(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Derive the Euler Equation of the dynamic programming problem:

[itex]\[max_{\{ z_t \}^\infty_{t=0}}

\sum_{t=0}^{\infty} \delta^t f(x_t, y_t, z_t)

\] [/itex]

subject to:

[itex] x_{t+1} = g_1(x_t, y_t, z_t),

\

y_{t+1} = g_2(x_t, y_t, z_t),

\

x_0 = x^0,

y_0 = y^0 [/itex]

and where [itex] \delta <1 [/itex]

2. Relevant equations

We can write the value function as:

[itex]V(x,y) = max_z [f(x, y, z) + \delta V(g_1(x, y, z), g_2(x, y, z))][/itex]

3. The attempt at a solution

The solution is characterized by 3 equations:

The first-order-condition at the optimal z* is:

[itex] \frac{\partial f(x, y, z^*)}{\partial z} + \delta[\frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_1(x, y, z^*)}\frac{\partial g_1(x, y, z^*)}{\partial z} + \frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_2(x, y, z^*)}\frac{\partial g_2(x, y, z^*)}{\partial z}] = 0

[/itex]

Differentiating the value function with respect to each state gives us:

[itex]

\frac{\partial V (x, y)}{\partial x} = \frac{\partial f(x, y, z^*)}{\partial x} + \delta[\frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_1(x, y, z^*)}\frac{\partial g_1(x, y, z^*)}{\partial x} + \frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_2(x, y, z^*)}\frac{\partial g_2(x, y, z^*)}{\partial x}]

[/itex]

and

[itex]

\frac{\partial V (x, y)}{\partial y} = \frac{\partial f(x, y, z^*)}{\partial y} + \delta[\frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_1(x, y, z^*)}\frac{\partial g_1(x, y, z^*)}{\partial y} + \frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_2(x, y, z^*)}\frac{g_2(x, y, z^*)}{\partial y}]

[/itex]

I think I should be able to derive an Euler Equation with the above 3 equations, but I'm not sure how to manipulate the equations to get a meaningful answer.

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# Homework Help: Dynamic Programming with 2 state variables

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