Question

Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers: 0.75 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 containers themselves yield no utility, express this utility function in terms of quantities of 0.75 -liter containers $(x)$ and 2 -liter containers $(y)$ b. State Ed's demand function for $x$ in terms of $p_x,p_y$ and $I$ c. Graph the demand curve for $x$, holding $I$ and $p_y$ constant. d. How do changes in $I$ and $p_y$ shift the demand curve for $x$ ? e. What would the compensated demand curve for $x$ look like in this situation?

Short answer

A: An increase in Ed's income would cause the demand curve for 0.75-liter containers to shift to the right, as he will be able to buy more containers of x.

Step-by-step solution

Utility Function for Thirsty Ed

We need to express the utility function in terms of the quantities of 0.75-liter containers (x) and 2-liter containers (y). Since they are perfect substitutes, we can simply sum up the total liters of water from both types of containers and express utility as a single variable. The utility function will be: $$U=0.75x+2y$$ #b. Demand Function for x#

Deriving the Demand Function for x

To derive the demand function of x, we first set up the budget constraint using prices and income. We know that: $$px\ast x+py\ast y=I$$ Where $px$ is the price of the 0.75-liter container, $py$ is the price of the 2-liter container, and $I$ is Ed's income. We can rearrange this equation to make y the subject: $$y=\frac{I-px\ast x}{py}$$ Now, substitute the expression for y into the utility function: $$U=0.75x+2\left(\frac{I-px\ast x}{py}\right)$$ To maximize utility, we will take the first derivative of U with respect to x and set it equal to zero: $$\frac{dU}{dx}=0.75-2\frac{px}{py}=0$$ Solving for x, we get the demand function for x: $$x=\frac{2I}{3px}$$ #c. Graphing the Demand Curve for x#

Demand Curve for x

To graph Ed's demand curve for x, we will use the demand function derived in step b. Assuming constant prices and income, the equation simplifies to: $$x=\frac{2I}{3px}$$ The graph will be a downward-sloping line, with x-axis representing the quantity of x and the y-axis representing the price of x. #d. Effect of Changes in Income and Price of 2-Liter Containers on Demand for x#

Shifts in the Demand Curve for x

Changes in income (I) and the price of 2-liter containers (py) will cause the demand curve to shift. - If I (income) increases, Ed will be able to buy more containers of x: $$x=\frac{2I}{3px}$$ The demand curve will shift to the right. - If I (income) decreases, the demand curve will shift to the left, as Ed will buy fewer containers of x. - If py (price of 2-liter containers) increases, Ed would likely buy more 0.75-liter containers as a substitute for the now more expensive 2-liter containers. The demand curve for x will shift to the right. - If py (price of 2-liter containers) decreases, Ed would likely buy fewer 0.75-liter containers, since the 2-liter containers become more affordable. The demand curve for x will shift to the left. #e. Compensated Demand Curve for x#

Compensated Demand Curve for x

The compensated demand curve refers to the relationship between the price of x and the quantity demanded of x, keeping the utility constant while adjusting the income to offset price changes. In this case, the compensated demand curve for x will be a vertical line, as the utility remains constant regardless of the price or quantity bought due to the perfect substitutes nature of the two container sizes.

Key concepts

Demand Function in Economics

In economics, a demand function is a mathematical expression that ties together the quantity demanded of a good with its price and other variables like consumer income and the price of related goods. It's an essential tool for understanding consumer behavior.

Consider Thirsty Ed's case with his two sizes of water containers. The demand function can be formulated as $x=\frac{2I}{3p_x}$, where $x$ represents the quantity of 0.75-liter water containers he purchases, $p_x$ is the price of these containers, and $I$ is Ed's income. This function shows how Ed's consumption of the containers is affected by his income and the price of the containers. If the price $p_x$ increases, the function tells us that the quantity demanded of $x$ will decrease, and vice versa, holding income constant.

In practice, to find the quantity that Ed would buy, we'd substitute the actual values of income and price into the equation. This is crucial for businesses and economists to predict consumer behavior in response to price changes or income fluctuations.

Budget Constraint

A budget constraint represents the combinations of goods and services a consumer can purchase given their income and the prices of those goods and services.

In our scenario, Ed's budget constraint is given by the equation $p_x\cdot x+p_y\cdot y=I$, where $p_x$ and $p_y$ are the prices for the 0.75-liter and 2-liter water containers, respectively, and $y$ is the quantity of 2-liter containers. The constraint illustrates how Ed's income ($I$) restricts the total amount of water he can buy in different container sizes. If he spends all his income on 0.75-liter containers, he can acquire $\frac{I}{p_x}$ units; similarly, he can buy $\frac{I}{p_y}$ units of the 2-liter containers if he spends his entire budget on them.

Understanding the budget constraint is pivotal when considering how Ed allocates his resources. It becomes the basis for his decisions, outlining the trade-offs he faces when choosing between different combinations of the two containers under financial limits.

Perfect Substitutes

When discussing perfect substitutes, we refer to goods that can completely replace each other in terms of utility to the consumer; the consumer is indifferent between these goods.

In Thirsty Ed's case, the 0.75-liter and 2-liter water containers are perfect substitutes because the utility he derives is solely from the amount of water, irrespective of the container size. This is captured in his utility function $U=0.75x+2y$. Each container contributes to his total water consumption without any preference for one size over the other. Given this perfect substitutability, Ed's purchase decision between the two container sizes will be driven purely by their prices and his available budget.

If one container becomes more expensive or if his income changes, Ed will freely substitute one container for the other without any loss in satisfaction. This concept is essential to understand market dynamics, especially under price changes, as consumers may readily switch from a higher-priced good to a less costly yet perfectly substitutable alternative, thus affecting the demand for each product.