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Chapter 2: Problem 1

Suppose \(U(x, y)=4 x^{2}+3 y^{2}\) a. 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, the utility function U(x, y) = 4x^2 + 3y^2 has partial derivatives 8x and 6y concerning x and y, respectively. The partial derivatives evaluated at x=1 and y=2 are 8 and 12. The total differential is given by dU = 8x dx + 6y dy. The rate of change of y concerning x when the utility is constant is dy/dx = -3y / 4x. The given utility value at x=1 and y=2 is U(1, 2) = 16. The change in x and y required to maintain the utility constant is -3:2. The utility function forms an ellipse shape, and the slope of its contour line is -3/2.

Step by step solution

01

Calculate the Partial Derivatives of U with respect to x and y.

To find the partial derivatives with respect to x and y, treat the other variable as a constant and differentiate. \(\frac{\partial U}{\partial x} = 8x\) \(\frac{\partial U}{\partial y} = 6y\)
02

Evaluate the partial derivatives at x=1 and y=2

Using the given x and y values, we will find the partial derivatives at these points. \(\frac{\partial U}{\partial x}(1, 2) = 8(1) = 8\) \(\frac{\partial U}{\partial y}(1, 2) = 6(2) = 12\)
03

Write the total differential for U

The total differential is given by adding the partial derivatives multiplied by their respective increments. \(dU = \frac{\partial U}{\partial x} dx + \frac{\partial U}{\partial y} dy\) \(dU = 8x\, dx + 6y\, dy\)
04

Calculate dy/dx for dU=0

To find the implied trade-off between x and y, holding U constant, follow these steps: 1. Set dU=0, which means there's no change in the utility function. 2. Solve the equation for dy/dx. \(0 = 8x\, dx + 6y\, dy\) \(\frac{-6y}{8x} = \frac{dy}{dx}\) \(\frac{dy}{dx} = \frac{-3y}{4x}\)
05

Show U=16 when x=1 and y=2

Substitute the given values of x and y into the utility function: \(U(1, 2) = 4(1)^2 + 3(2)^2\) \(U(1, 2) = 4 + 12 = 16\)
06

Determine the ratio of change for x and y holding U constant

To find the ratio of change, substitute x=1, y=2 into the trade-off equation \(\frac{dy}{dx} = \frac{-3y}{4x}\): \(\frac{dy}{dx}(1,2) = \frac{-3(2)}{4(1)}\) \(\frac{dy}{dx} = -\frac{3}{2}\) This means that x and y must change in a ratio of -3:2 to maintain a constant utility level of 16.
07

Analyze the shape of the U=16 contour line and its slope

The equation for the contour line is given as: \(U(x, y) = 4x^2 + 3y^2 = 16\) Dividing both sides by 16, we get: \(\frac{x^2}{4} + \frac{y^2}{\frac{16}{3}} = 1\) This is the equation of an ellipse. The slope of the tangent to this contour line can be found using the earlier trade-off ratio (-3/2).

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Key Concepts

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

Utility Function
Imagine a utility function as a mathematical way to represent a person's preferences for different commodities. The utility function we're exploring is defined as \(U(x, y)=4x^2+3y^2\). In this example, \(x\) and \(y\) could represent quantities of two different goods the person consumes.

The utility function's increase as more of one or both goods are consumed, illustrating the concept of 'more is better.' Our main goal with such a function is to determine how changes in the quantities of goods consumed affect the overall satisfaction level. This level of satisfaction is what economists call 'utility.' Understanding the utility function is crucial since it is the base from which we calculate other important economic measures like marginal utility and rates of substitution.
Total Differential
The total differential plays a pivotal role when we talk about changes in functions of multiple variables. It is a way to approximate the change in the function based on changes in its variables. For the utility function \(U(x, y)\), the total differential is expressed as \(dU = \frac{\partial U}{\partial x} dx + \frac{\partial U}{\partial y} dy = 8x\, dx + 6y\, dy\).

To put it simply, this expression helps us understand how a small change in the quantities of good \(x\) (indicated by \(dx\)) and good \(y\) (indicated by \(dy\)) would lead to a small change in utility (\(dU\)). The total differential is essential in predicting the behavior of the utility function under slight variations, thus serving as a basis for consumer choice theory in microeconomics.
Marginal Rate of Substitution
The marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. In our example, when we solve for \(dy/dx\) given that \(dU=0\), we find the MRS.

The calculation gives us \(dy/dx = -\frac{3y}{4x}\), which describes how much of good \(y\) we need to compensate for a small decrease in good \(x\) to keep the utility constant. Interestingly, this rate can change depending on the levels of \(x\) and \(y\) we start from. The negative sign indicates that the goods are substitutes–as you decrease one, you must increase the other to maintain utility. The MRS is a vital concept in understanding consumer behavior and how they make trade-offs between different goods.
Contour Lines
Understanding contour lines is integral in visualizing functions of two variables, such as our utility function. A contour line represents all the combinations of \(x\) and \(y\) that give the same level of utility. For the function \(U(x, y)=4x^2+3y^2\), the contour lines can be visualized by setting \(U(x, y)\) to a constant number.

As derived in step 7, the contour line for \(U=16\) is represented by the equation of an ellipse, indicating that the combinations of goods that provide a utility of 16 are elliptically distributed around the origin. The slope of the contour line \(-\frac{3}{2}\) is significant as it tells us how steeply the consumer needs to trade between goods \(x\) and \(y\) to stay on the same level of utility. Contour lines are not only fundamental in economics but also in fields like geography and engineering, where they help to understand topography and optimize various parameters.

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Most popular questions from this chapter

Because we use the envelope theorem in constrained optimization problems often in the text, proving this theorem in a simple case may help develop some intuition. Thus, suppose we wish to maximize a function of two variables and that the value of this function also depends on a parameter, \(a: f\left(x_{1}, x_{2}, a\right) .\) This maximization problem is subject to a constraint that can be written as: \(g\left(x_{1}, x_{2}, a\right)=0\) a. Write out the Lagrangian expression and the first-order conditions for this problem. b. Sum the two first-order conditions involving the \(x^{\prime}\) s. c. Now differentiate the above sum with respect to \(a\) - this shows how the \(x\) 's must change as \(a\) changes while requiring that the first-order conditions continue to hold. A. As we showed in the chapter, both the objective function and the constraint in this problem can be stated as functions of \(a: f\left(x_{1}(a), x_{2}(a), a\right), g\left(x_{1}(a), x_{2}(a), a\right)=0 .\) Differentiate the first of these with respect to \(a\). This shows how the value of the objective changes as \(a\) changes while keeping the \(x^{\prime}\) s at their optimal values. You should have terms that involve the \(x^{\prime}\) s and a single term in \(\partial f / \partial a\) e. Now differentiate the constraint as formulated in part (d) with respect to \(a\). You should have terms in the \(x\) 's and a single term in \(\partial g / \partial a\) f. Multiply the results from part (e) by \(\lambda\) (the Lagrange multiplier), and use this together with the first-order conditions from part (c) to substitute into the derivative from part (d). You should be able to show that \\[ \frac{d f\left(x_{1}(a), x_{2}(a), a\right)}{d a}=\frac{\partial f}{\partial a}+\lambda \frac{\partial g}{\partial a} \\] which is just the partial derivative of the Lagrangian expression when all the \(x^{\prime}\) 's are at their optimal values. This proves the envelope theorem. Explain intuitively how the various parts of this proof impose the condition that the \(x\) 's are constantly being adjusted to be at their optimal values. g. Return to Example 2.8 and explain how the envelope theorem can be applied to changes in the fence perimeter \(P\) -that is, how do changes in \(P\) affect the size of the area that can be fenced? Show that in this case the envelope theorem illustrates how the Lagrange multiplier puts a value on the constraint.

Consider the following constrained maximization problem: \\[ \begin{array}{ll} \text { maximize } & y=x_{1}+5 \ln x_{2} \\ \text { subject to } & k-x_{1}-x_{2}=0 \end{array} \\] where \(k\) is a constant that can be assigned any specific value. a. Show that if \(k=10\), this problem can be solved as one involving only equality constraints. b. Show that solving this problem for \(k=4\) requires that \(x_{1}=-1\) c. If the \(x^{\prime}\) s in this problem must be non-negative, what is the optimal solution when \(k=4 ?\) (This problem may be solved either intuitively or using the methods outlined in the chapter.) d. What is the solution for this problem when \(k=20 ?\) What do you conclude by comparing this solution with the solution for part (a)? Note: This problem involves what is called a quasi-linear function. Such functions provide important examples of some types of behavior in consumer theory-as we shall see.

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 Lagrange multiplier method.

One of the most important functions we will encounter in this book is the Cobb-Douglas function: \\[ y=\left(x_{1}\right)^{\alpha}\left(x_{2}\right)^{\beta} \\] where \(\alpha\) and \(\beta\) are positive constants that are each less than 1 a. Show that this function is quasi-concave using a "brute force" method by applying Equation 2.114 b. Show that the Cobb-Douglas function is quasi-concave by showing that any contour line of the form \(y=c\) (where \(c\) is any positive constant is convex and therefore that the set of points for which \(y>c\) is a convex set. c. Show that if \(\alpha+\beta>1\) then the Cobb-Douglas function is not concave (thereby illustrating again that not all quasiconcave functions are concave). Note: The Cobb-Douglas function is discussed further in the Extensions to this chapter.

The height of a ball that is thrown straight up with a certain force is a function of the time ( \(t\) ) from which it is released given by \(f(t)=-0.5 g t^{2}+40 t\) (where \(g\) is a constant determined by gravity). a. How does the value of \(t\) at which the height of the ball is at a maximum depend on the parameter \(g\) ? b. Use your answer to part (a) to describe how maximum height changes as the parameter \(g\) changes. c. Use the envelope theorem to answer part (b) directly. d. On the Earth \(g=32\), but this value varies somewhat around the globe. If two locations had gravitational constants that differed by \(0.1,\) what would be the difference in the maximum height of a ball tossed in the two places?
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