# Mathematical definition of perfect substitutes

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.)

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

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.

• Restrict to integers, as in, $n$ being an integer or $\text{domain}(U) = \mathbb{Z} \times \mathbb{Z}$? Feb 9 at 7:40
• Not just $U=(nx+y)^2$ but probably every $g(nx+y)$ where $g$ is non-decreasing considering the definition you gave. Feb 9 at 7:43
• Indeed they are. Feb 9 at 7:46
• Thanks. And what about the "restrict to integers"? Feb 9 at 7:48
• 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. Feb 9 at 8:11