Dynamic Programming with 2 state variables

[physicsguy142]

Homework Statement

Derive the Euler Equation of the dynamic programming problem:

$\[max_{\{ z_t \}^\infty_{t=0}} \sum_{t=0}^{\infty} \delta^t f(x_t, y_t, z_t) \]$

subject to:

$x_{t+1} = g_1(x_t, y_t, z_t), \ y_{t+1} = g_2(x_t, y_t, z_t), \ x_0 = x^0, y_0 = y^0$

and where $\delta <1$

Homework Equations

We can write the value function as:

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

The Attempt at a Solution

The solution is characterized by 3 equations:

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

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

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

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

and

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

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.

 

[physicsguy142]

bump. any help would be greatly appreciated!

 

[physicsguy142]

bumping this on the off-chance somebody will be able to help out.