Question

If the utility function of an individual takes the form $$U=U\left(x_{1}, x_{2}\right)=\left(x_{1}+2\right)^{2}\left(x_{2}+3\right)^{3}$$ where \(U\) is total utility, and \(x_{1}\) and \(x_{2}\) are the quantities of two commodities consumed: (a) Find the marginal-utility function of each of the two commodities. (b) Find the value of the marginal utility of the first commodity when 3 units of each commodity are consumed.

Short answer

The marginal utility function for the first commodity is \(MU_1 = 2(x_1 + 2)(x_2 + 3)^3\). For the first commodity at 3 units each, \(MU_1 = 2160\).

Step-by-step solution

Understanding Marginal Utility

Marginal utility refers to the additional satisfaction or utility that a person receives from consuming an additional unit of a good or service. It is the partial derivative of the utility function with respect to the good or service.

Differentiating Utility Function with Respect to x1

To find the marginal utility with respect to \(x_1\), we differentiate the utility function \(U(x_1, x_2) = (x_1 + 2)^2 (x_2 + 3)^3\) with respect to \(x_1\). Using the product rule:\[ MU_1 = \frac{\partial U}{\partial x_1} = 2(x_1 + 2)(x_2 + 3)^3 \]

Differentiating Utility Function with Respect to x2

Now, we differentiate the utility function \(U(x_1, x_2) = (x_1 + 2)^2 (x_2 + 3)^3\) with respect to \(x_2\). Again using the product rule, we have:\[ MU_2 = \frac{\partial U}{\partial x_2} = 3(x_1 + 2)^2(x_2 + 3)^2 \]

Substituting Given Values in MU1

To find the marginal utility of the first commodity when 3 units of each commodity are consumed, substitute \(x_1 = 3\) and \(x_2 = 3\) into the equation for \(MU_1\):\[ MU_1 = 2(3 + 2)(3 + 3)^3 = 2 \times 5 \times 6^3 = 2 \times 5 \times 216 = 2160 \]

Key concepts

Utility Function

In economics, a utility function is a mathematical representation that allows us to understand how different quantities of goods and services consumed affect an individual's total utility or satisfaction. This function is incredibly important in consumer theory, allowing us to model and understand choices.

For example, in the utility function provided, \(U(x_1, x_2) = (x_1 + 2)^2 (x_2 + 3)^3\), \(x_1\) and \(x_2\) represent the quantities of two distinct goods. The form of the utility function suggests that increases in either good increase total utility, though the rate of increase depends on each good's current quantity.

Utility functions enable economists to predict how changes in prices or income might influence consumer choices. By expressing preferences in mathematical terms, utility functions help in intently understanding the decision-making process.

Partial Derivative

The concept of a partial derivative is crucial in calculus and plays a significant role in economics, especially when dealing with functions of several variables, like our utility function. The partial derivative of a function with respect to one of its variables measures how the function changes as that one variable changes, holding the other variables constant.

In the context of our utility function, to find the marginal utility of a commodity, we take the partial derivative of the utility function with respect to the commodity's quantity. If differentiating with respect to \(x_1\), the result \( \frac{\partial U}{\partial x_1} \) gives us the marginal utility of the first commodity. This represents how much the total utility changes per additional unit of that commodity, assuming the amount of other goods stays unchanged.

The application of partial derivatives helps economists understand and calculate responsiveness, aiding in the formulation of various economic models.

Product Rule

In calculus, the product rule is essential for differentiation, especially when dealing with the derivative of a product of functions. This rule becomes particularly relevant when we differentiate complex utility functions.

The product rule states that if you have two functions, say \(f(x)\) and \(g(x)\), their product's derivative is \(f'(x)g(x) + f(x)g'(x)\). This principle is directly applied to find the marginal utilities in our utility function example as:

• Differentiate \((x_1 + 2)^2\) while keeping \((x_2 + 3)^3\) constant, and vice versa.
• The result allows us to effectively calculate how each good contributes to total utility.

In the utility function, the product rule simplifies and structures the process of taking these derivatives, which ultimately leads to the formulas for the marginal utility of each commodity.

Economics

Economics revolves around how individuals and societies choose to allocate resources. Within this field, understanding consumer behavior is key, which often involves studying utility and marginal utility.

Marginal utility, defined as the added satisfaction from consuming an additional unit of a good, helps analyze how consumers make decisions based on incremental changes. This ties back to the psychological aspect and the concept of diminishing returns, where each additional unit of a good typically provides less added utility than the previous one.

• Economists use utility functions to build models that offer insights into consumer behavior.
• These functions assist in understanding economic processes such as demand, pricing, and welfare analysis.

By focusing on how consumers derive utility and respond to changes, economists can craft policies and strategies that aim to optimize resource distribution and improve economic welfare.