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Consider the following passage in Sonnenschein (1973; full citation below):

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Perhaps because I do not know much about the uniform Lipschitz condition, I do not follow his exposition after the first two sentences of the proof. That is, do the definitions of $g_1^1$ and $g_1^2$ come from some standard result in mathematics? Any intuition here would be appreciated.

Furthermore, what is the source of $(g_1^1, I-pg_1^1)$ and $(g_2^1, I-pg_2^1)$?

Thank you.

Citation:

Sonnenschein, H. 1973. "Do Walras’ identity and Continuity Characterize the Class of Community Excess Demand Functions?" Journal of Economic Theory 6: 345-354.

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  • $\begingroup$ Is the question what a uniform Lipschitz condition is? The functions are constructed to be demand functions for the first good. Then the demand functions for both goods must be $(g_1^1, I-pg_1^1)$ and $(g^1_2,I−pg^1_2)$ comes from the consumer satisfying the budget constraint and the price of the second good being normalized to $1$. $\endgroup$
    – Michael Greinecker
    Jul 15, 2020 at 18:30
  • $\begingroup$ I am asking whether he is just taking the $g_1^1$ as a given (which then leads to the question of why that choice was made) or if it is derived from some standard piece of mathematics in the same way that we could write, say, the product rule in calculus using general notation. $\endgroup$
    – Ryan da Silva
    Jul 16, 2020 at 0:47
  • $\begingroup$ The construction is chosen because it works, and that is something you see if you read on how the functions are used later in the proof. $\endgroup$
    – Michael Greinecker
    Jul 16, 2020 at 6:05

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