Final answer:
The preferences represented by the utility function are partially concave in each good and convex overall. The Marshallian demands for this problem may not be unique. Good 2 is normal, but not strictly increasing in income. Good 1 is strictly increasing in income and strictly decreasing in its own price.
Step-by-step explanation:
According to the given utility function u^ (x)=x₁1/2 +x₂1/2, the preferences are partially concave in each good because the utility function exhibits decreasing marginal utility for each good as its consumption increases. However, the preferences are not partially strictly concave in each good because the utility function does not exhibit strictly diminishing marginal utility for each good.
The preference relation represented by this utility function is convex. This can be shown by taking any two bundles of goods, A and B, and any convex combination of these two bundles, say C = αA + (1-α)B, where α is a weight between 0 and 1. If the indifference curves between A and B are convex, then the indifference curve between C and any other bundle will be convex as well.
Marshallian demands for this problem may not be unique under certain conditions such as homogeneity of degree zero in prices and quasi-concavity of the utility function.
For these preferences, good 2 is normal because the marginal utility of good 2 increases as income increases. However, good 2 is not strictly increasing in income because the marginal utility of good 2 does not increase at a constant rate.
For these preferences, good 1 is strictly increasing in income because the marginal utility of good 1 increases at a constant rate. Good 1 is also strictly decreasing in p₁ because the marginal utility of good 1 decreases as the price of good 1 increases, and vice versa.