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In case of quasi-linear preference, why would one unit more of the numeraire good (good 1) give the same additional utility as spending an additional amount of wealth equal to the cost of one unit of good 1 on all other goods no matter how much unit of this good 1 have already been present in the consumption bundle? How would one prove it? (Interpretation would also be welcome.)

It seems that it has something to do with the fact that utility maximization implies identical marginal utility per unit currency spent on each good.

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This seems like a homework question, so I'll just give hints.

By definition, a quasilinear utility has the form $u(x, y) = x + v(y)$ where $y$ is a vector of all other goods and $v(\cdot)$ is strictly concave. In this case, $x$ is called the numeraire.

  • From utility maximization, what's the first order condition that relates $MU_x$, $MU_y$, $p_x$, and $p_y$? Write down the equation.
  • What's the utility gain from consuming one more unit of $x$?

You should be able to work things out from there.

Edit

$MU_x$ will be constant... more precisely, $MU_x$ is always one (by definition).

You can then plug this into FOC and get that $MU_y = 1/z$. where $z = p_x/p_y$ is simply how many $y$'s you could buy with "additional amount of wealth equal to the cost of one unit of good 1 on all other goods".

Since one more unit of good $y$ will get you $1/z$, you need (approximately) $z$ more of good $y$ to get one more unit of utility increase (which is the same as what you get when you increase $x$ by one unit).

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  • $\begingroup$ How would you interpret the statement? (It is not a homework question. I just saw the statement and want to gain more understanding about it.) $\endgroup$
    – Aqqqq
    Commented Oct 30, 2019 at 19:47
  • $\begingroup$ I am not sure whether my answer is correct: the marginal utility of good x will be constant no matter how many good x have already been consumed (differentiating the utility function against x). $\endgroup$
    – Aqqqq
    Commented Oct 30, 2019 at 20:00
  • $\begingroup$ As for "one unit more of the numeraire good (good 1) give the same additional utility as spending an additional amount of wealth equal to the cost of one unit of good 1 on all other goods": From utility maximization, I can get $MU_x/p_x = MU_y/p_y$. Hence $MU_x = MU_y/p_y*p_x$ I am not sure whether it indeed reflect "one unit more of the numeraire good ...all other goods. Wouldn't it be applicable without the assumption of quasi-linear preference? $\endgroup$
    – Aqqqq
    Commented Oct 30, 2019 at 20:00
  • $\begingroup$ Please see if my edited answer helps. $\endgroup$
    – Art
    Commented Oct 31, 2019 at 2:32
  • $\begingroup$ Thank you for your answer. But I still think that since $MUx=MU_y/p_y∗p_x$, "one unit more of the numeraire good ...all other goods" is also applicable for general case without the assumption of quasi-linear preference (assumption of "quasi-linear preference" only contributes to the part "no matter how much unit of this good 1 have already been present in the consumption bundle"). Am I correct and if I am not, what mistake did I made in my reasoning? $\endgroup$
    – Aqqqq
    Commented Nov 3, 2019 at 19:02

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