Question

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

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

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. \(\frac{\partial U}{\partial x} = 8x\) \(\frac{\partial U}{\partial y} = 6y\)

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. \(\frac{\partial U}{\partial x}(1, 2) = 8(1) = 8\) \(\frac{\partial U}{\partial y}(1, 2) = 6(2) = 12\)

Write the total differential for U

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

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. \(0 = 8x\, dx + 6y\, dy\) \(\frac{-6y}{8x} = \frac{dy}{dx}\) \(\frac{dy}{dx} = \frac{-3y}{4x}\)

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

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

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.

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).

Key concepts

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.