Question 1
Suppose we know that a consumer has a transitive, reflexive and complete
preference ordering over bundles of good 1 and 2. If we are given
indifference curves that are obtained from these preferences, can we
infer the preferences?
Correct: if we know that the preferences are strictly monotonic, then
yes!
False: yes, always!
Correct: if for any two indifference curves \(C\) and \(C^{\prime}\) we are
given some elements \(x\in C\) and \(x^{{}^{\prime}}\in C^{\prime}\) and whether \(x\succsim x^{\prime}\) or \(x^{\prime}\succsim x\), then yes!
Explanation:
In order to know the preferences, we need to give for any two elements \(x,y\) of the space of alternatives (in the case of this exercise the
space of alternatives is the set pairs of positive real numbers) whether \(x\succ y\) or \(x\prec y\) or \(x\sim y\). If we know the
indifference curves, this does not tell us how elements from different
indifference curves are ranked, so we need additional information. If
for the indifference curves \(C\) and \(C^{\prime}\) we are given some elements \(x\in C\) and \(x^{{}^{\prime}}\in C^{\prime}\) and whether \(x\succsim x^{\prime}\) or \(x^{\prime}\succsim x\), then this allows us by transitivity to deduce the
ranking of any other two elements of \(C\) and \(C^{\prime}\).
If the preferences are strictly monotonic, then we know that the
indifference curves are downward sloping and indifference curves further
noth-east have higher ranked elements. However, if the preferences are
not strictly monotonic, then we don’t know whether in moving north-east
we go to more or less preferred elements.