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This is the production function (two input factors: $x_1$ and $x_2$)

$$q=A[δx_1^ρ+(1-δ)x_2^ρ]^{\frac{1}{ρ}} $$

If distribution factor $δ$ is set to increase, what are the economic impacts on these two input factors? Substitution elasticity remains unchanged.

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In this answer you can see that $\delta$ becomes the exponent in the Cobb-Douglas function as $\rho \to 0$, i.e. as the elasticity of substitution becomes unity.
The exponent in the Cobb-Douglas production function is the output elasticity of the input(s), but also, it is the "output share" of each factor, under competitive conditions. This is why the scholars that introduced the C.E.S. function named this parameter "distribution parameter".
So you can start thinking from there.

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