Final answer:
To maximize utility given a budget constraint, we can derive the Walrasian and Hicksian demands for good x, where the marginal utility to price ratios are equalized. When income increases, all additional income is spent on the numeraire as the utility function for the necessity good x does not depend on income, indicating that x is a necessity with inelastic demand to income changes.
Step-by-step explanation:
To maximize utility given a consumption budget constraint, a consumer adjusts the consumption of goods based on the optimality condition where the ratio of marginal utility to price is equal across goods. Our consumer has a utility function represented as u(x,m) = ln(x) + m, with m being the numeraire (with price normalized to 1), and x being the quantity of a good with price p. To find the Walrasian demand (or Marshallian demand), we use the budget constraint p*x + m = I, where I is income, to maximize utility. The condition for utility maximization is MUx/p = MUm, where MUx is the marginal utility of good x(1/x), and MUm is the marginal utility of money (m), which is 1 as m is the numeraire. This gives us the optimal quantity of x as x* = I/(p+1). For the Hicksian demand (or compensated demand), we hold utility constant and vary prices to find the demand that maintains the initial level of utility.
When income increases, the additional income is completely spent on the numeraire good as the utility function does not depend on income level for good x (because the marginal utility of x does not depend on m), so all additional income affects only the amount of m consumed. This utility function models a necessity good, where demand does not change much with changes in income, as any extra income is not used to purchase additional quantities of x.