Effect of a change in prices when the consumer starts from a bundle of goods rather than wealth

Question

I'm struggling to make progress with the following problem:

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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?

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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.

Answer 1

If prices are equal to $(p_1, \ldots, p_n)$ and when an agent has endowments $e_1,\ldots, e_n$, her income equals the total worth of this endowment, which equals $p_1 e_1 + \ldots, p_n e_n$.

Now consider a case where all prices increase. Then buying goods becomes more costly, but on the other hand, her total income $\sum_i p_i e_i$ also increases. If income effects are positive (i.e. goods are normal) the second effect can compensate the first one.

The excess demand for good $1$ can be written as: $$ x_1(p_1,\ldots, p_n, \sum_{i = 1}^n p_i e_i) -e_1= x_1(p, p \cdot e) - e_1 $$ Now assume that prices move from $p$ to $p + tv$ where $v$ is a positive vector of size $n$ and $t \ge 0$. Then new demand equals. $$ x_1(p + tv, (p + tv)\cdot e) - e_1. $$ If we look at a tiny price increase, we can take the derivative with respect to $t$ and evaluate at $t = 0$ to get: $$ \sum_{i = 1}^n \frac{\partial x_1(p, p \cdot e)}{\partial p_j} v_j + \frac{\partial x_1(p, p\cdot e)}{\partial m} v\cdot e. $$ The second term is non-negative if goods are normal.

If $v$ is proportional to $p$ one can show (using homogeneity of degree zero of the demand function) that this expression equals zero. So demand will not change (in fact, the budget line will not move if all prices increase proportionally).

However, if relative prices change (say the price of good 1 becomes relatively cheaper), it could be the case that the demand for good 1 increases (at the expense of the other goods).