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 \((x_{1},x_{2})\).