Euler equation and time iterations¶
Description: First order conditions and Euler equation. Time iterations solution method. Euler residuals for measuring the accuracy of solution for consumption-savings model.
First order conditions in dynamic models with continuous choice¶
We can write FOCs for the maximization problem in Bellman equation! Is there any use?
- must be satisfied with optimal policy $ \Rightarrow $ solution method ideas
- backwards induction with equation solving instead of optimization
- time iterations in the infinite horizon problems
- must be satisfied even when the problem is solved by some other method $ \Rightarrow $ ideas of checking accuracy of numerical solutions
- Euler residuals accuracy check
- flat consumption path must be simulated with certain restrictions on parameters (Keane test)
- other known theoretical property of the solution must hold
FOCs in the consumption-savings problem (Deaton model)¶
$$ V(M)=\max_{0 \le c \le M}\big\{u(c)+\beta \mathbb{E}_{y} V\big(\underset{=M'}{\underbrace{R(M-c)+\tilde{y}}}\big)\big\} $$- discrete time, infinite horizon
- one continuous choice of consumption $ 0 \le c \le M $
- state space: consumable resources in the beginning of the period $ M $, discretized
- income $ \tilde{y} $, follows log-normal distribution with $ \mu = 0 $ and $ \sigma $
First order conditions for Deaton model¶
$$ V(M)=\max_{0 \le c \le M}\big\{u(c)+\beta \mathbb{E}_{y} V\big(\underset{=M'}{\underbrace{R(M-c)+\tilde{y}}}\big)\big\} $$FOC:
$$ u'(c^\star) - \beta R \mathbb{E}_{y} V'\big(R(M-c^\star)+\tilde{y}\big) = 0 $$- define implicit function $ c^\star(M) $ — policy function
- but $ V'(\cdot) $ requires special provisions in the numerical implementation..
Envelope theorem¶
$$ \text{Let } G(M,c)=u(c)+\beta \mathbb{E}_{y} V\big(\underset{=M'}{\underbrace{R(M-c)+\tilde{y}}}\big) $$so that the policy function $ c^\star(M) $ satisfies $ V(M)=G(M,c^\star(M)) $. Then
$$ V'(M) = \tfrac{\partial G(M,c^\star)}{\partial M} + \underset{=0\text{ by FOC}}{\underbrace{\tfrac{\partial G(M,c^\star)}{\partial c^\star}}} \tfrac{\partial c^\star(M)}{\partial M} = \tfrac{\partial G(M,c^\star)}{\partial M} = \beta R \mathbb{E}_{y} V'\big(R(M-c^\star)+\tilde{y}\big) $$Euler equation for Deaton model¶
$$ \text{(FOC) } u'(c^\star) = \beta R \mathbb{E}_{y} V'\big(R(M-c^\star)+\tilde{y}\big) $$$$ \text{(envelope theorem) } V'(M) = \beta R \mathbb{E}_{y} V'\big(R(M-c^\star)+\tilde{y}\big) $$Thus, we have $ u'(c^\star) = V'(M) $ in every period, and thus
$$ u'\big(c^\star(M)\big) = \beta R \mathbb{E}_{y} u'\big(c^\star\big(\underset{=M'}{\underbrace{R[M-c^\star(M)]+\tilde{y}}}\big)\big) $$Consumption smoothing¶
In deterministic model where $ \tilde{y} $ is fixed, if $ \beta R = 1 $ we have in every two consecutive periods
$$ u'\big(c^\star(M)\big) = u'\big(c^\star(M')\big) \Rightarrow c^\star(M) = c^\star(M') $$Perfect consumption smoothing!
This is one of the tests for the correct solution of the consumption-savings model!
Accuracy measure using Euler equation¶
Common in the literature is to use the average squared Euler residuals on a dense grid with $ K $ points as a measure of accuracy of the solution for Deaton model
$$ ER\big( c(M) \big) = u'\big(c(M)\big) - \beta R \mathbb{E}_{y} u'\big(c\big(R[M-c(M)]+\tilde{y}\big)\big) $$$$ Q \big( c(M) \big) = \frac{1}{K} \sum_{k=1}^{K} {ER}^2\big( c(M_k) \big) $$The closer $ Q\big( c(M) \big) $ is to zero, the better the solution
Time iterations¶
The idea of this solution method is to solve the Euler equation in the space of policy functions $ c(M) \in \mathcal{P} $ as a functional equation
$$ u'\big(c(M)\big) = \beta R \mathbb{E}_{y} u'\big(c[R(M-c(M))+\tilde{y}]\big) $$The solution is given by the fixed point of the Coleman-Reffett operator $ K(c)(M) $
- takes as input policy function $ c(M) \in \mathcal{P} $
- returns the updated policy function $ c'(M) \in \mathcal{P} $ that for every $ M $ satisfies
Contraction mapping?¶
Is Coleman-Reffett operator a contraction mapping? Yes (in the reasonable metric space with a specially defined norm)
📖 Huiyu Li, John Stachurski (2014, Journal of Economic Dynamics and Control) “Solving the income fluctuation problem with unbounded rewards”
- Existence and uniqueness of the solution!
- Successive approximations solver will deliver the solution
- Globally convergent
Time iteration algorithm¶
- Discretize the state space
- Set the initial policy $ c_0(M) $ at state grid
- Increment iteration counter $ i $ (initialize to 0)
- Solve Euler equation in every point of the the grid, i.e. plug $ c_{i-1}(M) $ to the
RHS of the Euler equation, and solve for the $ c $ in the LHS, it becomes new $ c_i(M) $
- Check for convergence in policy function space:
- If converged, output $ c_i(M) $
- Otherwise, return to step 3.
Accuracy and speed¶
How does time iteration solver compares to VFI (with explicit maximization)?
- theoretical complexity and convergence rates are the same
- policy functions are easier to interpolate (less curvature), so interpolation errors are smaller
- therefore time iterations usually converge faster
- for the same reasons time iterations deliver more accurate solution
Will do some experiments in the next practical video
Further learning resources¶
- Derivation of Euler equation in cake eating model https://python.quantecon.org/cake_eating_problem.html
- Time iterations on QuantEcon https://python.quantecon.org/coleman_policy_iter.html