question on chain rule

Question

I have a function $v(a_t, q_t)$ and the objective function is $\sum_t {v(a_t, q_t)}$ Here, q evolves according to $q_{t+1} = a_t + q_t$, and the cross-partial $v_{aq}(.)$ is positive, so $a_{t+1}$ is affected by a change in $a_t$ through $q_{t+1}$ as well. Now, I want to take the derivative of my objective function with respect to $a_t$. Suppose the end period T = 2. I am having troubles with seeing the following:

$\frac{\partial v (a_{t+2}, q_{t+2})}{\partial a_t} \neq \frac{\partial v }{\partial a_{t+2}}\frac{\partial a_{t+2} }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t+1}}\frac{\partial a_{t+1} }{\partial q_{t+1}}\frac{\partial q_{t+1} }{\partial a_{t}} + \frac{\partial v }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t+1}}\frac{\partial a_{t+1} }{\partial q_{t+1}}\frac{\partial q_{t+1} }{\partial a_{t}}$

But actually $\frac{\partial v (a_{t+2}, q_{t+2})}{\partial a_t} = \frac{\partial v }{\partial a_{t+2}}\frac{\partial a_{t+2} }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t}} + \frac{\partial v }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t}} $ since $q_{t+2}$ is a function of $(a_{t+1}, q_{t+1})$ which implies it is a function of $(\frac{\partial a_{t+1}}{q_{t+1}}\frac{q_{t+1}}{a_t}, (q_t + a_t))$

My question is why $\frac{\partial v }{\partial a_{t+2}}\frac{\partial a_{t+2} }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t+1}}\frac{\partial a_{t+1} }{\partial q_{t+1}}\frac{\partial q_{t+1} }{\partial a_{t}} + \frac{\partial v }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t+1}}\frac{\partial a_{t+1} }{\partial q_{t+1}}\frac{\partial q_{t+1} }{\partial a_{t}} \neq \frac{\partial v }{\partial a_{t+2}}\frac{\partial a_{t+2} }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t}} + \frac{\partial v }{\partial q_{t+2}}\frac{\partial q_{t+2} }{\partial a_{t}}$ ??

Answer 1

Let $a_{t+1}=f(a_t)$. Then,

\begin{equation} v(a_2,q_2)=v(f(f(a_0)),f(a_0)+a_0+q_0). \end{equation}

If we calculate the total derivative of the above function with respect to $a_0$, it should be

\begin{equation} \frac{d}{da_0}v(f(f(a_0)),f(a_0)+a_0+q_0)\\ =\frac{\partial v(a_2,q_2)}{\partial a_2}\cdot f'(a_1)\cdot f'(a_0)+\frac{\partial v(a_2,q_2)}{\partial q_2}\cdot (f'(a_0)+1). \end{equation}

I don't quite understand what you mean with your algebra.