Example 3.3 shows that the \(M R S\) for the Cobb-Douglas function \\[ U(x, y)=x^{a} y^{\beta} \\] is given by \\[ M R S=\frac{\alpha}{\beta}\left(\frac{y}{x}\right) \\] a Does this result depend on whether \(\alpha+\beta=1 ?\) Does this sum have any relevance to the theory of choice? b. For commodity bundles for which \(y=x\), how does the \(M R S\) depend on the values of \(\alpha\) and \(\beta\) ? Develop an intuitive explanation of why, if \(\alpha>\beta, M R S>1 .\) Illustrate your argument with a graph. c. Suppose an individual obtains utility only from amounts of \(x\) and \(y\) that exceed minimal subsistence levels given by \(x_{0}, y_{0}\) In this case, \\[ U(x, y)=\left(x-x_{0}\right)^{a}\left(y-y_{0}\right)^{\beta} \\] Is this function homothetic? (For a further discussion, see the Extensions to Chapter \(4 .\) )

Is the utility function accounting for minimal subsistence levels homothetic? Answer: The relevance of \(\alpha + \beta = 1\) in the MRS result is that it represents constant returns to scale and unitary elasticity of substitution, meaning that the consumer can replace one good with another at a constant rate while keeping their utility constant. When \(y=x\), MRS depends only on the ratio \(\frac{\alpha}{\beta}\), with the MRS being greater than \(1\) if \(\alpha>\beta\). This means that consumers derive more marginal utility from good \(x\) than good \(y\) and are willing to give up more than one unit of \(y\) for one additional unit of \(x\). The utility function accounting for minimal subsistence levels is not homothetic because it does not satisfy the homotheticity condition, implying that specific base amounts of goods are necessary for the utility function to be valid.