Question

Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers -75 liter and 2 liter. Because the water itself is identical, he regards these two "goods" as perfect substitutes. a. Assuming Ed's utility depends only on the quantity of water consumed and that the con tainers themselves yield no utility, express this utility function in terms of quantities of 75L containers \((X)\) and \(2 \mathrm{L}\) containers (F) b. State Ed's demand function for \(X\) in terms of \(P_{x}, P_{Y}\), and 1 c. Graph the demand curve for \(X\), holding /and \(P_{v}\) constant. How do changes in /and \(P\) shift the demand curve for \(X ?\) e. What would the compensated demand curve for Xlook like in this situation?

Short answer

Answer: The demand curve for 0.75-liter containers of water depends on the comparison of price per liter for both goods. If the price per liter for 0.75-liter containers is lower or equal to the price per liter for 2-liter containers, Ed will spend his entire income on 0.75-liter containers, resulting in a downward-sloping line. If the price per liter for 0.75-liter containers is higher than the price per liter for 2-liter containers, Ed will buy 2-liter containers only, resulting in a vertical line at X = 0. A change in income will shift the intercept on the vertical axis of the downward-sloping line, while a change in the price of 2-liter containers will change the "boundary" point between the two cases, vertically shifting it upwards or downwards. The compensated demand curve will be similar to the uncompensated demand curve but adjusted for the change in relative prices, keeping the total utility constant.

Step-by-step solution

Determine the Utility Function

First, let's express Ed's utility in terms of the quantity of water consumed. The total amount of water he consumes is given by the sum of the amount of water contained in the 0.75L containers (X) and the amount of water in the 2L containers (Y). His utility, therefore, would be the sum of 0.75 times X and 2 times Y: Utility = \(0.75X + 2Y\)

State the Demand Function for X

Now we need to state Ed's demand function for X in terms of the price of X (\(P_x\)), the price of Y (\(P_y\)), and his income (I). Since Ed regards both goods as perfect substitutes and the water in both containers is identical, his goal is to maximize the liters of water consumed under the constraint of his budget. Ed's budget constraint can be written as: \(I = P_xX + P_yY\) To find the demand function for X, we can express Y in terms of X, \(P_x\), \(P_y\), and I. Solve the budget constraint equation for Y: \(Y = \frac{I - P_xX}{P_y}\) Now, substitute this expression for Y into the utility function: Utility = \(0.75X + 2\left(\frac{I - P_xX}{P_y}\right)\) Since Ed wants to maximize the quantity of water consumed, he should compare the price per liter of both goods and choose the one with the lowest price per liter. The price per liter for goods X and Y are \(P_x / 0.75\) and \(P_y / 2\), respectively. The demand function for X will depend on the comparison of the price per liter for goods X and Y: If \(\frac{P_x}{0.75} \le \frac{P_y}{2}\), then \(X^* = \frac{I}{P_x}\) (Ed will buy only 0.75L containers) If \(\frac{P_x}{0.75} > \frac{P_y}{2}\), then \(X^* = 0\) (Ed will buy only 2L containers)

Graph the Demand Curve for X

To graph the demand curve for X, holding I and \(P_y\) constant, we can consider the two cases mentioned above. Case 1 (\(\frac{P_x}{0.75} \le \frac{P_y}{2}\)): The demand curve for X will be a downward-sloping line, with intercept at \(\frac{I}{P_x}\) and slope -1, as Ed will spend his entire income on 0.75L containers. Case 2 (\(\frac{P_x}{0.75} > \frac{P_y}{2}\)): The demand curve for X will be a vertical line at X = 0, as Ed will purchase only 2L containers. The resulting graph will be a piecewise function with downward-sloping and vertical lines.

Analyze Changes in I and Py

Now, let's analyze how changes in I and \(P_y\) will shift the demand curve for X: a. Changes in I (income): An increase in income will increase the intercept on the vertical axis of the downward-sloping line, while a decrease in income will lower it. b. Changes in \(P_y\) (price of Y): An increase in the price of Y will decrease the range of \(P_x\) values for which Ed is buying only 0.75L containers, therefore shifting the "boundary" between the two cases vertically upwards. A decrease in the price of Y will shift the boundary vertically downwards, expanding the range of \(P_x\) values for which Ed is buying only 0.75L containers.

Compensated Demand Curve for X

In the case of perfect substitutes, the compensated demand curve for X will be similar to the uncompensated demand curve described in steps 3 and 4. The only difference is that, on the compensated demand curve, any changes in income will be offset by adjusting the prices of goods proportionally to maintain the same level of utility. Consequently, when there are changes in I and \(P_y\), the shift in demand curve will be determined by the change in relative prices, keeping the total utility constant.

Key concepts

Perfect Substitutes

When discussing consumer choice in economics, the concept of perfect substitutes is central to understanding how decisions are made in the marketplace.

Perfect substitutes are different goods that a consumer can replace with one another without affecting their level of satisfaction. This means that the consumer views the goods as identical in terms of utility, despite any physical differences. An everyday example could be different brands of bottled water; if consumers view them as identical, these brands become perfect substitutes to each other in the eyes of the consumer.

In mathematical terms, when two goods, A and B, are perfect substitutes, the utility function can be represented as a linear combination of the quantities of each good. That is, utility can be expressed as:
Utility = aA + bB

where a and b are the marginal utilities of goods A and B, respectively.

In the context of Thirsty Ed's problem, he sees 75L and 2L bottles of water as perfect substitutes because drinking water from either sized bottle provides no additional utility over the other. His main concern is to maximize the total volume of water he can consume within his budget. This understanding helps inform his purchasing decisions and shapes the demand function for each product.

Demand Function

The demand function is a fundamental concept in economics that represents the relationship between the quantity of a good that consumers are willing to purchase and various factors influencing their decision, such as the price of the good, consumer income, and the price of other goods.

Typically, it is expressed with the quantity demanded as a function of price, assuming other influencing factors remain constant. A demand function might look something like:
Q_d = f(P)

where Q_d is the quantity demanded, P is the price, and f is the function depicting how the quantity demanded changes with price.

In the case of Thirsty Ed, his demand function for the 75L containers (X) was calculated considering that he views 75L and 2L bottles as perfect substitutes and by taking into account his budget constraint. In essence, his demand function reflects how he will allocate his spending between the two bottle sizes to maximize his total water consumption. If the price per liter of the 75L bottle is less or equal to the 2L bottle, his demand for the 75L bottle reaches its maximum. Conversely, if the 75L is relatively more expensive, he will not demand it at all. This example highlights how prices and budget constraints drive consumer choices, thereby shaping demand curves for products.

Budget Constraint

A budget constraint represents the combinations of goods and services a consumer can purchase given their limited income and the prices of the goods. This constraint is a key factor in the utility maximization problem that consumers face, as they aim to allocate their income in a way that provides the highest utility.

The budget constraint can be illustrated with the following equation:
I = P_AQ_A + P_BQ_B

where I represents income, P_A and P_B are the prices of goods A and B, and Q_A and Q_B are the quantities of goods A and B, respectively.

Thirsty Ed's budget constraint is what determines how many 75L and 2L bottles of water he can purchase. By solving his budget constraint for one of the goods, Ed can determine how his spending on one good affects the quantity he can purchase of the other. Ed's approach to spending his entire income on water indicates his utility-maximizing behavior. Across different purchasing scenarios, the budget constraint helps plot the consumption possibilities, demonstrating how a consumer's choices are limited by their budget.