Dynamic Consumption Theory October 2007 () Consumption October 2007 1 / 20
Two-Period Planning Horizon (discrete time, no uncertainty) Household utility: U ( c 1 , c 2 ) = u ( c 1 ) + β u ( c 2 ) ! u 0 ( c ) > 0, and u 00 ( c ) < 0 , 1 , ! subjective discount factor: β = 1 + ρ < 1 , ! ρ = rate of time preference. () Consumption October 2007 2 / 20
Dynamic budget constraints: A 1 = ( 1 + r 1 ) ( A 0 + w 1 � c 1 ) A 2 = ( 1 + r 2 ) ( A 1 + w 2 � c 2 ) Boundary condition A 2 � 0 ) intertemporal budget constraint: c 2 w 2 c 1 + � A 0 + w 1 + 1 + r 1 1 + r 1 () Consumption October 2007 3 / 20
The Euler Equation How should household allocate consumption across the two periods? Consider saving an additional $1 at the optimum : u 0 ( c 1 ) MC of extra $ saved = β ( 1 + r 1 ) u 0 ( c 2 ) . MB of extra dollar saved = , ! optimal allocation: u 0 ( c 1 ) = β ( 1 + r 1 ) u 0 ( c 2 ) Also intertemporal budget constraint must hold with strict equality. () Consumption October 2007 4 / 20
Diagrammatic Representation Slope of budget constraint: dc 2 = � ( 1 + r 1 ) , dc 1 Marginal rate of intertemporal substitution: � � = � u 0 ( c 1 ) dc 2 � β u 0 ( c 2 ) . � ¯ dc 1 U , ! optimal allocation u 0 ( c 1 ) β u 0 ( c 2 ) = 1 + r 1 which is the Euler equation . () Consumption October 2007 5 / 20
c 2 w 2 +(1+r 2 )w 1 A * c 2 slope = - (1+r 2 ) E w 2 c 1 * c 1 w 1 Figure: Optimal Consumption Choice () Consumption October 2007 6 / 20
Example — CES utility function 1 � θ + β c 1 � θ c 1 � θ 1 2 1 � θ . Euler equation c � θ = β ( 1 + r 1 ) c � θ 2 . 1 , ! re–arranges to c 2 1 θ . = [ β ( 1 + r 1 )] c 1 Taking natural logs of both sides we get ln c 2 = 1 θ ln β + 1 θ ln ( 1 + r 1 ) . c 1 , ! It follows that � � c 2 % change in c 2 d ln = 1 c 1 c 1 d ln ( 1 + r 1 ) = θ . % change in 1 + r 1 () Consumption October 2007 7 / 20
Three–Period Planning Horizon Discounted utility u ( c 1 ) + β u ( c 2 ) + β 2 u ( c 3 ) , Dynamic budget constraints: = ( 1 + r 1 ) ( A 0 + w 1 � c 1 ) A 1 A 2 = ( 1 + r 2 ) ( A 1 + w 2 � c 2 ) = ( 1 + r 3 ) ( A 2 + w 3 � c 3 ) A 2 Boundary condition: A 3 � 0. , ! intertemporal budget constraint c 2 c 3 w 2 w 3 c 1 + + ( 1 + r 1 )( 1 + r 2 ) � w 1 + + ( 1 + r 1 )( 1 + r 2 ) . 1 + r 1 1 + r 1 () Consumption October 2007 8 / 20
Optimal consumption allocation satis…es A 3 = 0 and u 0 ( c 1 ) = β ( 1 + r 1 ) u 0 ( c 2 ) = β 2 ( 1 + r 1 )( 1 + r 2 ) u 0 ( c 3 ) . or β ( 1 + r 1 ) u 0 ( c 2 ) u 0 ( c 1 ) = β ( 1 + r 2 ) u 0 ( c 3 ) u 0 ( c 2 ) = CES example: 1 θ c 1 c 2 = [ β ( 1 + r 1 )] 1 θ c 2 c 3 = [ β ( 1 + r 2 )] c 2 c 3 w 2 w 3 c 1 + + = w 1 + + 1 + r 1 ( 1 + r 1 )( 1 + r 2 ) 1 + r 1 ( 1 + r 1 )( 1 + r 2 ) () Consumption October 2007 9 / 20
T–Period Planning Horizon Discounted utility: T β t � 1 u ( c t ) ∑ t = 1 T dynamic budget constraints A t = ( 1 + r t ) ( A t � 1 + w t � c t ) 8 t 2 f 1 , ..., T g boundary condition A T � 0. () Consumption October 2007 10 / 20
Solution T � 1 Euler equations: u 0 ( c t ) = β ( 1 + r t ) u 0 ( c t + 1 ) 8 t 2 f 1 , ..., T � 1 g and an intertemporal budget constraint T T ∑ ∑ D t c t = A 0 + D t w t , t = 1 t = 1 where � � � � � � � � t � 1 1 1 1 1 ∏ D t = = . .... and D 1 + r τ 1 + r 1 1 + r 2 1 + r t � 1 τ = 1 () Consumption October 2007 11 / 20
Inde…nite Planning Horizon Boundary condition replaced by transversality condition , ! the present market value of household assets cannot be negative in the long run: T ! ∞ D T A T � 0 . lim ) intertemporal budget constraint (at the optimum): ∞ ∞ ∑ ∑ D t c t = A 0 + D t w t . t = 1 t = 1 () Consumption October 2007 12 / 20
Continuous Time (by analogy) Household utility: Z T e � ρ t u ( c ( t )) dt max 0 subject to ˙ A ( t ) = w ( t ) + r ( t ) A ( t ) � c ( t ) 8 t and D ( T ) A ( T ) � 0 , where R t 0 r ( s ) ds D ( t ) = e intertemporal budget constraint: Z T Z T D ( t ) c ( t ) dt � A ( 0 ) + D ( t ) w ( t ) dt . 0 0 () Consumption October 2007 13 / 20
Euler equation: R t u 0 ( c ( s )) = e s r ( τ ) d τ e � ρ ( t � s ) u 0 ( c ( t )) . Totally di¤erentiating w.r.t. to t we get c ( t ) = � u 0 ( c ( t )) 00 ( c ( t )) [ r ( t ) � ρ ] ˙ u CES example: � u 0 ( c ) c � θ θ c � θ � 1 = c 00 ( c ) = θ , u ) Euler equation in di¤erential form: c ( t ) = r ( t ) � ρ c ( t ) ˙ . θ () Consumption October 2007 14 / 20
Optimal Control Approach General problem: Z T max c ( t ) V ( 0 ) = v ( k ( t ) , c ( t ) , t ) dt subject to 0 ˙ k ( t ) = g ( k ( t ) , c ( t ) , t ) k ( T ) � 0 k ( 0 ) = k 0 Set up Lagrangian: Z T L = v ( k ( t ) , c ( t ) , t ) dt 0 Z T � � g ( k ( t ) , c ( t ) , t ) � ˙ + λ ( t ) k ( t ) dt + µ k ( T ) 0 () Consumption October 2007 15 / 20
Note that using integration by parts Z T Z T λ ( t ) ˙ k ( t ) dt = λ ( t ) dk ( t ) 0 0 Z T [ λ ( t ) k ( t )] T = 0 � k ( t ) d λ ( t ) 0 Z T k ( t ) ˙ = λ ( T ) k ( T ) � λ ( 0 ) k ( 0 ) � λ ( t ) dt 0 () Consumption October 2007 16 / 20
It follows that Z T L = [ v ( k ( t ) , c ( t ) , t ) + λ ( t ) g ( k ( t ) , c ( t ) , t )] dt 0 Z T k ( t ) ˙ + λ ( t ) dt � λ ( T ) k ( T ) + λ ( 0 ) k 0 + µ k ( T ) 0 The …rst–order conditions for a maximum are ∂ L ∂ c ( t ) + λ ( t ) ∂ g ∂ v = ∂ c ( t ) = 0 8 t 2 ( 0 , T ) ∂ c ( t ) ∂ L ∂ k ( t ) + λ ( t ) ∂ g ∂ v ∂ k ( t ) + ˙ = λ ( t ) = 0 8 t 2 ( 0 , T ) ∂ k ( t ) How do I remember all this junk? () Consumption October 2007 17 / 20
Hamiltonians Construct a "Hamiltonian function": H ( k , c , t , λ ) � v ( k , c , t ) + λ . g ( k , c , t ) Derive Hamiltonian conditions ∂ H = 0 ∂ c ∂ H � ˙ = λ ∂ k ∂ H ˙ = k ∂λ plus the boundary condition k ( T ) = 0. () Consumption October 2007 18 / 20
Application to consumer’s problem H ( A , c , t , λ ) = e � ρ t u ( c ) + λ [ w + rA � c ] Hamiltonian conditions ∂ H e � ρ t u 0 ( c ) � λ = 0 = ∂ c ∂ H λ r = � ˙ = λ ∂ A ∂ H w ( t ) + r ( t ) A ( t ) � c ( t ) = ˙ = A ∂λ () Consumption October 2007 19 / 20
Di¤erentiate the …rst one w.r.t. time � ρ e � ρ t u 0 ( c ) + e � ρ t u ( c ) 00 ˙ ˙ c = λ � ρ e � ρ t u 0 ( c ) + e � ρ t u ( c ) 00 ˙ � e � ρ t u 0 ( c ) r c = � ρ u 0 ( c ) + u ( c ) 00 ˙ � u 0 ( c ) r c = � u 0 ( c ( t )) c ( t ) = 00 ( c ( t )) [ r ( t ) � ρ ] ˙ u () Consumption October 2007 20 / 20
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