Existence, necessity and sufficiency: In general, to ensure that the Euler equation
characterizes the optimal path, one typically requires that the objective is finite (in this
example, u0 > 0) and that some feasible path exists.
Further, since Euler equations are first-order conditions, they are necessary but not sufficient conditions for an optimal dynamic path. Thus, theoretical results based only on Euler equations are applicable to a range of models. On the other hand, the equa- tions provide an incomplete characterization of equilibria. In the example, only by using the budget constraint also, can one solve for the time-path of consumption; its level is determined by the present value of income.
Dynamic analysis: More generally, complete characterization of optimal behavior uses the Euler equation as one equation in a system of equations. For example, replacing the budget constraint (equation (1)) with the capital-accumulation equation
kt+1 =f(kt)−ct +(1−δ)kt (3)
wherekiscapital,f(k)isoutput,f0 >0,f00 <0,f(0)=0,limk→0f0 >β−1−(1−δ), and limk→∞ f0 < β−1 − (1 − δ), and adding the constraints k1 given, kt ≥ 0, and ct ≥ 0, gives the basic Ramsey growth model. The constant real interest rate of equation (2) is replaced by the marginal product of capital in the resulting Euler equation
u0 (ct) = β (1 − δ + f0 (kt+1)) u0 (ct+1) . (4)
Further, since Euler equations are first-order conditions, they are necessary but not sufficient conditions for an optimal dynamic path. Thus, theoretical results based only on Euler equations are applicable to a range of models. On the other hand, the equa- tions provide an incomplete characterization of equilibria. In the example, only by using the budget constraint also, can one solve for the time-path of consumption; its level is determined by the present value of income.
Dynamic analysis: More generally, complete characterization of optimal behavior uses the Euler equation as one equation in a system of equations. For example, replacing the budget constraint (equation (1)) with the capital-accumulation equation
kt+1 =f(kt)−ct +(1−δ)kt (3)
wherekiscapital,f(k)isoutput,f0 >0,f00 <0,f(0)=0,limk→0f0 >β−1−(1−δ), and limk→∞ f0 < β−1 − (1 − δ), and adding the constraints k1 given, kt ≥ 0, and ct ≥ 0, gives the basic Ramsey growth model. The constant real interest rate of equation (2) is replaced by the marginal product of capital in the resulting Euler equation
u0 (ct) = β (1 − δ + f0 (kt+1)) u0 (ct+1) . (4)
Existence, necessity and sufficiency: In general, to ensure that the Euler equation characterizes the optimal path, one typically requires that the objective is finite (in this example, u0 > 0) and that some feasible path exists.
Further, since Euler equations are first-order conditions, they are necessary but not sufficient conditions for an optimal dynamic path. Thus, theoretical results based only on Euler equations are applicable to a range of models. On the other hand, the equa- tions provide an incomplete characterization of equilibria. In the example, only by using the budget constraint also, can one solve for the time-path of consumption; its level is determined by the present value of income.
Dynamic analysis: More generally, complete characterization of optimal behavior uses the Euler equation as one equation in a system of equations. For example, replacing the budget constraint (equation (1)) with the capital-accumulation equation
kt+1 =f(kt)−ct +(1−δ)kt (3)
wherekiscapital,f(k)isoutput,f0 >0,f00 <0,f(0)=0,limk→0f0 >β−1−(1−δ), and limk→∞ f0 < β−1 − (1 − δ), and adding the constraints k1 given, kt ≥ 0, and ct ≥ 0, gives the basic Ramsey growth model. The constant real interest rate of equation (2) is replaced by the marginal product of capital in the resulting Euler equation
u0 (ct) = β (1 − δ + f0 (kt+1)) u0 (ct+1) . (4)
period deviations can be considered, but they follow from sequences of one-period devi-
ations and so doing so does not provide additional information (u0 (ct) = β2R2u0 (ct+2)).
These equations imply that the optimizing agent equalizes the present-value marginal flow
benefit from the control across periods.
The canonical application of this problePm is to a household or representative agent: call c consumption, u utility, and let W1 = ∞t=1 R1−tyt, the present value of (exogenous) income, y. In this case, equations (2) imply the theoretical result that variations in income do not cause consumption to rise or fall over time. Instead, marginal utility grows or declines over time as βR ≷ 1; for βR = 1, consumption is constant.
Existence, necessity and sufficiency: In general, to ensure that the Euler equation characterizes the optimal path, one typically requires that the objective is finite (in this example, u0 > 0) and that some feasible path exists.
Further, since Euler equations are first-order conditions, they are necessary but not sufficient conditions for an optimal dynamic path. Thus, theoretical results based only on Euler equations are applicable to a range of models. On the other hand, the equa- tions provide an incomplete characterization of equilibria. In the example, only by using the budget constraint also, can one solve for the time-path of consumption; its level is determined by the present value of income.
Dynamic analysis: More generally, complete characterization of optimal behavior uses the Euler equation as one equation in a system of equ
The canonical application of this problePm is to a household or representative agent: call c consumption, u utility, and let W1 = ∞t=1 R1−tyt, the present value of (exogenous) income, y. In this case, equations (2) imply the theoretical result that variations in income do not cause consumption to rise or fall over time. Instead, marginal utility grows or declines over time as βR ≷ 1; for βR = 1, consumption is constant.
Existence, necessity and sufficiency: In general, to ensure that the Euler equation characterizes the optimal path, one typically requires that the objective is finite (in this example, u0 > 0) and that some feasible path exists.
Further, since Euler equations are first-order conditions, they are necessary but not sufficient conditions for an optimal dynamic path. Thus, theoretical results based only on Euler equations are applicable to a range of models. On the other hand, the equa- tions provide an incomplete characterization of equilibria. In the example, only by using the budget constraint also, can one solve for the time-path of consumption; its level is determined by the present value of income.
Dynamic analysis: More generally, complete characterization of optimal behavior uses the Euler equation as one equation in a system of equ
