Allocation of Price and Volume changes to a change in Rate

Consider a period $\boldsymbol{\tau} = (\mathbf{p}, \mathbf{v})$ with prices $\mathbf{p} = (p_1, \ldots, p_n)$ and volumes $\mathbf{v} = (v_1, \ldots, v_n)$. We may define the average price as the weighted average $$ \alpha(\mathbf{p}, \mathbf{v}) = \frac{p_1v_1 + \cdots p_nv_n}{v_1 + \cdots v_n}, $$ or as $$ \alpha(\mathbf{p}, \mathbf{w}) = p_1w_1 + \cdots p_nw_n, $$ where $w_i = v_i/v$ is a weighting factor. Here $v = v_1 + \cdots v_n$ is the total volume. Now consider two periods $\boldsymbol{\tau}^{1} = (\mathbf{p}^1, \mathbf{w}^1)$ and $\boldsymbol{\tau}^{2}= (\mathbf{p}^2, \mathbf{w}^2)$. You seek to describe the difference $\Delta \alpha = \alpha(\mathbf{p}^2, \mathbf{w}^2) - \alpha(\mathbf{p}^1, \mathbf{w}^1)$ as a sum involving ``impacts'' $$ \Delta \alpha = \sum_{i=1}^{n} (\Delta_{\mathbf{p}} \alpha)_{i} + \sum_{i=1}^{n} (\Delta_{\mathbf{w}} \alpha)_{i}, \tag{$\star$} $$ where $\Delta_{\mathbf{p}} \alpha$ are the price impacts and $\Delta_{\mathbf{w}} \alpha$ are the volume impacts. Moreover, you wish for the $i$th impact to only depend the $i$th product's values and the aggregrate information for the period. Since the gradient $\alpha$ is linear in $\mathbf{p}$ and $\mathbf{w}$, the key is to use what is known as the trapezoidal rule (applied to the gradient of $\alpha$). It states that (this is an exact equality) $$ (\Delta_{\mathbf{p}} \alpha)_i = \frac{1}{2}\left[\left.\frac{\partial \alpha}{\partial p_i}\right|_{\boldsymbol{\tau}^{1}} + \left.\frac{\partial \alpha}{\partial p_i}\right|_{\boldsymbol{\tau}^{2}}\right] \Delta p_i. $$ For $\mathbf{w}$, the situation is slightly more complicated since we must have that $w_1 + \cdots + w_n = 1$. To account for this(and give the qualitative behavior you desire), one takes the sum of the contribution to $\alpha$ given by $w_i$ due to the trapezoidal rule, and the contribution to $\alpha$ due to the rest of the $w_j$ changing to keep the sum equal to unity. Thus, $$ (\Delta_{\mathbf{w}} \alpha)_i = \frac{1}{2}\left[\left.\frac{\partial \alpha}{\partial w_i}\right|_{\boldsymbol{\tau}^{1}} + \left.\frac{\partial \alpha}{\partial w_i}\right|_{\boldsymbol{\tau}^{2}}\right] \Delta w_i + \frac{1}{2} \left[(\alpha^1+\alpha^2) \sum_{i \neq j} (w^2_j - w^1_j) \right]. $$ Simplifying, using the fact that the weights sum to unity, one has $$ (\Delta_{\mathbf{w}} \alpha)_i = \frac{1}{2}\left[\left.\frac{\partial \alpha}{\partial w_i}\right|_{\boldsymbol{\tau}^{1}} + \left.\frac{\partial \alpha}{\partial w_i}\right|_{\boldsymbol{\tau}^{2}}\right] \Delta w_i + \frac{1}{2} \left[(\alpha^1+\alpha^2) (w^1_i-w^2_i) \right] $$ It remains to explicitly compute these approximations. For price impacts, we have $\partial \alpha/\partial p_i = w_i$. Thus $$ (\Delta_{\mathbf{p}} \alpha)_i = \frac{(w^1_i + w^2_i) \cdot (p^2_i - p^1_i)}{2}. $$ For volume impacts, we have $\partial \alpha/\partial w_i = p_i$. Thus $$ (\Delta_{\mathbf{w}} \alpha)_i = \frac{[(p^1_i-\alpha^1) + (p^2_i-\alpha^2)] \cdot (w^2_i - w^1_i)}{2}. $$ It's not hard to see that $$ (\Delta_{\mathbf{p}} \alpha)_i + (\Delta_{\mathbf{w}} \alpha)_i = p^2_iw^2_i - p^1_iw^1_i + \frac{1}{2} \left[(\alpha^1+\alpha^2) (w^1_i-w^2_i) \right], $$ and hence we indeed obtain ($\star$), as the right term sums to zero and the left sums to $\alpha^2 - \alpha^1$. Moreover, the desired qualities you specify are fulfilled. If $\Delta p_i > 0$ then the price impact is positive and vice versa. Furthermore, if $p_i$ is less than the average price during both periods, then $(\Delta_{\mathbf{w}}\alpha)_i < 0$ if $\Delta w_i > 0$ and vice versa. To finish, we calculate the impacts in the example you give. For price impacts, one calculates $(\Delta_{\mathbf{p}} \alpha)_1 = \$0.2197$, $(\Delta_{\mathbf{p}} \alpha)_2 = -\$0.0265$, and $(\Delta_{\mathbf{p}} \alpha)_3 = \$0.0295$. For volume impacts, one calculates $(\Delta_{\mathbf{w}} \alpha)_1 = \$0.1773$, $(\Delta_{\mathbf{w}} \alpha)_2 = -\$0.0913$, and $(\Delta_{\mathbf{w}} \alpha)_3 = \$0.3535$. The sum of these values is $\$0.662 = \Delta \alpha$, as desired.