Question: Quasilinear preferences:

Suppose that preferences over goods 1,2 are represented by the utility function

where \(v\) is a function such that \(v^{\prime}\left(x_{1}\right)>0, v^{\prime \prime}\left(x_{1}\right)<0\) for all \(x_{1}>0\) (these preferences are convex).

1. Show that if \(v\left(x_{1}\right)=\ln x_{1}\) and \(p_{2}=1\), then a consumer with income \(m=2\) will buy exactly one more unit of good 2 than a consumer with income \(m=1\).
2. Show that if \(v\left(x_{1}\right)=\ln x_{1}\) and \(p_{2}>2\), then both of the consumers in part (b) will spend all of their income on good 1 .

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