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.