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I am currently working through problems to study for an upcoming exam. I am not seeking a solution per se. I am looking at the intertemporal choice model. I am looking at two periods where consumption in each is normal. Let $y_1, y_2>0$ be fixed endowments and $p_1=p_2=1$ be the prices of consumption in periods $1$ and $2$, respectively. Denote consumption by $c_1$ and $c_2$. Using the Slutsky Decomposition $$\frac{\partial c_t}{\partial r} = \frac{\partial h_t}{\partial r} + \frac{\partial c_t}{\partial W_1}\frac{(y_{1}-c_{1})}{1+r}$$ where $W_1 = y_1 + \frac{y_2}{1+r}$ is the net present value of wealth, $r$ is the interest rate, and $(y_1 - c_1)$ represents savings if positive, and borrowing if negative. The individual has access to capital markets in this respect.

We are given that for a drop in the interest rate, the person's consumption for $t=1$ does not change which (w/o proof) implies that the person is a saver.

The next part of the question states

suppose the drop in interest rate is due to the introduction of a capital gains tax without deductibility of capital losses. Given the person's behaviour above, is the tax efficient? Prove your answer graphically.

So this gives us a situation where only the portion of the resource constrain such that $c_1<y_1$ (individual is a saver). This gives a piece-wise resource constraint. Since there is no compensation mentioned, I assume the welfare measure to be used is Equivalent Variation (EV). My question is, how would we define whether a tax is efficient?

Would it be efficient if the Deadweight Loss is equal to zero? Or is it related to how large the tax revenue would be relative to EV? The answer to this question will give me enough information to be able to display this graphically.

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When there is a deadweight loss associated with a particular policy (e.g. a tax) the policy is said to be inefficient under the Pareto Efficiency definition. In the above circumstance, the capital gains tax could have an equivalent increase in government revenue as a lump-sum income tax but would cause a larger decrease in the welfare of those saving/investing. This implies that there is a way to make at least one party better off without making another worse off. This is what the differential between EV and tax revenue is capturing.

The tax is efficient $\iff$ $DWL=0$

under the Pareto Efficiency defintion.

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    $\begingroup$ Just to add to this - the Pareto Efficiency definition only makes sense in models for which all meaningful costs are internalized. $\endgroup$ – heh Jan 21 at 16:28

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