Please only do part a.

u 7 1. Consider an individual with the following utility function: u(x) = xi/z +x/2. (a) For each of the following properties, briey summarize the implications for preferences if the property holds and then show whether the preferences represented by this utility function satisfy each property: (i) local non-satiation, (ii) quasilinean'ty, and (iii) strict convexity. Briey justify your answers and then provide an example of another utility function that represents the same preferences. (b) Suppose p1 and p2 are the prices of goods x1 and x2, respectively, and that the consumer has income M. Derive Marshallian demands and the indirect utility function associated with this utility function. Then, show how Roy's Identity could be used to check your derivations (no need to actually apply it!). Next, derive the expenditure function and explain in words how to interpret both the indirect utility and expenditure functions and what we mean when we say they represent dual approaches to the consumer's problem (be sure to specify the arguments corresponding to each function). (e) Write down the Slutsky Equation that relates Marshallian to Hicksian demand and explain how it can be used to decompose changes in Marshallian demand in response to price changes into substitution and income effects. Use the Slutsky equation to verify that any Giffen good must be inferior. Then, determine whether either x1 or x2 are Giffen goods based on the utility function given. Is it always the case that goods must be normal if preferences satisfy monotonicity? Be sure to justify your responses. N Innrl. n\\