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Chapter 3: Problem 1

Laidback Al derives utility from 3 goods; music (M), wine \((W)\), and cheese (C). His utility function is of the simple linear form \\[ \text { utility }=\boldsymbol{U}(\boldsymbol{M}, \boldsymbol{W}, \boldsymbol{C})=\boldsymbol{M}+2 \boldsymbol{W}+\mathbf{3} \mathbf{C} \\] a. Assuming Al's consumption of music is fixed at \(10,\) determine the equations for the in difference curves for Wand Cfor \(U=40\) and \(U-70 .\) Sketch these curves. b. Show that Al's \(M R S\) of wine for cheese is constant for all values of Wand \(C\) on the indif ference 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

Expert verified
Answer: Al's marginal rate of substitution (MRS) of wine for cheese is constant at 2/3 for all values of W and C on his indifference curves. Even with an increase in his consumption of music, the MRS of wine for cheese remains constant at 2/3.

Step by step solution

01

a. Finding the indifference curves for wine and cheese#

Since Al's consumption of music is fixed at 10, we can rewrite the utility function as: U(W, C) = 10 + 2W + 3C Now, we need to find the indifference curves for wine and cheese for U = 40 and U = 70. An indifference curve represents all the combinations of wine and cheese that provide Al with the same level of utility. 1. For U = 40, we have: 40 = 10 + 2W + 3C 2. For U = 70, we have: 70 = 10 + 2W + 3C Now, let's solve these equations for C in terms of W: 1. For U = 40: C = (30 - 2W) / 3 2. For U = 70: C = (60 - 2W) / 3 These are the equations of the indifference curves for U = 40 and U = 70, respectively. To sketch these curves, plot C against W for each equation.
02

b. Showing that Al's MRS of wine for cheese is constant#

The marginal rate of substitution (MRS) of wine for cheese is the rate at which Al is willing to trade wine for cheese while maintaining the same level of utility. It is given by the ratio of the marginal utility of wine (MU_W) to the marginal utility of cheese (MU_C). So, we need to find the partial derivatives of the utility function with respect to W and C: MU_W = dU/dW = 2 MU_C = dU/dC = 3 Now, let's find the MRS of wine for cheese: MRS = MU_W / MU_C = 2 / 3 Since MRS is constant (2/3) for all values of W and C on the indifference curves, Al's MRS of wine for cheese is constant for all values of W and C on the indifference curves calculated in part (a).
03

c. Changes in Al's consumption of music#

If Al's consumption of music increases to 20, we can rewrite the utility function as: U(W, C) = 20 + 2W + 3C Now, we need to find the indifference curves for wine and cheese for U = 40 and U = 70: 1. For U = 40, we have: C = (20 - 2W) / 3 2. For U = 70, we have: C = (50 - 2W) / 3 The new indifference curves will be different from the ones we found in part (a). The increase in consumption of music shifts the indifference curves. Regarding the MRS of wine for cheese, the partial derivatives of the utility function with respect to W and C will remain the same: MU_W = dU/dW = 2 MU_C = dU/dC = 3 Therefore, the MRS of wine for cheese will still be constant at 2/3 for all values of W and C on the new indifference curves. The increase in consumption of music does not change the MRS of wine for cheese.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Marginal Rate of Substitution
In the realm of consumer choice theory, the Marginal Rate of Substitution (MRS) is a pivotal concept. It's concerned with how much of one good a consumer is willing to give up to acquire an additional unit of another good, keeping their utility level constant. In mathematical terms, it is expressed as the ratio of the marginal utilities of the two goods. In the exercise, Al's utility function is given as \( U(M, W, C) = M + 2W + 3C \). This means that the marginal utility of wine, \(MU_W\), is 2, and the marginal utility of cheese, \(MU_C\), is 3.

This translates the MRS of wine for cheese as \( \frac{MU_W}{MU_C} = \frac{2}{3} \). Al's MRS is constant, meaning he always values trading wine for cheese at the same rate, no matter how much of each he has. This steady MRS is typical in linear utility functions, representing constant trade-offs between goods.
Utility Function
A utility function is a mathematical representation that captures a consumer's preferences. It assigns numerical values to different combinations of goods, illustrating the satisfaction or happiness a consumer derives from consuming them. Al's utility function, \(U(M, W, C) = M + 2W + 3C\), is linear, which significantly simplifies the analysis. This form reflects that each unit of wine adds two units of utility, cheese adds three, and music adds one.
  • This function highlights the relative importance of each good to Al. Cheese, with the highest coefficient, is most valued, while music, with a coefficient of one, is least valued.
  • The indifference curves derived from this utility function, such as \(C = \frac{30 - 2W}{3}\) for \(U = 40\), show combinations of goods providing the same utility level, forming straight lines in linear functions.
These curves shift as the music consumption changes, showing different combinations of wine and cheese that maintain a constant utility despite an increase in music, reflecting Al's constant preferences.
Marginal Utility
Marginal utility (MU) delineates the additional satisfaction from consuming one more unit of a particular good. In the context of Al's utility function, we derive the marginal utilities by examining how changes in the quantities of each good affect overall utility. Since the utility function is \(U(M, W, C) = M + 2W + 3C\), marginal utility of each good is represented by the coefficient in front of each variable.

The marginal utility of wine, \(MU_W\), is 2, and of cheese, \(MU_C\), is 3. This suggests that for every additional unit of wine, Al's utility increases by 2, while each unit of cheese increases his utility by 3.
  • This discrepancy in marginal utilities indicates cheese's higher intrinsic value over wine to Al.
  • Marginal utility remains unchanged irrespective of Al's music consumption change from 10 to 20, indicating stable utilities from wine and cheese.
Understanding marginal utilities is crucial in ascertaining consumer preferences and making informed predictions about consumption patterns.

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Most popular questions from this chapter

Georgia always eats hot dogs in a bun together with 1 oz. of mustard. Each hot dog eaten in this way provides 15 units of utility, but any other combination of hot dogs, buns, and mustard is worthless to Georgia. a. Explain the nature of Georgia's utility function and indicate the form of her indifference curve map. b. Suppose hot dogs cost \(\$ 1,\) buns cost \(\$ .40,\) and mustard costs \(\$ .10\) per ounce. Show how Georgia's utility can be represented by the total amount of money she spends on these three items. c. How would your answer to part (b) change if the price of hot dogs rose to \(\$ 1.50 ?\)

Example 3.3 shows that the MRS for the Cobb-Douglas function \\[ U(X, Y)=X^{n} Y^{\wedge} \\] is given by \\[ M R S=_{P}^{\wedge}(Y / X) \\] a Does this result depend on whether \(a+(3=1 ?\) Does this sum have any relevance to the theory of choice? b. For commodity bundles for which \(Y=\mathrm{X}\), how does the \(M R S\) depend on the values of \(a\) and \(/ 3 ?\) Develop an intuitive explanation of why if \(a>(3, M R S>1\). Illustrate your argument with a graph. c. Suppose an individual obtains utility only from amounts of Xand Fthat exceed minimal subsistence levels given by \(\mathrm{X}_{c}, \mathrm{F}_{\mathrm{o} .}\) In this case, \(U(X, Y)=(X-X o)^{\circ}\left(Y-Y_{o}\right) e .\) Is this function homothetic? (For a further discussion, see the extensions to Chapter \(4 .)\)

Two goods have independent marginal utilities if \\[ \frac{d t U}{d Y d X} \quad \frac{d^{2} U}{d X d Y}=0 \\] Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing \(M R S\). Provide an example to show that the converse of this statement is not true. 3.10 a. Show that the CES function \\[ \begin{array}{c} x^{3} \\ 0 \end{array}+* \frac{Y}{T} \\] is homothetic. How does the \(M R S\) depend on the ratio \(\mathrm{F} / \mathrm{X}\) ? b. Show that your results from part (a) agree with Example 3.3 for the case \(6=1\) (perfect substitutes \()\) and \(5=0\) (Cobb-Douglas). c. Show that the \(M R S\) is strictly diminishing for all values of \(8<1\) d. Show that if \(X=Y\), the \(M R S\) for this function depends only on the relative sizes of a and j8. e. Calculate the \(M R S\) for this function when \(Y / X=.9\) and \(Y / X=1.1\) for the two cases \(8=5\) and \(8=-1 .\) What do you conclude about the extent to which the \(M R S\) changes in the vicinity of \(\mathrm{X}=\mathrm{F} ?\) How would you interpret this geometrically?

Consider the following utility functions: a. \(U(X, Y)=X Y\) b. \(U(X, Y)=X^{2} Y^{2}\) \(c \quad U(X, Y)=\ln X+\operatorname{In} Y\) Show that each of these has a diminishing \(M R S\), but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?

Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether they obey the assumption of a diminishing \(M R S\) ): a. \(U=3 X+Y\) b. \(\quad U=\mathrm{VX}-Y\) \(\mathrm{c} \quad U=\mathrm{VX}^{2}+\mathrm{P}\) d. \(U=\mathrm{VX}^{2}-\mathrm{Y}^{2}\) \(\mathbf{e}, \quad U=\mathbf{X}^{23} \mathbf{F}^{13}\) \(\mathrm{f}, \quad U=\log \mathrm{X}+\log Y\).
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