## 5 Questions

What does it mean for two utility indices to be related by an affine transformation?

Two utility indices are related by an affine transformation if the corresponding values of one index v satisfy a relationship of the form $v(x_i) = a u(x_i) + b$, for fixed constants a and b, when compared to the values of the other index u.

Explain the concept of cardinal utility function in economics and how it relates to preference orderings.

A cardinal utility function in economics is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value $u(x_i)$ of one index u, occurring at any quantity $x_i$ of the goods bundle being evaluated, the corresponding value $v(x_i)$ of the other index v satisfies a relationship of the form $v(x_i) = a u(x_i) + b$, for fixed constants a and b. Thus, the utility functions themselves are related by $v(x) = a u(x) + b$. The two indices differ only with respect to scale and origin.

Explain the implication of concavity in cardinal utility functions and its relation to diminishing marginal utility.

If one cardinal utility function is concave, so is the other, in which case there is often said to be diminishing marginal utility. This means that the rate of increase in utility decreases as the quantity of the goods bundle being evaluated increases.

How does the concept of cardinal utility function relate to the assumption of levels of absolute satisfaction?

The use of cardinal utility imposes the assumption that levels of absolute satisfaction exist, so that the magnitudes of incremental changes in utility can be measured.

What are the differences between two utility indices related by an affine transformation?

The two indices differ only with respect to scale and origin, and if one is concave, so is the other, implying that they have the same shape and diminishing marginal utility.

## Study Notes

### Cardinal Utility Function

- A cardinal utility function is a utility index that preserves preference orderings uniquely up to positive affine transformations
- Two utility indices are related by an affine transformation, meaning that the value of one index (u) can be converted to the value of another index (v) using a fixed linear transformation

### Affine Transformation

- An affine transformation is a relationship between two utility indices, where v(xi) = au(xi) + b, for fixed constants a and b
- This transformation affects the scale and origin of the utility function, but not its shape or concavity

### Properties of Cardinal Utility Functions

- If one utility function is concave, the transformed utility function will also be concave
- Cardinal utility functions imply the existence of absolute satisfaction levels
- The use of cardinal utility assumes that magnitudes of incremental satisfaction can be measured

Test your knowledge of cardinal utility functions and scales with this quiz. Explore concepts such as utility indices, preference orderings, and positive affine transformations in economics.

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