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

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

The production function for airplanes in Home is given by QLA = aLA * LA, where aLA = 1 and LA is the amount of labor used to produce airplanes. The production function for cars in Home is given by QLC = aLC * LC, where aLC = 5 and LC is the amount of labor used to produce cars.

Since Home has 900 units of labor, we can set up the following equations to find the maximum amount of airplanes and cars that Home can produce:

QLA = 1 * LA QLC = 5 * LC

We also know that LA + LC = 900.

Solving these equations simultaneously, we get:

LA = 450 LC = 450

Substituting these values back into the production functions, we get:

QLA = 1 * 450 = 450 airplanes QLC = 5 * 450 = 2250 cars

Now we can graph Home's PPF. Let's assume airplanes are on the x-axis and cars are on the y-axis. The points (450, 0) and (0, 2250) represent the maximum amounts of airplanes and cars that Home can produce respectively. Drawing a line connecting these points gives us Home's PPF.

The opportunity cost of airplanes is the amount of cars Home must give up to produce one more airplane. In this case, the opportunity cost of airplanes is the slope of the PPF, which is -5/1 = -5.

In the absence of trade, the price of airplanes in terms of cars would be the ratio of their opportunity costs, which is 1/5.

To determine how many airplanes and cars Home would choose to produce and consume in the absence of trade, we need to consider their utility function and find the consumption bundle that maximizes utility.

The utility function is given by U(DA, DC) = DA^(1/3) * DC^(2/3).

To maximize utility, we need to find the combination of DA and DC that satisfies the budget constraint and maximizes the utility function.

The budget constraint is given by pA * DA + pC * DC = wL, where pA is the price of airplanes, pC is the price of cars, and wL is the total income.

In the absence of trade, the price of airplanes in terms of cars is 1/5. Let's assume the price of cars is 1.

The budget constraint becomes (1/5) * DA + DC = wL.

To simplify the problem, let's assume wL = 1.

Substituting these values into the utility function, we get:

U(DA, DC) = DA^(1/3) * DC^(2/3) = DA^(1/3) * (1 - (1/5) * DA)^(2/3).

To find the optimal consumption bundle, we need to solve for the values of DA and DC that maximize this utility function.

Taking the partial derivatives of the utility function with respect to DA and DC, and setting them equal to zero, we can find the values of DA and DC that maximize utility.

Differentiating the utility function with respect to DA, we get:

dU/DA = (1/3) * DA^(-2/3) * (1 - (1/5) * DA)^(2/3) - (2/3) * DA^(1/3) * (1 - (1/5) * DA)^(-1/3) * (1/5) = 0.

Simplifying this equation, we get:

(1/3) * DA^(-2/3) * (1 - (1/5) * DA)^(2/3) - (2/3) * DA^(1/3) * (1 - (1/5) * DA)^(-1/3) * (1/5) = 0.

Multiplying through by 15DA^(-2/3) * (1 - (1/5) * DA)^(-1/3), we get:

5 * (1 - (1/5) * DA) - 2DA = 0.

Simplifying this equation, we get:

5 - DA - (2/5) * DA = 0.

Solving for DA, we get:

(7/5) * DA = 5.

DA = (5 * 5)/7 = 25/7.

Substituting this value back into the budget constraint, we can solve for DC:

(1/5) * (25/7) + DC = 1.

(5/35) + DC = 1.

DC = 35/35 - 5/35 = 30/35 = 6/7.

Therefore, in the absence of trade, the home country would choose to produce and consume 25/7 airplanes and 6/7 cars.