Question 5
Suppose that a person has a complete and transitive preference ordering over consumption bundles and suppose that the person always chooses a consumption bundle that is maximal in his preference ordering. Suppose we observe that when faced with prices \(p_{1},\ p_{2}\) and having income \(y\), the person chooses consumption bundle \((x_{1},x_{2})\). What can we deduce?
False: If \(p_{1}x_{1}+p_{2}x_{2}=y\) then \(\text{MR}S_{2,1}=\frac{p_{2}}{p_{1}}\)
Correct: If \(p_{1}x_{1}+p_{2}x_{2}=y\) and \(x_{1}>0,x_{2}>0\) then \(\text{MR}S_{2,1}=\frac{p_{2}}{p_{1}}\)
Correct: If \(p_{1}x_{1}+p_{2}x_{2}=y\) and \(x_{1}>0\) then \(\text{MR}S_{2,1}\geq\frac{p_{2}}{p_{1}}\)
Correct: if \(p_{1}x_{1}+p_{2}x_{2}<y\) and \(x_{1}>0,\ x_{2}>0\) then the preferences are not locally non-satiated.
False: If \(p_{1}x_{1}+p_{2}x_{2}=y\) then the preferences are not strictly monotone.
Explanation:
If \(p_{1}x_{1}+p_{2}x_{2}=y\) and \(x_{1}>0,x_{2}>0\) then it is feasible to move a bit along the budget line in either direction. Neither direction can lead to a preferred outcome by the optimality of the choice. Therefore, the indifference curve must be tangent to the budget line, so \(\text{MR}S_{2,1}=\frac{p_{2}}{p_{1}}\), as illustrated here: