Let a utility function for a consumer be defined as $u(x_{1},x_{2})=x_{1}^{1/2} x_{2}^{1/2}$. With the budget $x_{1}p_{1}+x_{2}p_{2}=m$. With values $p_1=p_2=1, m=32$. The state now adds a tax of unit 3 on $p_{1}$ (pr. Unit $x_1$)

How does it effect utility? What does the state earn?

  • I got the utility before to be 16 and after to be 8 with taxes correlating to 12 pr unit $x_1$

The state now considers an income tax such that the income is now $m-T$ How much will the state earn with the new system whilst keeping the consumer indifferent? Which system is better?

  • I figured that i solve for the Tax in the utility function under optimal demand conditions so that i kepy utility equal to 8. This gave me 16 units of income tax.

How does one do the last part mathematically. I figure the income tax is better for the consumer but how can i show it mathematically?


The intuition you have is correct. Mathematically you can show it by first deriving the optimal choices with the lump sum income tax. So you will set up the following lagrangian:

$$\mathcal{L} = x^{1/2}_1 x^{1/2}_2 - \lambda [x_1p_1+x_2p_2 - m + T] $$

This gives you 3 FOC's the budget constraint and:

$$ 0.5x_1^{-0.5} x_2^{0.5} = \lambda p_1 \\ 0.5x_2^{-0.5} x_2^{0.5} = \lambda p_2$$

Solve for optimal $x_1^*$ and $x_2^*$:

$$ x_1^* = \frac{m-T}{2p_1} = \frac{32-T}{2} \\ x_2^* = \frac{m-T}{2p_2} = \frac{32-T}{2}$$

Where here the second equalities take advantage of the assumptions that $p_1 = p_2=1$ and $m=32$.

Now you can just plug this into the utility function and assuming you made no mistake equate this to the utility with the consumption tax on $p_1$ so you will have:

$$ 8 = \left( \frac{32-T}{2}\right)^{0.5} \left( \frac{32-T}{2}\right)^{0.5} \\ T =16$$

So under the income tax regime government gets $T=16> t=12$ while consumer still has the same utility as under the consumption tax, meaning the income tax is better. The intuition for that is that income tax does not distort the relative prices only has an income effect whereas consumption tax has both income and substitution effects.

  • $\begingroup$ I have done everything up to " The intuition for that is that income tax does not distort the relative prices only has an income effect whereas consumption tax has both income and substitution effects." Can this quantity actually be shown I get the reasons but no idea on how to show it for this example? $\endgroup$ – user31331 Nov 23 '20 at 20:30
  • $\begingroup$ @bymathformath what do you mean by quantity? The substitution effect will be there with consumption tax because it disturbs the relative prices. With consumption price of 3 for $x_1$ the price of $x_1$ increases from $1$ to $4$ making the consumer want to substitute away from consuming $x_1$ to consuming $x_2$ in addition to consumer having lower income due to tax (as it changes the budget constraint from $x_1+x_2=32$ to $4x_1+x_2 =32$. Income tax only lowers the income without distorting the prices. You can quantify these effects using hicksian decomposition - but that was not part of the Q $\endgroup$ – 1muflon1 Nov 23 '20 at 20:35
  • $\begingroup$ "You can quantify these effects using hicksian decomposition" I was lacking the words to describe it, sorry. But yes - this. That is what I meant* $\endgroup$ – user31331 Nov 23 '20 at 20:37
  • $\begingroup$ How would one do that with this Cobb-Douglas? $\endgroup$ – user31331 Nov 23 '20 at 20:40
  • $\begingroup$ @bymathformath with hicksian decomposition, you can calculate the substitution effect by looking at what is the change in demand, holding utility constant and then the income effect will be rest. However, note this was not part of the question in the black so you should not just add that to an answer unless requested I was just giving you little bit of an tidbit of intuition how come that we got the result that we got. It is trivial to see that in the case of income tax there will be no substitution effect as that requires changes in prices. $\endgroup$ – 1muflon1 Nov 23 '20 at 20:45

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