Ricardian Model: Production, Opportunity Cost, and Consumption in Home Country

To graph Home's production possibility frontier (PPF), we need to find the maximum quantities of airplanes and cars that Home can produce given its labor resources.

Home's production function for airplanes is given by QA = aLA * LA, where QA represents the quantity of airplanes produced, aLA is the productivity of labor in producing airplanes, and LA is the amount of labor used in airplane production. Similarly, the production function for cars is QC = aLC * LC, where QC represents the quantity of cars produced, aLC is the productivity of labor in producing cars, and LC is the amount of labor used in car production.

Given that Home has 900 units of labor, we can substitute this value into the production functions to find the maximum quantities of airplanes and cars that can be produced:

QA = 1 * 900 = 900 QC = 5 * 900 = 4500

To graph the PPF, we can plot these maximum quantities on a graph with airplanes on the x-axis and cars on the y-axis. The PPF will be a straight line connecting the points (0, 4500) and (900, 0).

The opportunity cost of airplanes can be calculated by finding the slope of the PPF. In this case, the slope is -5/1 = -5, which means that for every additional airplane produced, Home must give up producing 5 cars.

In the absence of trade, the price of airplanes in terms of cars would be the ratio of their opportunity costs, which is 5. This means that Home would be willing to give up 5 cars to obtain 1 airplane.

To find the quantities of airplanes and cars Home would choose to produce and consume in the absence of trade, we need to consider the relative demand function. The relative demand for airplanes (DA) is given by the partial derivative of the utility function with respect to airplanes, divided by the partial derivative of the utility function with respect to cars:

DA/DC = (∂U/∂DA) / (∂U/∂DC) = (1/3)(DA^(-2/3))(DC^(2/3)) / (2/3)(DA^(1/3))(DC^(-1/3)) = (1/2)(DA/DC)

Setting PC = 1, we can equate the relative demand function to the relative price to find the quantities of airplanes and cars Home would choose to produce and consume. In this case, the relative price is 5 (opportunity cost of airplanes):

DA/DC = 5 DA = 5DC

Substituting this relationship into the PPF equation, we get:

5DC = 900 - DA 5DC = 900 - 5DC 10DC = 900 DC = 90

Substituting this value back into the relative demand equation, we find:

DA = 5 * DC = 5 * 90 = 450

Therefore, in the absence of trade, Home would choose to produce and consume 450 airplanes and 90 cars.