In a standard model of choice between consumption and free time, if a wage increase causes an individual's new optimal consumption-leisure bundle to be located to the left of their original bundle on the horizontal (free time) axis, the overall effect of the wage change is an increase in hours worked.

A function of the form v(t) = βt^α is used to model the value derived from a quantity 't', where β and α are positive parameters. For this function to exhibit the property of diminishing returns (where each additional unit of 't' adds less value than the previous one), its second derivative must be negative. The second derivative is given by v''(t) = α(α-1)βt^(α-2). Given that α, β, and t are all positive, which condition is necessary to ensure the function consistently shows diminishing returns?

Parameter Constraints and Function Shape

Parameter Constraints and Function Shape

Evaluating Economic Models of Satisfaction

A function of the form v(t) = βt^α is often used to model the value an individual receives from a quantity 't' of a good or activity. The shape of this function, which describes how the marginal value changes as 't' increases, depends critically on the parameter α. Match each range of α with the corresponding mathematical property of the function and its economic interpretation. (Assume β > 0 and t > 0).

A function representing the value derived from a quantity 't' is given by $v(t) = \beta t^\alpha$. Its second derivative is $v''(t) = \alpha(\alpha-1)\beta t^{\alpha-2}$. For this function to exhibit diminishing returns (i.e., be concave), its second derivative must be negative. Assuming the quantity 't' and the parameters $\alpha$ and $\beta$ are all positive, which statement correctly analyzes the components of the second derivative to establish the condition for concavity?

A function representing the value an individual gets from a quantity 't' of a good is given by $v(t) = 5t^{1.2}$. This function is a suitable model for an individual who experiences diminishing marginal value from each additional unit of the good.

Evaluating a Proposed Utility Function

Evaluating Competing Economic Models