I'm struggling to make progress with the following problem:
Assume that instead of income, the agent starts with an endowment of goods c (not necessarily her optimal bundle) & can buy and sell goods at p. Further assume that all goods are normal. Show that the agent's demand for the good 1 could be higher at prices p′ = (p′1, p2, ..., pn) than at prices p (with p′1 > p1). Under what condition(s) does this happen?
To me, this means that the agent's budget is given by p1x1,...,pkxk (i.e., the money he can make from selling the goods he has). From there, my intuition is to determine the agent's Marshallian demand for good x1, showing that under certain conditions it might be higher for prices p'1 (such that p'1>p1). However, this hasn't gotten me anywhere. I would appreciate any advice.