Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 5

Suppose \(U=(x, y)=4 x^{2}+3 y^{2}\) a. \(\quad\) Calculate \(\partial U / \partial x, \partial U / \partial y\) b. Evaluate these partial derivatives at \(x=1, y=2\) c. Write the total differential for \(U\) d. Calculate \(d y / d x\) for \(d U=0\) - that is, what is the implied trade-off between \(x\) and \(y\) holding \(U\) constant? e. Show \(U=16\) when \(x=1, y=2\) f. In what ratio must \(x\) and \(y\) change to hold \(U\) constant at 16 for movements away from \(x=1, y=2 ?\) g. More generally, what is the shape of the \(U=16\) contour line for this function? What is the slope of that line?

Short Answer

Expert verified
In summary, we found the partial derivatives of U with respect to x and y, evaluated them at the given point (x=1, y=2), and determined the total differential for U. Additionally, we found the implied trade-off between x and y to hold U constant, verified that U=16 for the given point, and calculated the shape of the U=16 contour line and its slope. The results show that at the point (x=1, y=2), the partial derivatives are 8 and 12 for x and y, respectively. The total differential is \(8x dx + 6y dy\), and the slope of the contour line at (x=1, y=2) is -2/3.

Step by step solution

01

Calculate \(\frac{\partial U}{\partial x}\)

To find the partial derivative of U with respect to x, treat y as a constant and differentiate U with respect to x: \(\frac{\partial U}{\partial x} = 8x\).
02

Calculate \(\frac{\partial U}{\partial y}\)

To find the partial derivative of U with respect to y, treat x as a constant and differentiate U with respect to y: \(\frac{\partial U}{\partial y} = 6y\). b. Evaluating Partial Derivatives
03

Evaluate \(\frac{\partial U}{\partial x}\) at \((x=1, y=2)\)

Evaluate the partial derivative with respect to x at the given point: \(\left.\frac{\partial U}{\partial x}\right\vert_{(x=1,y=2)} = 8(1) = 8\).
04

Evaluate \(\frac{\partial U}{\partial y}\) at \((x=1, y=2)\)

Evaluate the partial derivative with respect to y at the given point: \(\left.\frac{\partial U}{\partial y}\right\vert_{(x=1,y=2)} = 6(2) = 12\). c. Total Differential
05

Write the total differential for U

The total differential for U is given by: \(dU = \frac{\partial U}{\partial x} dx + \frac{\partial U}{\partial y} dy = 8x dx + 6y dy\). d. Implied Trade-off
06

Calculate \(\frac{dy}{dx}\) for \(dU=0\)

Set the total differential equal to zero, and solve for \(\frac{dy}{dx}\): \(0 = 8x dx + 6y dy \Rightarrow -8x dx = 6y dy \Rightarrow \frac{dy}{dx} = -\frac{4x}{3y}\). e. Show U=16
07

Verify that U=16 when \((x=1, y=2)\)

Substitute x=1 and y=2 into the function U: \(U=4(1)^2+3(2)^2=4+12=16\). f. Ratio of x and y
08

Find the ratio of x and y to hold U constant

Use the previously found formula for \(\frac{dy}{dx}\) to find the ratio for holding U constant: \(\frac{dy}{dx} = -\frac{4x}{3y}\). g. Contour Line and Slope
09

Find the shape of the U=16 contour line

The contour line for U=16 is the set of all points (x y) that satisfy the equation \(4x^2 + 3y^2 = 16\). This is an ellipse.
10

Find the slope of the U=16 contour line

Use the formula for \(\frac{dy}{dx}\) at the given point(x=1, y=2) to find the slope: \(\frac{dy}{dx} = -\frac{4(1)}{3(2)} = -\frac{2}{3}\). The slope of the contour line at the point (x=1, y=2) is -2/3.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The height of a ball \(t\) seconds after it is thrown straight up is \(-1 / 2 g t^{2}+40 t\) (where \(g\) is the acceleration due to gravity). a. If \(g=32\) (as on the earth), when does the ball reach a maximum height? What is that height? b. If \(g=5.5\) (as on the moon), when does the ball reach a maximum height and what is that height? Can you explain the reasons for the difference between this answer and the answer for part (a)? c. In general, develop an expression for the change in maximum height for a unit change in \(g .\) Explain why this value depends implicitly on the value of \(g\) itself.

For each of the following functions of one variable, determine all local maxima and minima and indicate points of inflection (where \(f^{\prime \prime}=0\) ): a. \(f(x)=4 x^{3}-12 x\) b. \(f(x)=4 x-x^{2}\) c. \(f(x)=x^{3}\)

Suppose that \(f(x, y)=x y .\) Find the maximum value for \(f\) if \(x\) and \(y\) are constrained to sum to \(1 .\) Solve this problem in two ways: by substitution and by using the Langrangian multiplier method.

Show that if \(f\left(x_{1}, x_{2}\right)\) is a concave function, it is also a quasi-concave function. Do this by comparing Equation 2.107 (defining quasi- concavity) to Equation 2.88 (defining concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasiconcave functions necessarily concave?

Taxes in Oz are calculated according to the formula \\[ T=.01 I^{2} \\] where \(T\) represents thousands of dollars of tax liability and \(I\) represents income measured in thousands of dollars. Using this formula, answer the following questions: a. How much tax do individuals with incomes of \(\$ 10,000, \$ 30,000,\) and \(\$ 50,000\) pay? What are the average tax rates for these income levels? At what income level does tax liability equal total income? b. Graph the tax schedule for Oz. Use your graph to estimate marginal tax rates for the income levels specified in part (a). Also show the average tax rates for these income levels on your graph. c. Marginal tax rates in \(\mathrm{Oz}\) can be estimated more precisely by calculating tax owed if persons with the incomes in part (a) get one more dollar. Make this computation for these three income levels. Compare your results by calculating the marginal tax rate function using calculus.
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free