The beginning of the parsha discusses the mitzvah of machtzis hashekel. There is a mitzvas assay for every man to give a half of a shekel yearly. We find something more stringent with this mitzvah than with most others. The Rambam (hilchos shikalim 1:1) writes that even a pauper who benefits from tzedakah is obligated in the mitzvah of machtzis hashekel, and is even required to sell the shirt off his back to pay for this mitzvah. We learn this from the pasuk of hadal lo yamit. There is a general rule regarding the amount one is obligated to spend for a mitzvah. The Gemarah in kesubos (50a) says one is not obligated to spend more than one fifth of his wealth on a mitzvah. Why is it that for the mitzvah of machtzis hashekel one must spend even more than a fifth of his wealth in order to perform the mitzvah?
The Shulchan Aruch Harav in his igeres hateshuva (chapter 3) writes, that although one should normally not spend more than a fifth of his wealth on a mitzvah, a bal teshuvah who wishes to give tzedakah to atone for his soul may spend more than a fifth of his wealth in doing so. He compares this to one that has medical expenses that he is allowed to spend whatever he can. Based on this p'sak the sefer Harirai Kedem suggests that we can understand why for the mitzvah of machtzis hashekel one must spend even more than a fifth of his wealth. Since the mitzvah of machtzis hashekel is also to atone for ones soul as it says lichaper al nafshosaichem, one may spend more than a fifth of his wealth in order to perform it
It should result from the p'sak of the Shulchan Aruch Harav, that for a korban chatas and other korbanos that bring atonement one should be obligated to bring them even if it will cost him more than a fifth of his wealth. I have not found anyone that discusses this point.
The Chofetz Chaim in the Be'ur Halacha (656 d"h afilu) says that the logic behind the spending limit of one fifth on mitzvos is so that one will not give away too much of his money and as a result become needy and be forced to rely on others. In a situation where a half of a shekel is more than one fifth of ones wealth, he is already needy and already relies on others. Since giving the machtzis hashekel will not change his financial situation, he is obligated to do so even though it is more than a fifth of his wealth.
Reb Moshe Feinstein (Igros Moshe Orach Chaim 5:41) explains that actually one is not exempt of any mitzvos even if they cost more than a fifth of his wealth. He explains that there are two types of mitzvos. One type is where the mitzvah is the spending of money i.e. pidyon haben. The second type of mitzvah is performed with an item i.e. essrog. With the second type, the money spent on obtaining that item is not part of the mitzvah. When one does not have the item required to perform a mitzvah, he must try to obtain it. Should there be no such items available he will be exempt from performing that mitzvah. The rule that one is not required to spend more than a fifth applies only in this type of mitzvah, where the spending is not part of the mitzvah, rather a means of obtaining it. Since one is not obligated to spend more than a fifth, we consider it as if there are none available. The rule is not an exemption to any of the mitzvos, it merely renders an item unavailable for purchase.
On the other hand with regard to a mitzvah in which the actual spending is part of the mitzvah, such as machtzis hashekel, there is no exemption, and he must spend the money even if it will cost more than a fifth of his wealth, for that is what the Torah has obligated us to do.
The following scenario signifies the difference between the p'sak of the Bear Halacha and the Igros Moshe. If a poor man who is currently not reliant on others for his finances, is obligated in the mitzvah of pidyon haben, and the five sla'im needed for the mitzvah amounts to more than a fifth of his wealth. According to the Igros Moshe he would not be exempt from the mitzvah, for the mitzvah is to spend the money. The Beur Halacha would however disagree and exempt the man from the mitzvah, since by spending money for this mitzvah he will become needy and reliant on others.
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