Answer:
The consumer's preferences represented by u(x1,x2)=ax1+bx2 are strictly monotonic and convex but not strictly convex. The indifference curves are straight lines, and the MRS, or slope of the indifference curves, is -a/b, indicating the rate of substitution between the two goods. Both goods are considered normal, but the definition of normality can be complicated without diminishing marginal utility.
Step-by-step explanation:
A. Convex and Strictly Monotonic Preferences
The preference relation ⪯ is represented by the utility function u(x1,x2) = ax1 + bx2, where a and b are positive constants. This utility function is strictly monotonic because an increase in either good x1 or x2 will increase utility, given that a, b > 0. It is convex because mixtures of goods are preferred or at least as good as extreme bundles, which is shown by the linear form of the utility function. However, it is not strictly convex, as the consumer is indifferent between mixtures and extremes due to the linearity of the utility function.
B. Indifference Curves
The indifference curves for this utility function are straight lines with a slope of -a/b. An explicit expression for an indifference curve with a given utility level ˉ is x2 = (uˉ - ax1)/b.
C. Marginal Rate of Substitution (MRS)
The MRS between good 1 and good 2 is given by the negative ratio of the marginal utilities of the two goods, which is -a/b. This coincides with the slope of the indifference curve because they both represent the rate at which the consumer is willing to substitute one good for another while maintaining the same level of utility.
D. Normal Goods and Complications
In this context, both goods are normal since an increase in income will lead to an increase in the consumption of both goods. However, defining normality can be complicated without additional constraints on the utility function, such as diminishing marginal utility, which is not present in a linear utility function like this one.