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

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

The production function for airplanes in Home is aLA = 1, which means that 1 unit of labor produces 1 airplane. Similarly, the production function for cars in Home is aLC = 5, which means that 5 units of labor produce 1 car.

Let's denote the quantity of airplanes produced by Home as QA and the quantity of cars produced by Home as QC.

The labor constraint in Home is given by: QA + 5QC ≤ 900

Solving for QC, we get: QC ≤ (900 - QA)/5

Now, let's calculate the opportunity cost of airplanes in Home. The opportunity cost of airplanes is the amount of cars that Home must give up to produce one more airplane. It is equal to the slope of the PPF.

The slope of the PPF is given by the ratio of the marginal products of labor for cars and airplanes: MPLC/MPLA = aLC/aLA = 5/1 = 5

Therefore, the opportunity cost of airplanes in Home is 5 cars.

In the absence of trade, the price of airplanes in terms of cars would be equal to the opportunity cost, which is 5 cars per airplane.

To find the quantities of airplanes and cars that Home would choose to produce and consume in the absence of trade, we need to find the point on the PPF that maximizes Home's utility.

Home's utility function is U(DA, DC) = DA^(1/3) * DC^(2/3)

Let's denote Home's consumption of airplanes as DA and consumption of cars as DC.

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: PADA + PCDC = Total income

Since PC = 1, the budget constraint simplifies to: PA*DA + DC = Total income

Substituting the prices of airplanes and cars, we get: PA*DA + DC = Total income

To simplify the calculation, we can assume that Total income is equal to the value of production in Home, which is the total labor available: Total income = 900

Now, let's solve for DA and DC by maximizing the utility function subject to the budget constraint:

Maximize U(DA, DC) = DA^(1/3) * DC^(2/3) subject to PA*DA + DC = 900

Using Lagrange multipliers, we set up the following equations:

∂U/∂DA = λ∂(PADA + DC - 900)/∂DA ∂U/∂DC = λ∂(PADA + DC - 900)/∂DC

Taking the partial derivatives and setting them equal to 0, we get:

(1/3) * DA^(-2/3) * DC^(2/3) = λ (2/3) * DA^(1/3) * DC^(-1/3) = λ

Dividing these two equations, we get:

(1/3) * (DA/DC) * (DC/DA) = λ^2

Simplifying, we find:

(1/3) = λ^2

Solving for λ, we get:

λ = ± (√3)/3

Substituting λ back into the first equation, we find:

(1/3) * DA^(-2/3) * DC^(2/3) = (√3)/3

Simplifying, we get:

DA^(-2/3) * DC^(2/3) = √3

Taking the cube root, we get:

(DA/DC)^(2/3) = (√3)^(1/3)

Simplifying, we find:

DA/DC = (√3)^(1/2)

Squaring both sides, we get:

(DA/DC)^2 = (√3)

Simplifying, we find:

DA/DC = √3

Therefore, the ratio of airplanes to cars that maximizes Home's utility is 1:√3.

Since the production possibility frontier represents all the possible combinations of airplanes and cars that Home can produce, the point (QA, QC) on the PPF that satisfies the ratio DA/DC = √3 is the combination that Home would choose to produce and consume in the absence of trade.

To find QA and QC, we can use the labor constraint and the ratio DA/DC = √3:

QC = (900 - QA)/5 DA/DC = √3

Substituting QC into the second equation, we get:

DA/((900 - QA)/5) = √3

Simplifying, we find:

DA = (√3)*(900 - QA)/5

Substituting this into the labor constraint, we get:

QA + 5*((√3)*(900 - QA)/5) ≤ 900

Simplifying, we find:

QA + (√3)*(900 - QA) ≤ 900

Expanding, we get:

QA + (√3)*900 - (√3)*QA ≤ 900

Simplifying, we find:

QA*(1 - √3) ≤ (√3)*900 - 900

Dividing both sides by (1 - √3), we get:

QA ≤ [(√3)*900 - 900]/(1 - √3)

Simplifying, we find:

QA ≤ 447.21

Therefore, in the absence of trade, the home country would choose to produce and consume approximately 447 airplanes and the remaining labor would be allocated to producing cars.