 1 β ( 1 + R ) = ⇒ c 2 = β ( 1 + R ) c 1 c 1 c 2 c 2 = β ( 1 + R ) . c 1 Master en Econom´ ıa Internacional (UAM) Consumption and Savings Decisions Fall 2014 22 / 75
Recommend
More recommend
