Hello,

I have the following problem. There are known dependencies between variables.

$\displaystyle a = a1 + a2.$

$\displaystyle C = C1 + C2.$

$\displaystyle C = \frac{a}{{{{\left( {1 + d} \right)}^t}}},C1 = \frac{{a1}}{{{{\left( {1 + d1} \right)}^t}}},C2 = \frac{{a2}}{{{{\left( {1 + d2} \right)}^t}}}.$

The desired quantity is d. In this simple case it can be easily expressed using C1, C2, C, d1, d2. The solution is following

$\displaystyle {\left( {1 + d} \right)^t} = \frac{a}{C} = \frac{{a1 + a2}}{C} = \frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}.$

$\displaystyle d = {\left( {\frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}} \right)^{\frac{1}{t}}} - 1.$

But I am confused with the similar, but a bit more complex problem.

The known dependencies between variables are following

$\displaystyle {a_t} = a{1_t} + a{2_t}.$

$\displaystyle {C_t} = C{1_t} + C{2_t}.$

$\displaystyle {C_{t - 1}} = \frac{{{a_t}}}{{{{\left( {1 + {d_t}} \right)}^t}}} + \frac{{{C_t}}}{{\left( {1 + {d_t}} \right)}},C1 = \frac{{a{1_t}}}{{{{\left( {1 + d{1_t}} \right)}^t}}} + \frac{{C{1_t}}}{{\left( {1 + d{1_t}} \right)}},C2 = \frac{{a2{}_t}}{{{{\left( {1 + d{2_t}} \right)}^t}}} + \frac{{C{2_t}}}{{\left( {1 + d{2_t}} \right)}}.$

t - time index.

The desired quantity is $\displaystyle {d_t}$ .

The goal is to express $\displaystyle {d_t}$

using C1, C2, C, d1, d2, a1 and a2. The solution of the equation seems cumbersome. Is it possible to reduce the resulting formula to elegant form, which would resemble $\displaystyle d = {\left( {\frac{{{{\left( {1 + d1} \right)}^t}C1 + {{\left( {1 + d2} \right)}^t}C2}}{C}} \right)^{\frac{1}{t}}} - 1$?