Question

a. $$ Mr. Odde Ball enjoys commodities $X$ and $Y$ according to the utility function $$U(X,Y)=\sqrt{X^2+Y^2}$$ Maximize Mr. Ball's utility if $P_x=\mathrm{\$}3,P_Y=\mathrm{\$}4,$ and he has $\mathrm{\$}50$ to spend. Hint: It may be easier here to maximize $U^2$ rather than $U.$ Why won't this alter your results? b. Graph Mr. Ball's indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball's behavior? Have you found a true maximum?

Short answer

Answer: The optimal quantities of X and Y that maximize Mr. Ball's utility are X = 150/13 and Y = 100/13.

Step-by-step solution

1. Write the budget constraint

We will start by writing the budget constraint of Mr. Ball, given the prices of X and Y and his income. The budget constraint can be expressed as: $$P_{x}X+P_{y}Y=I$$ Substitute the prices and income into the equation: $$3X+4Y=50$$

2. Set up the utility maximization problem

Now set up the utility maximization problem. Since it's easier to maximize U^2 instead of U (as hinted), and it won't alter the results, we'll work with U^2. Our objective is to maximize U^2 subject to the budget constraint: $$\underset{X,Y}{max}{U}^2(X,Y)=X^2+Y^2\text{subject to}3X+4Y=50$$

3. Find the optimal solution for X and Y

To solve the maximization problem, we can use the method of Lagrange multipliers. The Lagrangian function is given by: $$\mathcal{L}(X,Y,\lambda )=X^2+Y^2-\lambda (3X+4Y-50)$$ Now, we'll compute the first-order conditions by taking the partial derivatives with respect to X, Y, and λ: $$\begin{gathered} \frac{\partial \mathcal{L}}{\partial X}=2X-3\lambda =0 \\ \frac{\partial \mathcal{L}}{\partial Y}=2Y-4\lambda =0 \\ \frac{\partial \mathcal{L}}{\partial \lambda }=3X+4Y-50=0 \end{gathered}$$ Solve the system of equations to find X, Y, and λ. We get: $$\begin{gathered} X=\frac{3}{2}\lambda \\ Y=\lambda \\ 3\left(\frac{3}{2}\lambda \right)+4\lambda =50 \\ \frac{13}{2}\lambda =50 \\ \lambda =\frac{100}{13} \end{gathered}$$ Now, find the optimal values of X and Y: $$\begin{gathered} X=\frac{3}{2}\lambda =\frac{3}{2}\times \frac{100}{13}=\frac{150}{13} \\ Y=\lambda =\frac{100}{13} \end{gathered}$$

4. Graph the indifference curve and budget constraint

We have found that Mr. Ball maximizes his utility when consuming X = 150/13 and Y = 100/13. Now, draw the graph using the budget constraint and the equation of the indifference curve, U(X, Y) = k, where k is a constant: The budget constraint: $$3X+4Y=50$$ The indifference curve: $$\begin{gathered} U(X,Y)=\sqrt{X^2+Y^2}=k \\ {X}^2+Y^2=k^2 \end{gathered}$$ By plotting the budget constraint and the indifference curve passing through the optimal consumption point (150/13, 100/13), we can observe that they are tangent at this point, which confirms that we have found a true maximum. The graph also tells us that Mr. Ball is spending his entire budget on X and Y since the budget constraint is binding. Additionally, it shows that Mr. Ball's preferences exhibit diminishing marginal utility for both commodities, as the indifference curve is convex to the origin.

Key concepts

Lagrange Multipliers

Lagrange multipliers is a powerful mathematical tool used in optimization problems, like maximizing Mr. Ball's utility. Imagine you have a goal to maximize a certain function, such as utility, under some constraints, like a budget. Here, Lagrange multipliers help to find the best possible solution that adheres to these constraints.
The method involves introducing an auxiliary variable (lambda, $\lambda$) that helps to incorporate the constraint into the utility function. In this case, the function to maximize is $X^2+Y^2$, subject to the budget constraint $3X+4Y=50$. The Lagrangian function combines these as follows:

• $\mathcal{L}(X,Y,\lambda )=X^2+Y^2-\lambda (3X+4Y-50)$

We solve for $X$, $Y$, and $\lambda$ by setting partial derivatives of the Lagrangian to zero, leading to optimal values that lie on the given constraint curve. This ensures Mr. Ball maximizes his utility while spending within his budget.

Indifference Curve

An indifference curve represents combinations of two goods that provide the consumer with the same level of satisfaction or utility. In Mr. Ball's case, the utility function is $U(X,Y)=\sqrt{X^2+Y^2}$. The indifference curve derived from this function shows different bundles of goods $X$ and $Y$ that result in the same utility level.
When graphed, indifference curves typically have a convex shape, indicating that as you consume more of one good, the additional satisfaction you get from consuming more of that good decreases. In Mr. Ball's situation, the point where the indifference curve is tangent to the budget line is where he maximizes utility. That's because any movement from this point would result in a lower utility without more spending.

Budget Constraint

The budget constraint outlines the combinations of goods a consumer can purchase given their income and the prices of the goods. For Mr. Ball, the budget constraint is expressed as $3X+4Y=50$. Here, 3 and 4 are the prices of goods $X$ and $Y$ respectively, and 50 is the total money available for spending.
The budget line graphically represents this constraint. It will slope downwards, showing the trade-off between the two goods – as you spend more on one, you must spend less on the other to stay within your budget. Mr. Ball's task is to reach the highest indifference curve possible while still touching his budget line, which leads to utility maximization at the tangency point.

Diminishing Marginal Utility

Diminishing marginal utility is the principle that as a person consumes more units of a good, the additional satisfaction or utility gained from each additional unit decreases.
In simpler terms, the more of a good you have, the less you crave another unit. For instance, the first slice of pizza might bring much joy, but the fourth slice not as much.

• In Mr. Ball's utility function $U(X,Y)=\sqrt{X^2+Y^2}$, this concept implies that as he consumes more of either good $X$ or $Y$, the additional satisfaction he obtains from each extra unit reduces.

This is why the indifference curve is convex. Mr. Ball will balance his consumption so that the extra utility per dollar spent on each good is equal, which leads him to maximize total utility under the given budget. This behavior reflects what economists describe as equilibrium in consumption.