Question

Suppose the production possibility frontier for guns \((x)\) and butter \((y)\) is given by \\[ x^{2}+2 y^{2}=900 \\] a. Graph this frontier. b. If individuals always prefer consumption bundles in which \(y=2 x,\) how much \(x\) and \(y\) will be produced? c. At the point described in part (b), what will be the \(R P T\) and hence what price ratio will cause production to take place at that point? (This slope should be approximated by considering small changes in \(x\) and \(y\) around the optimal point.) d. Show your solution on the figure from part (a).

Short answer

Answer: The optimal production point under the given preferences is (10, 20), and the relative price trade at this point is 4.

Step-by-step solution

Graph the equation

To graph the equation \(x^2+2y^2=900\), we first notice that it represents a general ellipse with the center at the origin. Divide through by 900, obtaining: \(\frac{x^2}{900}+\frac{y^2}{450}=1\). Thus, the semi-major axis (responsible for width) is \(\sqrt{900} = 30,\) and the semi-minor axis (responsible for height) is \(\sqrt{450} = 15\sqrt{2}\). An intuitive way to graph this is to create a set of axes, then plot points horizontally and vertically from the origin that respectively reach the semi-major and semi-minor distances. Then, draw a smooth curve connecting these points. #b. Find x and y#

Substitute y with 2x

In order to find the values of x and y when individuals prefer consumption bundles such that \(y = 2x\), we will substitute this relationship into the equation for the PPF: \(x^2 + 2y^2 = 900\). This substitution will allow us to solve for x, which can then be used to find the value of y. Replacing y with 2x results in \(x^2 + 2(2x)^2 = 900\).

Solve for x

Now, solve the equation \(x^2+8x^2=900\) for x: \\[ x^2 + 8x^2 = 900 \\ 9x^2 = 900 \\ x^2 = 100 \\ x = 10 \\]

Solve for y

Since \(y=2x\), we can now find the value of y: \\[ y = 2(10) \\ y = 20 \\] Therefore, x = 10 and y = 20 when individuals prefer consumption bundles such that \(y = 2x\). #c. Calculate RPT and price ratio#

Determine the derivative of PPF

To find the RPT and price ratio, we need to calculate the slope of the PPF at the point (10, 20). The slope can be found by taking the derivative of the original equation with respect to x: \\[ \frac{d}{dx}(x^2+2y^2) = \frac{d}{dx}(900) \\] Differentiating both sides yields: \\[ 2x + 4y\frac{dy}{dx} = 0 \\]

Find slope at point (10, 20)

We now plug our values for x and y into the differentiated equation and solve for \(\frac{dy}{dx}\): \\[ 2(10) + 4(20)\frac{dy}{dx} = 0 \\ 20 + 80\frac{dy}{dx} = 0 \\ 80\frac{dy}{dx} = -20 \\ \frac{dy}{dx} = -\frac{1}{4} \\] At the point (10, 20), the RPT (i.e., the slope of the PPF) is -1/4. Thus, the price ratio will be the negative reciprocal of the slope, which is 4. #d. Show the solution on the figure# Place the point (10, 20) on the graph created in Step 1. The point should be on the ellipse, representing the optimal production point under the given preferences. Then, draw the tangent line to the ellipse at this point that has a slope of -1/4. This represents the RPT.