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Chapter 4: Problem 4

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 \(p_{x}=\$ 3, p_{y}=\$ 4,\) and he has \(\$ 50\) to spend. Hint: It may be easier here to maximize \(U^{2}\) rather than \(U\). Why will this not 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

Expert verified
Answer: The optimal consumption bundle for Mr. Odde Ball to maximize his utility given his budget is to consume 25/2 units of commodity x and 25/4 units of commodity y.

Step by step solution

01

Rewrite the utility function into \(U^2\)

To simplify the problem, we will maximize \(U^2\) instead of \(U\). Since the utility function is monotonously increasing, maximizing \(U^2\) will still yield the same results as maximizing \(U\). The rewritten utility function is: \[U^2(x, y) = x^2 + y^2.\]
02

Write down the budget constraint

Mr. Ball's budget constraint is given by the fact that the amount he spends on commodities x and y cannot exceed his total budget of $50: \[p_x x + p_y y = 3x + 4y = 50.\]
03

Express y in terms of x from budget constraint

Solve the budget constraint for y to use in the utility function. Rearrange the budget constraint to get: \[y = \frac{50 - 3x}{4}.\]
04

Substitute y in the utility function

Substitute the expression found for y in the \(U^2\) function: \[U^2(x) = x^2 + \left(\frac{50 - 3x}{4}\right)^2.\]
05

Find the optimal x by taking the derivative of the utility function

Take the first derivative of utility function with respect to x and set it to zero: \[\frac{d}{dx}U^2(x) = 2x - \frac{9}{4}(50-3x) = 0.\] Solve this equation for x to find the optimal consumption of x: \[x = \frac{25}{2}.\]
06

Find the optimal y

Plug the optimal x into the equation for y: \[y = \frac{50 - 3\left(\frac{25}{2}\right)}{4} = \frac{25}{4}.\]
07

Confirm the solution is a maximum

We can check that the second derivative of the utility function with respect to x is negative, which confirms we found a maximum: \[\frac{d^2}{dx^2}U^2(x) = 2-\frac{27}{4} < 0.\]
08

Graph the indifference curve and budget constraint

Graph the indifference curve (involving \(U = \sqrt{x^2+y^2}\)) and the budget constraint (involving \(3x+4y=50\)). Observe that the indifference curve is tangent to the budget constraint at the point \((x,y) = \left(\frac{25}{2},\frac{25}{4}\right)\). This shows us that Mr. Ball is maximizing his utility given his budget. The graph demonstrates that Mr. Ball's behavior will include consuming \(\frac{25}{2}\) units of commodity x and \(\frac{25}{4}\) units of commodity y, which maximizes his utility given his budget. By analyzing the graph, we can see that we have found a true maximum, as shown by the tangent point between the indifference curve and budget constraint. The second derivative test also confirms this by showing a negative value.

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Most popular questions from this chapter

Suppose individuals require a certain level of food \((x)\) to remain alive. I et this amount be given by \(x_{0}\). Once \(x_{0}\) is purchased, individuals obtain utility from food and other goods \((y)\) of the form \\[U(x, y)=\left(x-x_{0}\right)^{\alpha} y^{\beta},\\] where \(\alpha+\beta=1\). a. Show that if \(I>p_{x} x_{0}\) then the individual will maximize utility by spending \(\alpha\left(I-p_{x} x_{0}\right)+p_{x} x_{0}\) on good \(x\) and \(\beta\left(I-p_{x} x_{0}\right)\) on good \(y,\) Interpret this result. b. How do the ratios \(p_{x} x / I\) and \(p_{x} y / I\) change as income increases in this problem? (See also Extension E4.2 for more on this utility function.)

a. A young connoisseur has \(\$ 600\) to spend to build a small wine cellar. She enjoys two vintages in particular: a 2001 French Bordeaux \(\left(w_{F}\right)\) at \(\$ 40\) per bottle and a less expensive 2005 California varietal wine \(\left(w_{C}\right)\) priced at \(\$ 8\). If her utility is \\[U\left(w_{\mathrm{F}}, w_{\mathrm{C}}\right)=w_{F}^{2 / 3} w_{C}^{1 / 3},\\] then how much of each wine should she purchase? b. When she arrived at the wine store, this young oenologist discovered that the price of the French Bordeaux had fallen to \(\$ 20\) a bottle because of a decrease in the value of the euro. If the price of the California wine remains stable at \(\$ 8\) per bottle, how much of each wine should our friend purchase to maximize utility under these altered conditions? c. Explain why this wine fancier is better off in part (b) than in part (a). How would you put a monetary value on this utility increase?

Two of the simplest utility functions are: 1\. Fixed proportions: \(U(x, y)=\min [x, y]\). 2\. Perfect substitutes: \(U(x, y)=x+y\). a. For each of these utility functions, compute the following: \(\bullet\)Demand functions for \(x\) and \(y\). \(\bullet\)Indirect utility function \(\bullet\)expenditure function b. Discuss the particular forms of these functions you calculated-why do they take the specific forms they do?

Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies \((t)\) and soda \((s),\) and these provide him a utility of \\[\text { utility }=U(t, s)=\sqrt{t s}.\\] a. If Twinkies cost \(\$ 0.10\) each and soda costs \(\$ 0.25\) per cup. how should Paul spend the \(\$ 1\) his mother gives him to maximize his utility? b. If the school tries to discourage Twinkie consumption by increasing the price to \(\$ 0.40,\) by how much will Paul's mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)?

In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by \\[U(x, y)=\left(x^{8}+y^{\delta}\right)^{1 / 8}.\\] [in this function the elasticity of substitution \(\sigma=1 /(1-\delta)]\) a. Show that the indirect utility function for the utility function just given is \\[\begin{array}{r} V=I\left(p_{x}^{\prime}+p_{y}^{\prime}\right)^{-1 / r} \\ \text { where } r=\delta /(\delta-1)=1-\sigma \end{array}\\] b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by \\[E=V\left(p_{x}^{r}+p_{y}^{r}\right)^{1 / r}.\\] f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.
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