I'm studying for an exam and facing a question about the relationship between the $\alpha$ of the Cobb-Douglas function and the loss of utility of imposing taxes.

If I have understood correctly, given a function ${X}^{\alpha}Y^{1-\alpha}$, the DWL because of a tax on good X is defined as the lump-sum tax the agent would pay to avoid the tax minus the income from the tax.

The lump-sum tax is $m(1-\frac{p}{p+t}^\alpha)$ where $m$ is the income and $p$ the before-tax price and the income from the tax is $\alpha \frac{m}{p+t} t$ where $\alpha \frac{m}{m+t}$ is the quantity of good X consumed and $t$ is the tax income by unit of x.

Deriving the DWL in Wolfram Alpha I get $\frac{d}{da}=-m\frac{p}{p+t}^\alpha log\frac{p}{p+t} - \frac{mt}{p+t}$, where the first term is always positive for a tax since $\frac{p}{p+t} < 1$.

I don't know how to go from here. Would the answer be that there is no clear effect for a change in $\alpha$ in the DWL? My intuition is that a high value of $\alpha$ means a higher lose of utility for the individual but also a higher income from the tax because MRS is higher and therefore good X can't be easily substituted by good Y. Is that correct?


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