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If $X$ and $Y$ are perfect substitutes such that a unit of $X$ can be replaced by $n$ units of $Y$, how do we get the mathematical equation from it? I know the equation is of the form $ax+by$ (and $U = b(nx+y)$ for this case), but how do we arrive at that?

If the utility function is $U(x,y)$, can we say that any function $U$ satisfying $U(x+1,y) = U(x,y+n)$ will describe this relation between $X$ and $Y$? (Assume that $U$ is increasing to ensure the non-satiation property.)

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  • $\begingroup$ Is this a history-of-economic-thought question, are you asking who first defined perfect substitutes and how? $\endgroup$
    – Giskard
    Feb 9, 2022 at 7:24
  • $\begingroup$ @Giskard Not really. $\endgroup$ Feb 9, 2022 at 7:45

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  1. If $X$ and $Y$ are perfect substitutes such that a unit of $X$ can be replaced by $n$ units of $Y$ [...] I know [...] $U=b(nx+y)$ for this case

This is a possible representation but not the only one. E.g., $\tilde{U}=(nx+y)^2$ represents the same preferences.

  1. how do we arrive at that?

Perfect substitutes (with a substitution ratio of $1:n$) defines a preference relation $\succeq$ over the baskets of goods. A possible definition of the perfect substitutes preference relation over $\mathbb{R}^2$ is that the relation fulfills the following two properties:
i) $(x_1,y_1) \sim (x_2,y_2) \text{ iff } nx_1 + y_1 = nx_2 + y_2$
ii) monotonicity

Though ii) is generally implied, it is not strictly necessary - e.g., two bads can also be perfect subtitutes.

As I already pointed it out above, there are many utility functions that represent this preference.

  1. can we say that any function $U$ satisfying $U(x+1,y) = U(x,y+n)$ will describe this relation [...]?

If you restrict yourself to integer numbers, sure. A possible reformulation of $$(x_1,y_1) \sim (x_2,y_2) \text{ iff } nx_1 + y_1 = nx_2 + y_2$$ is $$(x_1,y_1) \sim (x_2,y_2) \text{ iff } n(x_1-x_2) = y_2-y_1$$ which is indeed fulfilled by the equation above.

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  • $\begingroup$ Restrict to integers, as in, $n$ being an integer or $\text{domain}(U) = \mathbb{Z} \times \mathbb{Z}$? $\endgroup$ Feb 9, 2022 at 7:40
  • $\begingroup$ Not just $U=(nx+y)^2$ but probably every $g(nx+y)$ where $g$ is non-decreasing considering the definition you gave. $\endgroup$ Feb 9, 2022 at 7:43
  • $\begingroup$ Indeed they are. $\endgroup$
    – Giskard
    Feb 9, 2022 at 7:46
  • $\begingroup$ Thanks. And what about the "restrict to integers"? $\endgroup$ Feb 9, 2022 at 7:48
  • $\begingroup$ I may have misunderstood you there. I thought you meant for $x,y,n$ all to be integers, in which case your definition only covers $U$'s domain if it is $\mathbb{Z}^2$. If $x,y,n$ can have any real value then your definition seems general enough. $\endgroup$
    – Giskard
    Feb 9, 2022 at 8:11

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