Symbolic partial differentiation of a Lagrangian function including a summation

Question

I'm using Mathematica to double-check that I correctly derived first order conditions for the models I'm researching. Before I get started, I wanted to actually teach myself how to use Mathematica, and check that I can derive FOC's for a model I know. It's the (fairly simple) stochastic growth model with additive labor. I'm solving the model by setting up the following Lagrangian function.

In[24]:= sgmodEq1 = 
 util[c[t], l[t]] == (c[t]^(1 - \[Phi])/(1 - \[Phi])) + 
   Subscript[\[Chi], 0] z[t] ((1 - l[t])^(1 - \[Chi])/(1 - \[Chi]))

In[25]:= sgmodEq2 = 
 f[k[t], l[t]] == a[t] *k[t]^\[Alpha]*(z[t]*l[t])^(1 - \[Alpha])

In[27]:= sgmodEq3 = \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(t = 0\), \(\[Infinity]\)]\(
\*SuperscriptBox[\(\[Beta]\), \(t\)]*\((util[c[t], 
      l[t]] + \[Lambda][
       t]*\((f[k[t], l[t]] - c[t] + \((1 - \[Delta])\)*k[t] - 
        k[t + 1])\))\)\)\)

In[28]:= sgmodEq4 = D[sgmodEq3, k[t + 1]]

Out[28]= \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k[1] = 0\), \(\[Infinity]\)]\(
\*SuperscriptBox[\(\[Beta]\), \(K[1]\)] \[Lambda][K[1]] \((\(-
\*SubscriptBox[\(\[Delta]\), \(t, K[1]\)]\) + \((1 - \[Delta])\) 
\*SubscriptBox[\(\[Delta]\), \(1 + t, K[1]\)] + 
\*SubscriptBox[\(\[Delta]\), \(1 + t, K[1]\)] 
\(\*SuperscriptBox[\(f\), \((1, 0)\)]\)[k[K[1]], l[K[1]]] - 
    c[t])\)\)\)

If you're not familiar with the problem, you should know this is not the correct answer by the fact that $\delta$ is a function of K[1], which is now capitalized for some reason?

Before I set up the Lagrangian, I checked that I was able to correctly take partial derivatives of the utility function and production function. Afterward, I also checked that if I break the problem up into pieces, without the summation, I could get the correct answer, as follows:

In[29]:= sgmodEq5 = \[Beta]^
  t*(util[c[t], 
     l[t]] + \[Lambda][
      t]*(f[k[t], l[t]] - c[t] + (1 - \[Delta])*k[t] - k[t + 1]))

In[30]:= sgmodEq6 = \[Beta]^(
  t + 1)*(util[c[t + 1], 
     l[t + 1]] + \[Lambda][
      t + 1]*(f[k[t + 1], l[t + 1]] - 
       c[t + 1] + (1 - \[Delta])*k[t + 1] - k[t + 2]))

In[31]:= sgmodEq7 = D[sgmodEq5, k[t + 1]]

Out[31]= -\[Beta]^t \[Lambda][t]

In[32]:= sgmodEq8 = D[sgmodEq6, k[t + 1]]

Out[32]= \[Beta]^(1 + t) \[Lambda][1 + t] (1 - \[Delta] + 
\!\(\*SuperscriptBox[\(f\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[k[1 + t], l[1 + t]])

The solution to sgmodEq4 is sgmodEq7+sgmodEq8.

Also, I wish I could attach a notebook, because the code looks super messy. Basically, I'm doing this:

$ \begin{align*} \mathscr{L} &= \sum_{t=0}^\infty \beta^t \left( \left( \frac{C_t^{1-\phi}}{1-\phi} + \chi_0 Z_t \frac{(1-L_t)^{1-\chi}}{1-\chi} \right) \right. \\[1ex] &+\left. \lambda_t \left( A_t K_t^{\alpha} (Z_t L_t)^{1-\alpha} - C_t + (1-\delta) K_t - K_{t+1}\right) \right) \end{align*} $

And this is the result I should be getting:

$ \frac{\partial \mathscr{L}}{\partial k_{t+1}}:\ \lambda_t = \beta \ \lambda_{t+1} \left( \alpha A_t K_t^{\alpha-1} (Z_t L_t)^{1-\alpha} + (1-\delta) \right) $

And, instead, I am getting this nonsense:

$ \sum_{K[1]=0}^\infty \beta^{K[1]} \lambda[K[1]] \left( -\delta_{t,K[1]} + (1-\delta) \delta_{1+t,K[1]} + \delta_{1+t,K[1]} f^{(1,0)}[k[K[1]],l[K[1]]] \right) $

But, I can get the right answer if I break the problem into parts. So, my questions are: (1) Why do I get the wrong answer (and such a strange answer) when I put all the pieces together?

And, (2) how can I adjust my code so that I can correctly take the partial derivative of the Lagrangian function, including the summation, and get the right answer?

Answer 1

Changing Subscript[\[Chi], 0] to \[Gamma], and using Assuming[t ∈ NonNegativeIntegers, D[sgmodEq3, k[t + 1]]] gives me this result:

$ -\beta^t \left( \lambda_t - \beta \lambda_{t+1} + \beta \delta \lambda_{t+1} - \beta \lambda_{t+1} f^{(1,0)}(k_{t+1},l_{t+1}) \right) $

which is the correct answer when I set it equal to zero, rearrange and simplify.

(I guess now I just need to learn how to get Mathematica to group terms, keep terms as I originally grouped them and simplify.)

Thank you all!