Question

Laid back Al derives utility from 3 goods: music \((M),\) wine \((W),\) and cheese (C). His utility function is of the simple linear form \\[ \text { utility }=U(M, W, C)=M+2 W+3 C \\] a. Assuming Al's consumption of music is fixed at \(10,\) determine the equations for the indifference curves for \(W\) and \(C\) for \(U=40\) and \(U=70 .\) Sketch these curves. b. Show that Al's MRS of wine for cheese is constant for all values of Wand Con the indifference curves calculated in part (a). c. Suppose Al's consumption of music increases to \(20 .\) How would this change your answers to parts \((a)\) and \((b) ?\) Explain your results intuitively.

Short answer

Answer: The equations for the indifference curves when the utility is 40 and 70, with a fixed consumption of music are: $$ \begin{cases} 3C = 30 - 2W, & \text{for } U=40 \\ 3C = 60 - 2W, & \text{for } U=70 \end{cases} $$ The MRS of wine for cheese is constant at 2. If Al's consumption of music increases to 20, the equations for the indifference curves would change to: $$ \begin{cases} 3C = 20 - 2W, & \text{for } U=40 \\ 3C = 50 - 2W, & \text{for } U=70 \end{cases} $$ However, the MRS of wine for cheese would still be constant at 2.

Step-by-step solution

a. Calculate indifference curves with utility of 40 and 70 with fixed consumption of music

Since Al's consumption of music is fixed at \(10,\) we can plug this value into the utility function: $$U(M, W, C) = 10 + 2W + 3C$$ When \(U = 40,\) we set \(U(M, W, C) = 40\) and solve for C: $$40 = 10 + 2W + 3C$$ $$30 = 2W + 3C$$ Similarly, when \(U = 70,\) set \(U(M, W, C) = 70\) and solve for C: $$70 = 10 + 2W + 3C$$ $$60 = 2W + 3C$$ Now we have the equations for indifference curves: $$ \begin{cases} 3C = 30 - 2W, & \text{for } U=40 \\ 3C = 60 - 2W, & \text{for } U=70 \end{cases} $$ These equations describe Al's indifference curves when consuming 10 units of music, with different levels of utility (40 and 70). Sketch these curves with Wine (W) on the horizontal axis and Cheese (C) on the vertical axis, using graphing software or by plotting points and connecting them.

b. Calculate and show the MRS of wine for cheese is constant

To find the marginal rate of substitution of wine for cheese, we need to find the slope of the indifference curve. We will differentiate the indifference curves equations with respect to W and use the negative of the quotient as the MRS. Differentiate the first indifference curve equation with respect to W: $$\frac{d(3C)}{dW} = -2$$ Differentiate the second indifference curve equation with respect to W: $$\frac{d(3C)}{dW} = -2$$ Now we can see that for both of the indifference curves, the MRS is constant: $$\text{MRS}_{WC} = -\frac{dC}{dW} = 2$$ The MRS of wine for cheese is constant for all values of W and C on the indifference curves from part (a).

c. Evaluate changes when consumption of music increases to 20

Now, if Al's consumption of music increases to 20, plug this new value into the utility function: $$U(M, W, C) = 20 + 2W + 3C$$ When \(U = 40,\) set \(U(M, W, C) = 40\) and solve for C: $$40 = 20 + 2W + 3C$$ $$20 = 2W + 3C$$ For \(U = 70,\) set \(U(M, W, C) = 70\) and solve for C: $$70 = 20 + 2W + 3C$$ $$50 = 2W + 3C$$ Now we have the new equations for indifference curves: $$ \begin{cases} 3C = 20 - 2W, & \text{for } U=40 \\ 3C = 50 - 2W, & \text{for } U=70 \end{cases} $$ We can differentiate these new indifference curve equations and show that MRS is still constant: $$\frac{d(3C)}{dW} = -2$$ Thus, when the consumption of music increases to 20, the equations for the indifference curves change, but the MRS of wine for cheese remains constant at 2. Intuitively, this result means that Al's preference for trading wine for cheese does not change when his consumption of music increases. He will still give up the same amount of wine for each additional unit of cheese, regardless of his music consumption.

Key concepts

Understanding the Utility Function

A utility function represents a consumer's preference for various goods and services. In the context of our exercise, Al's utility function is given by \( U(M, W, C) = M + 2W + 3C \). This linear formula suggests that Al derives value—the term 'utility' reflects this value—from consuming music (M), wine (W), and cheese (C). The coefficients before each good indicate how much utility Al gets from each additional unit. Here, cheese clearly holds the most value for Al, as each unit of cheese adds three times as much utility than a unit of music.

Utility functions help economists understand consumer behaviors. The linear form is simple and implies that Al's satisfaction increases at a constant rate with more consumption of each good. Utility doesn't have a tangible unit of measure; instead, it provides a way for us to quantify and compare Al's preferences, aiding in predicting his choices among different 'consumption bundles', or sets of goods.

Marginal Rate of Substitution (MRS)

The Marginal Rate of Substitution (MRS) is a crucial concept in understanding how consumers make trade-offs between different goods. As shown in the step-by-step solution, when we calculate the MRS of wine for cheese, we essentially measure the slope of the indifference curve at a given point. Since the curve represents all combinations of W and C that provide Al the same level of utility, the MRS tells us how many units of cheese Al is willing to give up to get one more unit of wine, while keeping his utility constant.

In Al's case, we see a constant MRS of 2, which means that his willingness to trade wine for cheese does not change regardless of how much wine or cheese he currently has. This constant MRS represents perfect substitutability between wine and cheese in Al's perception - an assumption that holds in linear utility functions. In a more complex utility function, the MRS would typically change as the quantities of goods consumed change.

Interpreting Consumption Bundles

A 'consumption bundle' refers to the combination of different goods and services that a consumer chooses. For example, Al's consumption bundle consists of music, wine, and cheese. When Al's consumption of music is fixed at 10 units, the step-by-step solution has shown us how to apply his utility function to determine indifference equations for given utility levels, such as 40 and 70. These indifference curves illustrate Al's potential consumption bundles, each providing him with equal satisfaction.

When Al's music consumption increases from 10 to 20 units, the resulting change in the indifference equations reflects a shift in these curves. Even though the equations for the indifference curves change, Al's preferences between wine and cheese, captured by the MRS, stay the same. In real-world scenarios, consumption bundles help economists and businesses analyze consumer choices, plan product bundles, and understand market trends. Al's case simplifies complex human choices into a model that assumes consistent preferences and a linear trade-off between goods, providing a clear-cut insight into his consumption patterns.