Question

Suppose individuals require a certain level of food \((x)\) to remain alive. Let this amount be given by \(x_{0}\). Once \(x_{0}\) is purchased, individuals obtain utility from food and other goods \((y)\) of the form $$U(x, y)=\left(x-x_{0}\right)^{\alpha} y^{\beta}$$ where \(\alpha+\beta=1\) a. Show that if \(I>p_{x} x_{0}\) then the individual will maximize utility by spending \(\alpha\left(I-p_{x} x_{0}\right)+p_{x} x_{0}\) on good \(x\) and \(\beta\left(I-p_{x} x_{0}\right)\) on good \(y\). Interpret this result. b. How do the ratios \(p_{x} x / I\) and \(p_{y} y / I\) change as income increases in this problem? (See also Extension E4.2 for more on this utility function.)

Short answer

Answer: Using the method of Lagrange multipliers, an individual will maximize their utility by allocating a fraction \(\alpha\) of their disposable income (income left after purchasing the necessary amount of good \(x\)) to good \(x\) and a fraction \(\beta\) of their disposable income to good \(y\). This means that the individual will spend \(\alpha(I-p_x x_0) + p_x x_0\) on good \(x\) and \(\beta(I-p_x x_0)\) on good \(y\). As income increases, the proportion of income spent on good \(x\) and good \(y\) will converge to constant proportions, \(\alpha\) and \(\beta\), respectively.

Step-by-step solution

Formulate the utility maximization problem with a budget constraint

Consider an individual with income \(I\). This person wants to maximize their utility function \(U(x, y) = (x-x_{0})^{\alpha}y^{\beta}\), given the constraint that their total spending on goods \(x\) and \(y\) cannot exceed their income \(I\). The budget constraint can be written as: \(p_x x + p_y y \leq I\), where \(p_x\) and \(p_y\) are the prices of goods \(x\) and \(y\), respectively.

Solve the utility maximization problem using the method of Lagrange multipliers

To solve this utility maximization problem, we will use the method of Lagrange multipliers. We will formulate the Lagrangian function as: \(L(x, y, \lambda) = (x - x_{0})^{\alpha}y^{\beta} + \lambda (I - p_x x - p_y y)\) Now, we need to find the partial derivatives of the Lagrangian function with respect to \(x\), \(y\), and \(\lambda\), and set them equal to zero: 1. \(\frac{\partial L}{\partial x} = \alpha(x-x_{0})^{\alpha-1}y^{\beta} - \lambda p_x = 0\) 2. \(\frac{\partial L}{\partial y} = \beta(x-x_{0})^{\alpha}y^{\beta-1} - \lambda p_y = 0\) 3. \(\frac{\partial L}{\partial \lambda} = I - p_x x - p_y y = 0\)

Solve the system of equations obtained in Step 2

From step 2, we have three equations, which form a system of equations. To find the optimal allocation of income/spending, we need to solve this system of equations. First, we will eliminate \(\lambda\) by dividing the first equation by the second equation: \(\frac{\alpha(x-x_{0})^{\alpha - 1}y^{\beta}}{\beta(x-x_{0})^{\alpha}y^{\beta - 1}} = \frac{p_x}{p_y}\) Solving this equation, we get: \((x-x_{0}) = \frac{\alpha}{\beta}y\) Next, substituting the obtained equation into the budget constraint (equation 3 from step 2): \(I - p_x x - p_y y = 0\) \(I - p_x(x_0 + \frac{\alpha}{\beta}y) - p_y y = 0\) Through a series of algebraic calculations, we obtain: \(y = \beta(I - p_x x_0)\) And: \(x = \alpha(I-p_x x_0) + p_x x_0\)

Interpret the results

The optimal allocation of income indicates that the individual will maximize their utility by spending \(\alpha(I-p_x x_0) + p_x x_0\) on good \(x\) and \(\beta(I-p_x x_0)\) on good \(y\). This means that the individual will allocate a fraction \(\alpha\) of their disposable income (income left after purchasing the necessary amount of good \(x\)) to good \(x\) and a fraction \(\beta\) of their disposable income to good \(y\). This result is consistent with the Cobb-Douglas utility function, where the individuals allocate a constant proportion of their income to each good. Now we'll move to part b:

Analyze the ratios \(p_x x / I\) and \(p_y y / I\) as income increases

Notice that as income, \(I\) increases, the spending on each of the goods also increases, so the terms in the ratio will change accordingly. Using the results from part a, we can write: \(\frac{p_x x}{I} = \frac{p_x (\alpha(I-p_x x_0) + p_x x_0)}{I} = \alpha - \frac{\alpha p_x x_0}{I} + \frac{p_x^2 x_0}{I}\) Similarly, we have: \(\frac{p_y y}{I} = \frac{p_y \beta(I-p_x x_0)}{I} = \beta - \frac{\beta p_x x_0}{I}\) As income \(I\) increases, the terms \(\frac{\alpha p_x x_0}{I}\) and \(\frac{\beta p_x x_0}{I}\) go to zero. Therefore, in the limit, as income increases: \(\lim_{I \to \infty} \frac{p_x x}{I} = \alpha\) \(\lim_{I \to \infty} \frac{p_y y}{I} = \beta\) This means that as income increases, the share of income spent on good \(x\) will converge to a constant proportion, \(\alpha\), and the share of income spent on good \(y\) will also converge to a constant proportion, \(\beta\). The proportion spent on both goods together will be equal to one, which is consistent with the Cobb-Douglas utility function of two goods.

Key concepts

Cobb-Douglas Utility Function

The Cobb-Douglas utility function is a widely used formula in economics that captures the relationship between two or more goods and the satisfaction or utility a consumer derives from consuming them. It has a mathematical form given by
\(U(X_1, X_2, ..., X_n) = A \times X_1^{a_1} \times X_2^{a_2} \times ... \times X_n^{a_n}\),
where \((X_1, X_2, ..., X_n)\) represent quantities of different goods consumed, \((a_1, a_2, ..., a_n)\) are constants showing the relative importance of each good, and \(A\) is a constant. For a two-good case, such as in the exercise, the function takes the simplified form of \(U(x, y) = (x - x_0)^{\alpha}y^{\beta}\), where \((\alpha + \beta = 1)\). This utility function imposes the idea that consumers allocate their spending between goods in a particular way, adhering to their preferences and the limited resources they have available.

Understanding the Cobb-Douglas function is essential for solving utility maximization problems because it defines the utility obtained from any combination of goods. The parameters \(\alpha\) and \(\beta\) show the elasticity of substitution between goods \(x\) and \(y\). If a consumer has a preference for a balanced consumption of goods, these parameters will be similar in value. Suppose an individual's utility follows this structure. In that case, the outcome of the optimization process suggests that consumers will spend constant proportions of their income on each good, as shown in the textbook exercise.

Budget Constraint

In economics, a budget constraint represents the combinations of goods and services that a consumer can purchase given their limited income and the prices of those goods and services. It's formulated as
\[p_x x + p_y y = I\],
where \(x\) and \(y\) are the quantities of goods purchased, \(p_x\) and \(p_y\) are the prices of these goods, and \(I\) is the income available to spend. The budget constraint is a fundamental concept as it limits the choices that consumers can make.

In the provided exercise, the budget constraint takes the form
\[p_x x + p_y y \leq I\],
and it is implied that a certain minimum quantity \(x_0\) of good \(x\) is essential for survival. The budget constraint must accommodate this minimum requirement. When solving the utility maximization problem, the budget constraint is critical as it ensures that the solution is not only the highest utility but also attainable within the consumer's financial limits. As income \(I\) increases, the terms of the budget constraint adjust, reflecting that a consumer may spend more on goods while always respecting the budget constraint.

Method of Lagrange Multipliers

The method of Lagrange multipliers is an optimization technique used to find the maximum or minimum value of a function subject to constraints. In utility maximization problems, like the one in this exercise, the method introduces an extra variable, the Lagrange multiplier (often denoted by \(\lambda\)), to convert a constrained problem into one without constraints.

The Lagrangian for our problem is as follows:
\[L(x, y, \lambda) = (x - x_{0})^{\alpha}y^{\beta} + \lambda (I - p_x x - p_y y)\]
The next step involves taking partial derivatives of the Lagrangian with respect to each of the variables, including the Lagrange multiplier, and setting them equal to zero. The resulting system of equations provides conditions for optimization. By solving this system, one can deduce how much of each good should be consumed to maximize utility.

This method conveniently transforms a complex problem of finding utility-maximizing quantities within a budgetary limitation into a more manageable form. Its application in the exercise shows that the optimal utility occurs when consuming specific amounts of each good, which is proportional to the income available for consumption after meeting the essential needs. The Lagrange multiplier itself can be interpreted as the rate at which the total utility would increase if the consumer had an additional unit of income to spend, showing its significance as a tool for understanding economic behaviors and constraints.