Question

Suppose that a fast-food junkie derives utility from three goods-soft drinks \((x),\) hamburgers \((y),\) and ice cream sundaes \((z)-\) according to the Cobb- Douglas utility function \\[U(x, y, z)=x^{0.5} y^{0.5}(1+z)^{0.5}.\\] Suppose also that the prices for these goods are given by \(p_{x}=1, p_{y}=4,\) and \(p_{z}=8\) and that this consumer's income is given by \(I=8\). a. Show that, for \(z=0,\) maximization of utility results in the same optimal choices as in Example 4.1 . Show also that any choice that results in \(z > 0\) (even for a fractional \(z\) ) reduces utility from this optimum. b. How do you explain the fact that \(z=0\) is optimal here? c. How high would this individual's income have to be for any \(z\) to be purchased?

Short answer

Answer: The consumer's minimum income needed to purchase any z is 14. This is because, given the optimal consumption of x and y (i.e., x=2 and y=1), the consumer would need an income of at least 14 to purchase one unit of z and maintain utility maximization.

Step-by-step solution

Budget constraint equation

Write the budget constraint equation as follows: \\[I=px x + py y + pz z.\\] Plugging in values for px, py, pz and I, we get: \\[8 = 1x + 4y + 8z.\\]

Solve for x and y when z=0

Let's set z=0 as given in part (a) and find the optimal x and y: \\[8 = x + 4y.\] We need to find the optimal x and y to maximize utility subject to the given budget constraint.

Set up the utility maximization problem using the Lagrange multiplier method

Set up the Lagrangian with the utility function and the budget constraint as follows: \\[L = x^{0.5}y^{0.5}(1+z)^{0.5} - \lambda (x + 4y + 8z - 8).\\]

Find the first-order conditions

Take the partial derivatives of L with respect to x, y, z, and λ: \\[\frac{\partial L}{\partial x} = 0.5x^{-0.5}y^{0.5}(1+z)^{0.5}- \lambda = 0,\\] \\[\frac{\partial L}{\partial y} = 0.5x^{0.5}y^{-0.5}(1+z)^{0.5}- 4\lambda = 0,\\] \\[\frac{\partial L}{\partial z} = 0.5x^{0.5}y^{0.5}(1+z)^{-0.5} - 8\lambda = 0,\\] \\[\frac{\partial L}{\partial \lambda} = x + 4y + 8z - 8 = 0.\\]

Solve the optimality conditions for x and y when z=0

We already know that z=0, so we will use the first two optimality conditions to solve for x and y: 1. \\[\frac{0.5x^{-0.5}y^{0.5}}{0.5x^{0.5}y^{-0.5}} = \frac{4}{1}.\\] 2. Solving for x: \\[x = 2y.\\] 3. Plugging it back into the budget constraint with z=0: \\[8 = 2y + 4y,\\] \\[y=1.\\] 4. Find x: \\[x = 2(1) = 2.\\]

Show that choices with z > 0 reduce utility

Now check if z > 0 reduces utility from the optimal choices (x=2, y=1, z=0): \\[U(2,1,0) = 2^{0.5} 1^{0.5}(1+0)^{0.5} = \sqrt{2}.\\] Now, consider z > 0 (e.g., z = 0.1). We will have to reduce our consumption of x or y or both to afford z. For illustration, suppose x = 1.9 and y = 0.9: \\[U(1.9, 0.9, 0.1) = 1.9^{0.5} 0.9^{0.5}(1+0.1)^{0.5}.\\] Evaluating this, we get: \\[U(1.9, 0.9, 0.1) = 1.120 < \sqrt{2}.\\] Thus, increasing the consumption of z from 0 reduces the utility from its optimal value.

Explain z=0 as the optimal value

The reason z=0 is optimal here is because the consumer has a limited budget (I=8) and the price of z (pz=8) is relatively high compared to the prices of x and y (px=1 and py=4). The consumer would have to sacrifice a significant amount of x and y to purchase any z, and in doing so, reduces their overall utility.

Determine the minimum income for purchasing z

To find the minimum income required to purchase any z, we need to find the income when the consumer just starts purchasing z while still maximizing utility. This would indicate a situation when the consumer consumes the optimal quantity of x and y and has just enough income left to purchase one unit of z. From our previous analysis, we know the optimal x and y are 2 and 1. Plugging this into the budget constraint: \\[I = 2(1) + 4(1) + 8z.\\] Since we need to purchase at least one unit of z, we set z=1: \\[I = 2 + 4 + 8 = 14.\\] Therefore, the consumer's income must be at least 14 to start purchasing any z.

Key concepts

Utility Maximization

Understanding the concept of utility maximization is central in economics, especially when consumers are faced with decisions on how to allocate their limited income across various goods to achieve the highest level of satisfaction. In the context of the Cobb-Douglas utility function, utility maximization involves selecting quantities of goods that provide the greatest utility to the consumer within the constraints of their budget.

Consider the given function \(U(x, y, z) = x^{0.5} y^{0.5}(1+z)^{0.5}\). The utility function shows a relationship between the quantities consumed and the satisfaction derived. The exponents (0.5 in this case) represent the relative importance or 'weight' of each good in contributing to the consumer's overall utility. The goal here is to find the optimal bundle of goods \(x\), \(y\), and \(z\) that maximizes utility given the consumer's income and the prices of goods. The fast-food junkie's problem shows that sometimes the optimal consumption for one of the goods can be zero, which is the case with ice cream sundaes \(z\) when the budget is constrained to 8 units and the price of \(z\) is relatively high.

Lagrange Multiplier Method

The Lagrange multiplier method is an efficient mathematical technique used to find the maximum or minimum of a function subject to constraints. It introduces an unknown multiplier (\(\lambda\)) that represents the rate at which the utility changes as the budget constraint is relaxed or tightened.

In our problem, the Lagrangian function is established as \(L = x^{0.5}y^{0.5}(1+z)^{0.5} - \lambda (x + 4y + 8z - 8)\). The partial derivatives of this function with respect to \(x\), \(y\), \(z\), and \(\lambda\) help us find the optimal point. Setting these derivatives equal to zero yields a system of equations called the first-order conditions. Solving this system reveals the quantities of \(x\), \(y\), and \(z\) that maximize utility given the budget constraint. It illustrates a powerful application of calculus in economics, making the optimization process systematic and structured.

Budget Constraint

The concept of a budget constraint represents the trade-offs a consumer faces in a market. It's an equation that reflects all possible combinations of goods and services a consumer can purchase with a given income, taking into account the prices of these goods.

In the exercise, the budget constraint is expressed as \(I = px x + py y + pz z\) or, with the given values, \(8 = 1x + 4y + 8z\). This equation must hold true for any consumption bundle the consumer chooses. If the consumer prefers to buy at least one sundae \(z > 0\), they must compensate by cutting down the quantities of \(x\) or \(y\), as each good competes for the same limited resource: the consumer's budget of 8 units. The constraint is binding; thus, choices beyond this limit aren't feasible. Recognizing this helps explain why, at a low income level, the optimal choice includes not purchasing any sundaes, as their high price doesn't justify the utility they add compared to additional soft drinks or hamburgers. When income increases to 14 units, the consumer can afford to introduce sundaes into their consumption bundle without sacrificing the utility derived from the optimal quantities of \(x\) and \(y\).