# Question:gradient in terms of sum set of indexed variables of sum defined in terms of those variables

## Question:gradient in terms of sum set of indexed variables of sum defined in terms of those variables

How does one take the gradient of a expression involving sums in terms of a set of indexed variables.

I am trying to find the maxima for the satisfaction for a public good from a tax code. The sum represents the total revenue. The tax code is defined as a piecewise linear function of income and indexed variables. The indexed variables represent the points of the kinks and the marginal tax rate for each income bracket. (The set of indexed variables will vary based upon other parameters of the problem--and I want to write the solution in terms of these values. That is I want a solution in terms of M sub p and N sub p. Ultimately, I will use Maple's code generation package to transfer the results to our simulation written in Java.

tried several approaches by creating loops to do the differentiation with respect to each indexed variable. I also tried the suggestion of Alex Mihailov in Maple Primes to the question by Paul Weiss. None have worked; can anyone suggest the best approach?

Dr. Laurence Leff Western Illinois University, Macomb IL 61455 ||(309) 298-1315 Stipes 447 Assoc. Prof. of Computer Sci. Pager: 309-367-0787 FAX: 309-298-2302

``` Tax:=(p) ->
> sum  (R[i+1]*k[L], i=M[p]+1..N[p]) + R[M[p]]*(k[M[p]]-Inb[p])
> +R[N[p]]*(Ine[p]-k[N[p]]);
/    N[p]                  \
|   -----                  |
|    \                     |
Tax := p -> |     )       R[i + 1] k[L]| + R[M[p]] (k[M[p]] - Inb[p])
|    /                     |
|   -----                  |
\i = M[p] + 1              /

+ R[N[p]] (Ine[p] - k[N[p]])

> TT := sum (Tax(p)*Pop[p],p=1..P);
P    /    N[p]                  \
-----  |   -----                  |
\     |    \                     |
TT :=   )   (|     )       R[i + 1] k[L]| + R[M[p]] (k[M[p]] - Inb[p])
/     |    /                     |
-----  |   -----                  |
p = 1  \i = M[p] + 1              /

+ R[N[p]] (Ine[p] - k[N[p]])) Pop[p]

# this is wht we optimize
> ToP:=m[i]*TT^2/2+b[i]*TT - Tax[i];
P    /    N[p]                  \
-----  |   -----                  |
\     |    \                     |
ToP := 1/2 m[i] (  )   (|     )       R[i + 1] k[L]|
/     |    /                     |
-----  |   -----                  |
p = 1  \i = M[p] + 1              /

+ R[M[p]] (k[M[p]] - Inb[p]) + R[N[p]] (Ine[p] - k[N[p]])) Pop[p])^2 +

P    /    N[p]                  \
-----  |   -----                  |
\     |    \                     |
b[i] (  )   (|     )       R[i + 1] k[L]| + R[M[p]] (k[M[p]] - Inb[p])
/     |    /                     |
-----  |   -----                  |
p = 1  \i = M[p] + 1              /

+ R[N[p]] (Ine[p] - k[N[p]])) Pop[p]) - Tax[i]

> print(`hello`.ToP);
P    /    N[p]                  \
-----  |   -----                  |
\     |    \                     |
hello.(1/2 m[i] (  )   (|     )       R[i + 1] k[L]|
/     |    /                     |
-----  |   -----                  |
p = 1  \i = M[p] + 1              /

+ R[M[p]] (k[M[p]] - Inb[p]) + R[N[p]] (Ine[p] - k[N[p]])) Pop[p])^2 +

P    /    N[p]                  \
-----  |   -----                  |
\     |    \                     |
b[i] (  )   (|     )       R[i + 1] k[L]| + R[M[p]] (k[M[p]] - Inb[p])
/     |    /                     |
-----  |   -----                  |
p = 1  \i = M[p] + 1              /

+ R[N[p]] (Ine[p] - k[N[p]])) Pop[p]) - Tax[i])```

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