An indirect utility function is a mathematical representation of a consumer's utility as a function of their income and the prices of the goods and services they consume. It is called "indirect" because it expresses utility in terms of monetary variables (income and prices) rather than the goods and services quantities. The indirect utility function is often used in economics to study consumer behavior and to analyze the effects of changes in income or prices on a consumer's well-being.
Properties Of Indirect Utility Function
The properties of an indirect utility function include
1. Homogeneity of degree zero:
The indirect utility function is homogeneous of degree zero in prices and income, meaning that a change in the units of measurement for prices or income does not affect the level of utility.
2. Quasi-concavity:
The indirect utility function is quasi-concave, meaning that it has a single maximum value, and as the consumer's income increases, so does the maximum level of utility.
3. Monotonicity:
The indirect utility function is monotonically increasing in income and decreasing in prices, meaning that as income increases or prices decrease, the level of utility increases.
4. Continuity:
The indirect utility function is a continuous function of prices and income.
5. Invertibility:
The indirect utility function can be inverted to give the direct utility function, which expresses utility as a function of the quantities of goods and services consumed.
6. Walras' Law:
The sum of the marginal utilities of all goods is equal to the marginal utility of money, which is equal to the inverse of the price of the good.
7. Transitivity:
If a consumer prefers bundle A to bundle B and bundle B to bundle C, then the consumer will prefer bundle A to bundle C.
These properties are useful for analyzing consumer behavior, and for studying the effects of changes in income or prices on a consumer's well-being
How Do You Calculate Indirect Utility?
The indirect utility function is typically represented mathematically as V = V(p,m), where V is the level of utility, p is a vector of prices for goods and services, and m is the consumer's income. To calculate the indirect utility, you need to know the specific functional form of the utility function and the values of the prices and income.
The standard functional form of the indirect utility function is V(p,m) = U(x(p,m)) + m, where U(x) is the utility function of the consumer's choice of a bundle of goods and services. X is a vector of the quantities of goods and services consumed, which is a function of prices and income m. The function x(p,m) is called the compensated demand function, it gives the optimal bundle of goods and services for given prices and income m.
Once you have the functional form of the indirect utility function and the values for prices and income, you can plug them into the function and solve for the level of utility. For example, if V(p,m) = U(x(p,m)) + m, then to calculate the indirect utility, you would need to find the optimal bundle of goods and services x(p,m) by solving the consumer's budget constraint, and then substitute it into U(x(p,m)) + m.
It is also worth noting that there are different methods to find the indirect utility function, such as using the expenditure function, and the Hicksian and Slutsky compensated demand functions, among others.
How To Find A Utility Function From The Indirect Utility Function?
The utility function can be found from the indirect utility function by inverting the function. The process of inverting the function is called the "expenditure function method" or "Hicksian decomposition". The method consists in expressing the indirect utility function in terms of the expenditure function E(p,u)=m.
The expenditure function is defined as the minimum amount of money needed to achieve a certain level of utility, given the prices of the goods and services.
The utility function u(x) can be obtained by solving the equation E(p,u)=m for u, and then replacing it back into V(p,m) = u(x(p,m)) + m.
Here's an example:
V(p,m) = U(x(p,m)) + m
E(p,u) = p'x(p,m) = m
u = U(x(p,E^-1(p,u)))
where E^-1(p,u) is the inverse function of E(p,u) with respect to m, and x(p, E^-1(p,u)) is the optimal bundle of goods and services for a given price vector, and level of utility.
It's worth noting that this method is only valid if the indirect utility function is differentiable and strictly increasing. Furthermore, if the utility function is not unique, the method will only provide one of the possible utility functions that generate the same indirect utility function.
How to find Marshallian demand from the indirect utility function? Explain With an Example.
The Marshallian demand is the set of quantities of goods consumed by a consumer that maximizes their utility, given the prices of the goods and their income. The indirect utility function is a function that relates the consumer's utility to their income and the prices of goods. To find the Marshallian demand, we take the derivative of the indirect utility function with respect to the quantity of each good and set the derivative equal to zero, subject to the budget constraint.
Example:
Suppose a consumer has the following indirect utility function:
V(p,m) = sqrt(m/2) * (1 + p1^(-0.5) + p2^(-0.5))
where p1 and p2 are the prices of goods 1 and 2, and m is the consumer's income. To find the Marshallian demand for each good, we take the partial derivative of V with respect to each good and set it equal to zero:
∂V/∂q1 = 0.5 * (m/2)^(-0.5) * p1^(-1.5) = 0
∂V/∂q2 = 0.5 * (m/2)^(-0.5) * p2^(-1.5) = 0
Solving for q1 and q2, we find the Marshallian demands each good.
Hicksian Demand From Indirect Utility Function
The Hicksian demand is the set of goods consumed by a consumer that maximizes their utility, given the prices of the goods and their income, and holds the utility level constant. The indirect utility function is a function that relates the consumer's utility to their income and the prices of goods. To find the Hicksian demand, we take the derivative of the indirect utility function with respect to the quantity of each good and set it equal to the negative of the ratio of the price of the good to the marginal utility of income.
Example:
Suppose a consumer has the following indirect utility function:
V(p,m) = sqrt(m/2) * (1 + p1^(-0.5) + p2^(-0.5))
where p1 and p2 are the prices of goods 1 and 2, and m is the consumer's income. To find the Hicksian demand for each good, we take the partial derivative of V with respect to each good and set it equal to the negative of the ratio of the price of the good to the marginal utility of income:
∂V/∂q1 = -p1 / (dV/dm) = -p1 / (sqrt(m/2) * 0.5) = 0
∂V/∂q2 = -p2 / (dV/dm) = -p2 / (sqrt(m/2) * 0.5) = 0
Solving for q1 and q2, we find the Hicksian demands for each good.
Indirect vs Direct Utility Function
The indirect utility function and the direct utility function are two different ways of representing a consumer's preferences.
The direct utility function (U) is a function that maps a bundle of goods (q1, q2, ..., qn) directly to the consumer's utility. It represents the consumer's preferences over the consumption of goods and is a measure of their satisfaction from consuming those goods.
The indirect utility function (V), on the other hand, is a function that relates the consumer's utility to their income (m) and the prices of goods (p1, p2, ..., pn). It is defined as V(p, m) = max{U(q1, q2, ..., qn) | p1 * q1 + p2 * q2 + ... + pn * qn <= m}. The indirect utility function gives us a way to compute the consumer's utility for a given set of prices and income, but it does not provide information about the consumer's preferences over different bundles of goods.
In other words, the direct utility function gives us information about what the consumer likes and dislikes, while the indirect utility function gives us information about how the consumer's utility changes with changes in prices and income.
Conclusion
.An indirect utility function is a useful tool in microeconomic theory for understanding consumer behavior. It provides a way to calculate the consumer's utility given their income and the prices of goods and can be used to analyze the effects of changes in these variables on the consumer's consumption decisions. Understanding the indirect utility function is important for analyzing consumer choice, market demand, and general market behavior. By linking the consumer's utility to their income and prices, it provides a framework for studying consumer behavior in different market conditions.
