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General Form of a Quasi-Linear Utility Function
Condition for Convexity in Quasi-Linear Preferences
Equivalence of Convex Quasi-Linear Preferences and Concavity of v(t)
For quasi-linear preferences, the convexity of the indifference curves is equivalent to the function being concave. This 'if and only if' relationship means that preferences are convex if is concave, and conversely, if preferences are convex, then must be concave. The convexity of preferences implies a diminishing Marginal Rate of Substitution (MRS), which for this utility form is . A diminishing MRS means decreases as increases, which is the definition of a concave function . The formal mathematical test for the concavity of is a negative second derivative, . This condition ensures the indifference curve equation, , represents a convex curve.
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xand moneym(representing all other consumption) are described by the utility functionu(x, m) = 10√x + m. By analyzing the properties of this function, which statement accurately describes the consumer's behavior or preferences?Consider a consumer whose preferences for a specific good,
x, and money available for all other goods,m, can be represented by the utility functionu(x, m) = 20 * ln(x) + m. According to this model, if the consumer's income increases, their willingness to pay for an additional unit of goodxwill also increase.A utility function of the form u(x, m) = v(x) + m is said to represent 'well-behaved' quasi-linear preferences. For this to be true, the utility from good x must be increasing (meaning its first derivative, v'(x), is positive), and there must be diminishing marginal utility for good x (meaning its second derivative, v''(x), is negative), for all x > 0. Which of the following specifications for v(x) satisfies both of these conditions?
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Modeling Consumer Preferences for Different Goods
An individual's preferences are modeled by a utility function
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u(x, m) = v(x) + mis referred to as 'quasi-linear' because while it is typically non-linear with respect to goodx, it is perfectly linear with respect to the variable ______, which represents an individual's income available for other goods.Evaluating Model Suitability for Different Goods
Critical Evaluation of the Quasi-Linear Utility Model
Equivalence of Convex Quasi-Linear Preferences and Concavity of v(t)
Learn After
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 functionv(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 thevfunction,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 functionv(t)must be concave. Which of the following functional forms forv(t)would result in preferences that are not convex for allt > 0?