Strict Convexity:
Given \(x\neq y\) and \(z\in X\):
if \(x\curlyeqsucc z\) and \(y\curlyeqsucc z\) then \(\text{tx}+1-\text{ty}\succ z\)
Consider any consumption bundle \((x_{1},x_{2})\). The marginal rate of
substitution, denoted by \(\text{MR}S_{1,2}(x_{1},x_{2})\), is defined
to be the rate at which the consumer is ready to give up consuming good
2 in exchange for an increase in the consumption of good 1, while
maintaining the same level of utility. When preferences are represented
by a utility function, we have the following way of computing the
marginal rate of substitution:
\(\begin{equation}MRS_{1,2}\left(x_{1},x_{2}\right)=\frac{\frac{\partial u}{\partial x_{1}}}{\frac{\partial u}{\partial x_{2}}}\nonumber \\ \end{equation}\)
We now give a more general definition of the marginal rate of
substitution. The notions of limits involved here is not required for
the course, so feel free to skip over this definition if you do not find
it helpful for your intuitive understanding:
Now we define the marginal rate of substitution: Consider a given
consumption bundle \((x_{1},x_{2})\). Suppose we give the consumer \(dx_{1}\) more of good 1. We can ask: By how much could we decrease the
amount of good 2 so as to leave the consumer indifferent? We call this
amount \(dx_{2}(d_{x_{1}})\). Now if by making \(dx_{1}\) closer and
closer to 0, \(\frac{dx_{2}\left(d_{x_{1}}\right)}{dx_{1}}\) converges
to some number, then we call this number the marginal rate of
substitution . We say in this case that ‘the marginal rate of
substitution is well-defined at \((x_{1},x_{2})\) and thus define:
\(\begin{equation}MRS_{1,2}(x_{1},x_{2})=\frac{dx_{2}\left(d_{x_{1}}\right)}{dx_{1}}\nonumber \\ \end{equation}\)
\(\text{MR}S_{1,2}(x_{1},x_{2})\) is called the marginal rate of
substitution at the consumption bundle \((x_{1},x_{2})\).