Suppose a consumer has a consumption space with 4 goods, so that we are in $\mathbb R_+^4$. Suppose further that the consumer's endowment is $(10,7,1,25)$ and that se faces the price vector $(6,3,9,2)$.
- What is the equation of the budget hyperplane (ignoring the restriction to positive consumption for now)?
- What would be the easiest way to denote the set of all feasible consumption vectors (the frontier of the budget set of the consumer)?
- Is the consumption $(8,11,3,15)$ affordable? What about $(9,12,0,25)$?
It's been a while since I had to deal with vectors, if someone could help me along with 1.1 that would be very much appreciated.
add a)
Imagine the consumer is selling his endowment. The proceeds would be $\sum_{i=1}^4 p_i\cdot x_i = 6\cdot 10 + 3 \cdot 7 + 9\cdot 1+2\cdot 25=140$
The consumer can buy different amounts of goods, wihich satisfy the following constraint:
$6\cdot x_1 + 3 \cdot x_2 + 9\cdot x_3+2\cdot x_4\leq 140$ (budget hyperplane)
add b)
If the goods are not divisible, then you have to add the condition
$x_i \in \mathbb N_0$
add c)
Insert the values for $x_i$ in the inequality (budget hyperplane). If the inequality holds, then the consumption is affordable. This can be done for both consumption bundles.