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 consumption mix is (10, 20) with an RPT of 1/4, and the corresponding price ratio is 1:4, meaning the price of 1 unit of \(x\) is 1/4 the price of 1 unit of \(y\).

Step-by-step solution

Graph the PPF

To graph the Production Possibility Frontier (PPF) given by the equation \(x^2 + 2y^2 = 900\), we can rewrite the equation in terms of \(y\) as follows: \[y = \sqrt{\frac{900-x^2}{2}}\] Graph this equation to create the PPF, which will resemble an ellipse.

Optimal consumption mix

We are given a consumption preference equation: \(y = 2x\). To find the optimal consumption mix with respect to the PPF, we can substitute the consumption equation into the PPF equation: \[x^2 + 2(2x)^2 = 900\] Solving for \(x\): \[x^2 + 8x^2 = 900\] \[9x^2 = 900\] \[x^2 = \frac{900}{9}\] \[x^2 = 100\] \[x = 10\] Now, we can substitute the value of \(x\) back into the consumption equation to find the value of \(y\): \[y = 2(10)\] \[y = 20\] The optimal consumption mix will be \((x, y) = (10, 20)\).

Calculate RPT and price ratio

To find the RPT (Relative Price Tangency) at the optimal consumption mix \((x, y) = (10, 20)\), we need to find the slope of the PPF at that point. We can do that by differentiating the PPF equation with respect to \(x\) and plugging in the optimal consumption mix: \[\frac{d y}{d x} = -\frac{x}{2y}\] At \((x, y) = (10, 20)\), the slope is: \[\frac{d y}{d x} = -\frac{10}{2\cdot 20}\] \[\frac{d y}{d x} = -\frac{1}{4}\] The RPT is equal to the absolute value of the slope at the optimal consumption mix: \[RPT = \frac{1}{4}\] Therefore, the price ratio that will cause production to take place at the optimal point is given by the RPT, which is \(1:4\), or \(\frac{1}{4}\), i.e., the price of 1 unit of \(x\) is 1/4 of the price of 1 unit of \(y\).

Show the solution on the graph

On the graph with the PPF from Step 1, plot the point \((10, 20)\) as the optimal consumption mix. Draw a tangent line with slope \(-\frac{1}{4}\) at this point, illustrating the RPT and price ratio as they relate to the PPF and consumption preference. The tangent line represents the trade-offs between producing \(x\) and \(y\) at the optimal point.

Key concepts

Microeconomic Theory

Microeconomic theory is the branch of economics that focuses on the actions of individuals and firms, as well as the outcome of these actions on the markets for goods and services. It analyzes how these entities make decisions to allocate limited resources, typically aiming to optimize their gains or benefits. One key aspect of microeconomics is the concept of the production possibility frontier (PPF), which represents the various combinations of two goods or services that can be produced within an economy, given fixed resources and technology.

The PPF provides insights into trade-offs and the concept of opportunity cost—the cost of forgoing the next best alternative when making a decision. For example, as shown in the textbook exercise, an economy that produces more guns (x) would have to produce less butter (y), since the resources are limited, which is depicted by the specific shape of the PPF curve. The curve is typically bowed outwards because resources are not perfectly adaptable to the production of both goods, which reflects the increasing opportunity cost.

Optimal Consumption Mix

The optimal consumption mix refers to the combination of goods and services that maximizes a consumer’s satisfaction or utility, given their income and the prices of goods and services. In the context of the PPF, the optimal consumption point is where a consumer's preference, or indifference curve, is tangent to the PPF curve. This represents the highest level of satisfaction obtainable without exceeding the productive capabilities of the economy.

In the exercise provided, the optimal mix is identified by the preference equation, where the consumer prefers twice as much butter (y) as guns (x), indicated by the equation \(y = 2x\). By solving this preference equation along with the PPF equation, we determined that the optimal consumption mix for this particular preference is 10 units of guns (x) and 20 units of butter (y). This mix lies on the PPF, ensuring that it's a producible and efficient combination.

Relative Price Tangency

Relative price tangency occurs when the slope of the PPF, which represents the opportunity cost of one good in terms of another, is equal to the slope of a consumer's indifference curve, which represents their marginal rate of substitution (MRS). In mathematical terms, the point of tangency reflects the balance where the trade-off between the goods in production matches the consumer's preference trade-off.

In the exercise, the relative price tangency is calculated as the relative price of one gun to one unit of butter. This can be determined using the concept of the Relative Price Tangency (RPT), which in our example is the slope of the PPF at the optimal consumption point (10, 20). By differentiating the PPF equation and substituting the optimal values, we find that the RPT and hence the price ratio at this point is \(1:4\), meaning that in this economy, producing one additional unit of guns costs four units of butter.

PPF Graphing

Graphing a PPF involves plotting the possible quantities of two different goods that an economy can produce with its available resources and technology. The shape of the PPF is determined by the production function outlined in the given scenario. The PPF curve provides a visual tool for understanding production constraints, efficiency, and the trade-offs between the two goods.

In the exercise, the PPF is graphed using the equation \(x^2 + 2y^2 = 900\), resulting in an elliptical shape due to the differing coefficients of x and y in the equation. This indicates that the resources are not equally efficient in producing both goods. To display the solution graphically, one would illustrate point (10, 20) on the curve and draw a tangent line at that point. The slope of the tangent line (RPT) reflects the opportunity cost of the goods at the point of optimal consumption, providing a clear visual representation of the price ratio needed to sustain that production point in the economy.