Homogeneous production and average cost

I have discussed before that constant returns to scale (CRS) function with a single factor is of the form F(x) = cx where c is a constant. In this post, I wanted to show that if the two−factor production function has CRS, then the average cost is independent of the output level. Then I decided that such a result is too boring to deserve a post, so I am going to discuss the relationship between homogeneity of production function and average cost in full generality, following Sandmo (1970).

A function F: Rⁿ → R is homogeneous of degree k if a^kF(x) = F(ax) for all a > 0 and x in Rⁿ. A production function has constant returns to scale if it is homogeneous of degree 1. It has increasing returns to scale if homogeneous of degree k > 1, and decreasing returns to scale if homogeneous of degree k < 1. These properties can be defined locally (that is, find k as a function of x) by taking the derivative of the identity equation above with respect to a and then solving for k (assuming F is differentiable): ka^k−1F(x) = Σ F_ix_i and since a^k−1F(x) = F(ax)/a, we get k = aΣ (F_ix_i/F(ax)). Evaluating at a = 1, k = Σ (F_ix_i/F(x)).

Assume perfect competition in factor markets so that the factor prices are exogenously determined. Let r_i be the price of factor i and let r be the vector of prices. The total cost is then Σ r_ix_i. For fixed output level Q, the firm is cost minimizing with regard to x subject to its production function F(x) = Q. The Lagrangian is L = Σ r_ix_i − λ(F(x) − Q) and the first order conditions are r_i − λF_i = 0 for i = 1,..., n and F(x) − Q = 0. These n+1 equations yield the optimal x* as a function of r and Q and we can find the optimal cost C as a function of r and Q: C(r,Q) = Σ r_ix_i*.

While we are talking about cost function, let's mention that the Lagrange multiplier in cost minimization problem can be interpreted as the marginal cost. The derivative of this minimum cost function with regard to Q then is the marginal cost: dC/dQ = Σ r_i(dx_i*/dQ). By the first order conditions, r_i = λF_i. Totally differentiating F(x) = Q condition, we get Σ F_i(dx_i*/dQ) = 1. Thus, λ = dC/dQ.

Going back to the average cost problem, let A(r,Q) = C(r,Q)/Q be the average cost. To show the behavior of average cost as Q varies, take dA/dQ = Q⁻²(dC/dQ × Q − C). Thus the direction of average cost in Q depends on the sign of dC/dQ × Q − C or dC/dQ − C/Q. We showed C = Σ r_ix_i = λΣ F_ix_i and λ = dC/dQ. By the constraint, Q = F(x). Thus dC/dQ − C/Q = λ(1 − Σ (F_ix_i/F(x))). But we have also shown that k = Σ (F_ix_i/F(x)). Therefore, the average cost is increasing in Q if k > 1 (decreasing returns to scale), decreasing in Q if k < 1 (increasing returns to scale), and constant in Q if k = 1 (CRS).

The Basic Ricardian Model

Given that Ricardian insight of comparative advantage is introduced in the very introductory economics classes and that the model is introduced quite early in the intro microeconomics, one would expect that I should know at least the simple Ricardian model inside out. Well, sadly this is not the case; I find going over the model in detail surprisingly nontrivial. In this post I am going to outline the basic Ricardian trade model.

Set Up

In the simple Ricardian model, there are two countries, home and foreign. It seems a literature tradition to postfix foreign variables with ^* so I will follow the tradition. There are two goods, 1 and 2, and single production factor, labor (L). The total labor endowments, L and L^*, are exogenously fixed (immobile across countries). Within each country, labor is perfectly mobile between the industries for good 1 and 2 (L₁ + L₂ = L).

The production function has constant returns to scale with marginal product of labor equal to a_i at home and a_i^* for i = 1, 2. In other words, to produce one unit of good 1 at home, it takes 1/a₁ units of labor, etc. As discussed earlier, CRS with single factor implies that the production function is linear: Q_i(L) = a_iL.

At this point we can find the production-possibility frontier: {(Q₁, Q₂) s.t. Q₁ = a₁L₁, Q₂ = a₂L₂, and L₁ + L₂ = L} = {(Q₁, Q₂) s.t. Q₁/a₁ + Q₂/a₂ = L}. Graphed on Q1-Q2 plane, the PPF is a straight line with slope -a₂/a₁.

Autarky Equilibrium

Let's consider the autarky equilibrium. Let p_i denote price in each industry and let p = p₁/p₂ be the relative price (of good 1). The model assumes perfect competition, so each industry makes zero profit: Π_i = p_iQ_i - w_iL_i = 0, where w_i denotes the wage for industry i. Solving for the wage, we get w_i = p_iQ_i/L_i = p_ia_i.

Suppose consumer preference is such that at any price both goods are demanded. For both goods to be produced, we require w₁ = w₂, implying p = a₂/a₁. We cannot determine the exact autarky equilibrium without a specific demand function, but we know it lies on some interior point of the PPF found above.

Trade Equilibrium

Allow the countries to trade with each other. Assume the foreign country has a comparative advantage in good 2. That is, a₂/a₁ < a₂^*/a₁^*, implying that home autarky relative price of good 1 is lower than the foreign's: p_a < p_a^*. To find the equilibrium price with trade, we consider the supply and demand as usual.

The world relative supply (of good 1) is 0 when both countries specialize in good 2. This occurs when the world relative price p is less than both the home and foreign autarky relative prices. On the other hand, both countries specialize in good 1 when p is greater than both of the autarky prices. If the world price lies strictly in between the autarky prices, so that p_a < p < p_a^*, then home specializes in good 1 while the foreign country specializes in good 2. The relative supply in this region is then (La₁)/(L^*a₂^*).

There are end points to consider. When p = p_a, the home country may produce both goods while the foreign country specializes in good 2, so that the world relative supply is less than or equal to (La₁)/(L^*a₂^*). When p = p_a^*, the foreign country may produce both goods while the home country specializes in good 1, so that the supply is greater than or equal to (La₁)/(L^*a₂^*).

Assuming identical and homothetic tastes across the countries so that the relative demand is decreasing in the relative price p, there are three possible cases. The equilibrium price may equal to the home autarky price or the foreign autarky price, or it my lie in between. Let's focus on the last case in which both countries specialize. Consider the new PPF of the home country. Since it specializes in good 1, it has total income of p₁a₁L. So the new PPF is {(Q₁, Q₂) s.t. p₁Q₁ + p₂Q₂ = p₁a₁L} = {(Q₁, Q₂) s.t. Q₁/a₁ + (1/p)Q₂/a₁ = L}. Since 1/p < 1/p_a = a₁/a₂, it follows that the old PPF is a proper subset of the new subset (and since we assumed the preference is such that both goods are demanded, the home country is strictly better off). Graphically, the PPF pivots outward centered at (La₁, 0) since the slope of the curve is now p > p_a. Similarly, for the foreign country which specializes in good 2, the PPF includes the end point (0, L^*a₂^*) but now has the slope of p < p_a^* so again it pivots outward.

So in the trade equilibrium, both countries are better off than in autarky. The Econ 101 punchline of the Ricardian model: mutually beneficial trade occurs when there exists a comparative advantage and the direction of the comparative advantage determines the specialization and direction of the trade. In particular, even if a country has no absolute advantage in either good, mutually beneficial trade occurs. For example, even if a₁ < a₁^* and a₂ < a₂^*, comparative advantage can lead to a trade (of course, if one country has a comparative advantage in one good, the other country has the advantage in the other good; if there is no comparative advantage, then autarky prices of both countries are the same, so there will be no gain from trading). How can the home country export when it has lower MPL for both goods?

The answer is that the wages are adjusting to the productivity. In the home country (specializing in good 1), the wage level is w = p₁a₁ and in the foreign country, it is w^* = p₂a₂^* > p₁a₁^* since p = p₁/p₂ < a₂^*/a₁^* = p_a^*. So a₁ < a₁^* implies w < w^*.

It is tempting to close the model with such an optimistic and insightful conclusion, but we have not finished examining the model since we have skipped the end point cases. The only thing to note here is that in the case both countries specialize, we can determine the relative output (La₁)/(L^*a₂^*) and use this to determine the equilibrium price given the demand (without demand specified, we can only put a bound). In the case only one country fully specializes, we know the price (the autarky price of the other country) but we have to determine the quantity from a specific demand function from the price. The point is, the model is analytically rather cumbersome even at the basic setting.

Thus the post-Ricardian models have focused on expanding the model to allow multiple countries and factors for empirical studies. Indeed, many international trade models seem to be characterized by three parameters: the number of countries, the number of industries, and the number of factors. Other than those, the most important part of Ricardian model would be the technological difference across countries, which allows gain from trade.

The Envelope Theorem

I learned in school the Kuhn-Tucker conditions and the envelope theorem but I always felt I didn't have a complete understanding of them. One thing confusing about the envelope theorem is that there are different varieties of it, depending on the flavor of the optimization problem. I decided to review envelope theorem today and keep the notes in the blog.

Consider a constrained optimization problem on the function f(x,a) with regard to x subject to a g(x,a) = 0, where x is an n-vector and a is a scalar. Let M(a) be the solution to the problem; that is,

M(a) = max_x f(x,a) s.t. g(x,a) = 0.

The Lagrangian is then L = f(x,a) − λg(x,a), giving n+1 first-order conditions (or, n first-order conditions and m complementary slackness conditions for more general constrained optimization problem with m inequality constraints). These conditions yield the optimizing argument x*(a) and the solution M(a) = f(x*(a), a).

The envelope theorem states that if x*(a) is a C¹ function and the usual constraint qualification is satisfied (∇g(x*(a))≠0), then M'(a) = ∂L(x,a)/∂a evaluated at x = x*(a). To put crudely, to differentiate the solution with regard to a parameter (say for comparative statics), one only needs to differentiate the Lagrangian with respect to the parameter and then "plug in" the solution x* rather than explicitly find M(a) and then differentiate it.

The proof for this version of envelope theorem is a straight-forward calculation. Since M(a) = f(x*(a), a), it follows

M'(a) = Σ(∂f/∂x_i)(∂x_i/∂a) + ∂f/∂a

for i = 1,..., n.

By the first order conditions, ∂f/∂x_i = λ∂g/∂x_i for each i. It follows

M'(a) = λ Σ(∂g/∂x_i)(∂x_i/∂a) + ∂f/∂a

Identically, it must be that g(x*(a), a) = 0. Differentiating this equality with respect to a yields Σ(∂g/∂x_i)(∂x_i/∂a) + ∂g/∂a = 0. So we get

M'(a) = -λ∂g/∂a + ∂f/∂a evaluated at x = x*(a).

But of course, -λ∂g/∂a + ∂f/∂a = ∂L/∂a

The more general case with m inequality constraints have a similar flavor of proof: differentiate the maximum function with respect to the parameter, collect the terms, and then use the first order conditions and complementary slackness conditions to cancel out the terms. The theorem can be further generalized into the case in which the parameter is a k-vector rather than a scalar.