Closed Economy vs. Open Economy

The idea here is to compare the best situation in a closed economy with that in an open economy. This problem is similar to one's occupational choice problem. There are thousands of different jobs, and most people usually choose at most one or two jobs, and in the latter case it is called part time or moonlighting.

Closed Economy

Autarky (auto = self, arkein = rule, defend in Greek) = Self sufficiency

1. Problem:

maximize u(x₁,x₂)
subject to (s.t.): x_i = y_i, ( x_i = consumption of good i, y_i = production of good i)
a_L1 y₁ + a_L2 y₂ = L (L = labor supply)

Figure 2. Equilibrium in a closed economy

How does one solve the utility maximization problem of a closed economy?

As the above diagram shows, the maximum welfare is achieved at the tangency point between the highest indifference curve and the PPF. Thus, when a maximum is attained, the two curves must have the same slopes. Marginal cost of good 1, measured in terms of the resources used up or the other good sacrificed is represented by the slope of the PPF, and called the marginal rate of transformation (MRT), i.e.,

Consumers' willingness to pay for a good is often represented by the slope of the indifference curve is called the marginal rate of substitution (MRS),

Thus, the consumption or production point which satisfies the equilibrium condition,

is the solution to this problem.

2. An Example of Autarky

Given u = x₁x₂, MRS = x₂/x₁
a_L1 = 2, a_L2 = 1, L = 120, find the solution for the closed economy's problem.

First, note that MRT (Marginal Rate of Transformation) = slope of PPF is

Note that MRT is the relative price of good 1 in autarky.

In a closed economy, production must be equal to consumption in each industry. That is,

x₁ = y₁, and x₂ = y₂ .

Thus, a closed economy's problem is to choose y₁ and y₂ to

maximize u = y₁y₂ s.t. PPF: 2y₁ + y₂ = 120.

Recall that the equilibrium condition for this problem is: MRS = MRT, or

Substitute this equilibrium condition (y₂ = 2y₁) into the PPF to obtain.

2y₁ + 2y₁= 120,

or y₁ = 30, and hence y₂ = 60. It follows that u^A = y₁ × y₂ =1800.

Open Economy

The open economy's problem is to choose y₁, y₂, x₁ and x₂ to maximize u(x₁, x₂), subject to a constraint that the total value of consumption is equal to that of production, i.e.,

(income = expenditure)

This is a complex problem of choosing 4 unknown variables, subject to a constraint. However, it can be decomposed into two simpler problems.

Step I: maximize (NDP = net domestic product)

s.t. (PPF)

To simplify the problem, eliminate one variable, say y₂ by expressing it in terms of y₂. This problem is equivalent to:

max

which expresses income as a function of only y₁: NDP(y₁).

The slope of the NDP function: (prime ' means “slope”)

The slope is positive, negative, or zero iff

(Note that iff stands for "if and only if.")

1. If (world relative price of good 1 > autarky relative price of good 1 in autarky),

then choose a maximum y₁ = L/a_L1, as shown below. Accodingly, y₂ = 0.

Figure 3.

2. If then choose a minimum y₁ = 0. In this case, y₂ = L/a_L2.

Figure 4.

3. If then any output is OK. (That is, y₁ = 0, or 1 or 2 is feasible, but y₂ must satisfy the full employment constraint.)

Figure 5.

Conclusion: In a Ricardian model, unless specialize in one product!

Step II: maximize u(x₁,x₂)

s.t. I = p*₁x₁ + p*₂x₂. (s.t. = subject to)

Equilibrium Condition: MRS = p*₁/p*₂

Figure 3. Utility maximization: U^F is higher than U^A

An Example of an Open Economy

Given u = x₁x₂;
MRS (slope of Indifference curve) = x₂/x₁,
p*₁ = 3, p*₂ = 1, a_L1 = 2, a_L2 = 1, and L = 120.

Step 1: maximize 3y₁ + y₂, subject to 2y₁ + y₂ = 120.

Note that

That is, NDP is rising with y₁. Thus, the country should specialize in good 1.

(NDP = net domestic product).

y₁ = 60 = L/a_L1, y₂ = 0.

I (national income) = 3 × 60 + 1 × 0 = 180.

In the next step, use this income to maximize utility by choosing the right consumption bundle.

Step 2: maximize u = x₁x₂, subject to 3x₁ + x₂ = 180

Equilibrium condition is: or

x₂ = 3x₁.

Substitute the above equilibrium condition into the budget line p*₁x₁ + p*₂x₂ = 180 to get

3x₁ + x₂ = 6x₁ = 180.

x₁ = 30, x₂ = 90, u^F = 30 × 90 = 2700.

U^F = 2700 > U^A = 1800.

(You are half-way through the math in this course).

Homework 1

Given: U = x₁x₂, MRS = x₂/x₁, L = 300, a_L1 = 3, a_L2 = 2, p*₁ = 2, p*₂ = 1.

1. Closed Economy: Find y₁, y₂, and U^A

2. Open Economy: Find y₁, y₂, x₁, x₂, U^F.

3. Compare U^A and U^F and sketch solutions in 1 and 2.

	Pudong, Shanghai GM has a joint venture with Shanghai Automobile to produce GM cars.
	One day in Jade Buddha Temple, Shanghai.(See the new Around the World in 80 days.)
	The Reclining Buddha, as in the latest Indiana Jones movie.