A consumer's preferences are described by a quasi-linear utility function of the form u(c, t) = c + v(t), where 'c' is the consumption of a composite good and 't' is the consumption of another good. For these preferences to be convex, which implies a diminishing marginal rate of substitution, which of the following functional forms for v(t) would be appropriate? Assume t > 0 and that more of good t is always preferred.

Consistency of a Utility Model

A consumer's preferences for a composite good 'c' and a specialized product 't' are represented by the quasi-linear utility function u(c, t) = c + 8√t.

True or False: These preferences are convex for any positive quantity of the specialized product 't'.

Linking Utility Shape to Substitution Behavior

Explaining the Link Between Utility Function Shape and Preference Convexity

A consumer's preferences are represented by a quasi-linear utility function of the form u(c, t) = c + v(t). Match each economic property or concept related to these preferences with its direct mathematical equivalent condition on the function v(t).

For preferences represented by a utility function of the form u(c, t) = c + v(t), the economic principle of a diminishing marginal rate of substitution holds if and only if the second derivative of the v function, v''(t), is strictly _________.

Evaluating a Proposed Utility Model

Interpreting Consumer Behavior in a Quasi-Linear Model

A consumer's preferences for a composite good 'c' and a specialized product 't' can be represented by a quasi-linear utility function u(c, t) = c + v(t). For these preferences to be convex, which requires a diminishing marginal rate of substitution, the function v(t) must be concave. Which of the following functional forms for v(t) would result in preferences that are not convex for all t > 0?