Question

Suppose the utility function for two goods, \(X\) and \(Y\), has the Cobb-Douglas form \\[ \text { utility }=U(X, Y)=\sqrt{X \cdot Y} \\] a. Graph the \(U=10\) indifference curve associated with this utility function. b. If \(X=5,\) what must \(Y\) equal to be on the \(U=10\) indifference curve? What is the \(M R S\) at this point? c. In general, develop an expression for the \(M R S\) for this utility function. Show how this can be interpreted as the ratio of the marginal utilities for \(X\) and \(Y\) d. Consider a logarithmic transformation of this utility function: \\[ U^{\prime \prime}=\log U \\] where log is the logarithmic function to base \(10 .\) Show that for this transformation the \(U^{\prime}=1\) indifference curve has the same properties as the \(U=10\) curve calculated in parts (a) and (b). What is the general expression for the \(M R S\) of this transformed utility function?

Short answer

Answer: The MRS for the given Cobb-Douglas utility function at the point X=5 and Y=20 on the U=10 indifference curve is -4.

Step-by-step solution

Write the utility function with U=10

The given Cobb-Douglas utility function is: \\[ U(X, Y)=\sqrt{X\cdot Y} \\] We need to find the indifference curve for U=10, so we plug in U=10: \\[ 10=\sqrt{X\cdot Y} \\]

Find the implicit function for the indifference curve

Isolate Y in the above equation: \\[ Y=\frac{100}{X} \\] This is the equation representing the U=10 indifference curve.

Plot the indifference curve

To plot the indifference curve with the equation \(Y=\frac{100}{X}\), we can generate a table of values for X and Y. For example: - X = 2, Y = 50 - X = 4, Y = 25 - X = 5, Y = 20 - X = 10, Y = 10 - X = 20, Y = 5 Plot these points on a graph, and connect the dots to form the indifference curve. b. If X=5, what must Y equal to be on the U=10 indifference curve? What is the MRS at this point?

Calculate Y for X=5 on the U=10 indifference curve

Using the equation for the U=10 indifference curve, plug in X=5: \\[ Y=\frac{100}{5}=20 \\] So, when X=5, Y must be 20 to be on the U=10 indifference curve.

Calculate MRS for X=5 and Y=20

To find the MRS, we need to find the negative of the ratio of marginal utilities: \\[ MRS=-\frac{MU_X}{MU_Y} \\] The marginal utility of X is: \\[ MU_X = \frac{\partial U}{\partial X} = \frac{1}{2} \cdot \frac{Y}{\sqrt{X\cdot Y}} \\] The marginal utility of Y is: \\[ MU_Y = \frac{\partial U}{\partial Y} = \frac{1}{2} \cdot \frac{X}{\sqrt{X\cdot Y}} \\] So, the MRS is: \\[ MRS = -\frac{\frac{1}{2} \cdot \frac{Y}{\sqrt{X\cdot Y}}}{\frac{1}{2} \cdot \frac{X}{\sqrt{X\cdot Y}}} = -\frac{Y}{X} \\] At the point X=5 and Y=20, MRS is: \\[ MRS = -\frac{20}{5} = -4 \\] c. In general, develop an expression for the MRS for this utility function. The general expression for the MRS has been found in Step 5: \\[ MRS = -\frac{Y}{X} \\] d. Consider a logarithmic transformation of the utility function, and show that the U'=1 indifference curve has the same properties as the U=10 curve. What is the general expression for the MRS of this transformed utility function?

Calculate the logarithmic transformation

Given the logarithmic transformation: \\[ U''=\log U = \log \sqrt{X\cdot Y} \\] For the U''=1 indifference curve, we have: \\[ 1 = \log\sqrt{X\cdot Y} \\]

Prove that U'=1 has the same properties as U=10

To show that the U''=1 indifference curve has the same properties as the U=10 curve, we will show that the relationship between X and Y is the same. \\[ 10^1 = \sqrt{X\cdot Y} \\] The above expression is the same as the expression found for the U=10 indifference curve in Step 2. Therefore, the U'=1 and U=10 curves have the same properties.

Find the MRS for the transformed utility function

To find the MRS for the transformed utility function, we first need to find the marginal utilities of X and Y with respect to the transformed utility function U''. \\[ MU''_X = \frac{\partial U''}{\partial X} = \frac{1}{2} \cdot \frac{Y}{\sqrt{X\cdot Y}} \\] \\[ MU''_Y = \frac{\partial U''}{\partial Y} = \frac{1}{2} \cdot \frac{X}{\sqrt{X\cdot Y}} \\] The MRS for U'' is calculated as the negative ratio of marginal utilities as before: \\[ MRS'' = -\frac{MU''_X}{MU''_Y} = -\frac{Y}{X} \\] So the general expression for MRS for the transformed utility function is the same as for the original utility function: \\[ MRS'' = -\frac{Y}{X} \\]

Key concepts

Indifference Curve

Imagine you are standing in an ice cream shop, deciding between two delicious flavors: chocolate and vanilla. Here's a fun fact: when you can't decide because you enjoy both equally, you are essentially on an indifference curve. In economics, an indifference curve represents all the combinations of two goods or services that provide a consumer with the same level of satisfaction. For instance, in the context of the Cobb-Douglas utility function \( U(X, Y) = \sqrt{X \cdot Y} \), for a utility level of 10 (\( U = 10 \)), the curve illustrates different pairs of \( X \) and \( Y \) that keep your happiness constant. The equation for this curve is \( Y = \frac{100}{X} \), which shows that if you have more chocolate (\( X \) increasing), you need less vanilla (\( Y \) decreasing) to stay equally content.

To make learning more engaging, imagine plotting your choices in the ice cream shop on a graph. Once plotted, all the combinations where you’re just as happy form a smooth curve sloping downwards, embodying the trade-off between the scoops of chocolate and vanilla. These graphical representations are incredibly handy for visualizing how you'd substitute one good for another while maintaining the same level of delight.

Marginal Rate of Substitution (MRS)

Now, let's say you're enjoying your ice cream, but you're considering swapping a scoop of vanilla for an extra scoop of chocolate. The rate at which you're willing to make this trade without changing your satisfaction level is the Marginal Rate of Substitution (MRS). In economic terms, it's the rate at which you're willing to give up units of one good (vanilla) to obtain one more unit of another good (chocolate), remaining on the same indifference curve. It technically represents the slope of the indifference curve at a particular point.

In the exercise with the Cobb-Douglas utility function, the MRS is calculated by finding the negative ratio of the marginal utilities of \( X \) and \( Y \) (\[ MRS = -\frac{MU_X}{MU_Y} \]). This gives us an expression \[ MRS = -\frac{Y}{X} \], which simplifies the comparison of enjoyment between chocolate and vanilla scoops. When \( X = 5 \) and \( Y = 20 \) scoops, you would be willing to give up 4 scoops of vanilla for one additional scoop of chocolate, which is an intuitive way to understand your preferences for more chocolatey goodness.

Marginal Utility

The concept of marginal utility is similar to deciding just how much more enjoyable one more sprinkle of toppings is on your ice cream. Marginal utility refers to the additional satisfaction you gain from consuming one more unit of a good or service. It’s like the little extra happiness from one more bite of ice cream.

In our textbook solution, marginal utility is a calculus concept used to measure the additional pleasure (utility) you get from one more unit of goods \( X \) or \( Y \). The formulas for these are derived from the initial utility function, \( MU_X = \frac{1}{2} \cdot \frac{Y}{\sqrt{X\cdot Y}} \) and \( MU_Y = \frac{1}{2} \cdot \frac{X}{\sqrt{X\cdot Y}} \). Explaining it in simpler terms, if you have one more scoop of chocolate, the marginal utility tells you how this changes your happiness considering the amount of vanilla you already have.

Logarithmic Transformation

You might be wondering what a logarithmic transformation has to do with enjoying your ice cream. Picture this: when you take logarithms (like a fun math selfie!), you are simply transforming your original experience (just like applying a cool filter) to see it from a different perspective. In mathematics, we often apply a logarithm to simplify complex multiplication into easier addition, making it simpler to analyze.

In our exercise, a logarithmic transformation is applied to the utility function \( U(X, Y) = \sqrt{X \cdot Y} \) to become \( U'' = \log U \). For the transformed utility function, \( U''=1 \) suggests the same level of satisfaction as \( U=10 \) from the original function, as shown by \( 10^1 = \sqrt{X\cdot Y} \). It reveals that the relationships among goods don't change, just as your preference for ice cream won't change whether you calculate your happiness in scoops or in 'log scoops.' This transformation retains the same MRS, confirming that the satisfaction landscape is preserved, just viewed through a new lens.