Increasing Costs

1.  Why consider a general model

Why a general model

(i) Although the Ricardian model provides a rough and ready explanation why Portuguese might export wine and Colombians export coffee, it is overly simplistic.

Simplistic, because labor is assumed the only factor of production and the production process is linear. In the real world, there are other factors, such as capital, land, and other natural resources that are used in production.

(ii) Although it is possible to consider a general model with many factors of production and many outputs, it would be useful, at least for the purpose of this course, to consider a model with two factors and two goods, and to relax the restrictive assumption that production process is linear.

Purpose of this section: show Gains from Trade

Gains from Trade exists in the general model.

The purpose of this chapter is to demonstrate that there are gains from trade even when a country does not specialize in only one good but produces many goods. In other words, a country benefit from trade when it is cheaper to import a product than to produce it in the home market using domestic resources.

First maritime traders

Phoenicians were master shipbuilders. Boards were interlocked with pegs, similar to the modern mortise and tenon method. They were strong and had rounded hulls, and hence can easily land on shores. They were not small vessels either. They were as big as the ships Columbus used to navigate to America.

A Phonician mask. Together with Philistines, Phoenicians were a sea faring people, i.e., they were the first maritime traders. Phoenicians were known for glass and purple dye, which was made from a sea snail, murex. They also traded slaves captured on the lands they traveled (See Amos 1:9 below). They landed on the coast of the current Lebanon and Syria around 1200 BC (after Moses's time), and built cities like Tyre, Sidon and Bybylos. The Tyrian purple was highly valued by European royalties. (Dyed textiles and glass were their main exports.) An inscription of Rameses III at Medina Habu recorded that the sea people included Peleset (Philistines), Lukka (Lycians) and Denyen (Danaen Greeks). They may have included Teresh (Greek name for Etruscans). They also invented the alphabet of 22 letters. They also taught slavery and crucifixion to the Romans.

Amos 1:9 This is what the LORD says:
"For three sins of Tyre,
even for four, I will not turn back my wrath .
Because she sold whole communities of [Hebrew] captives to Edom,
disregarding a treaty of brotherhood,

10 I will send fire upon the walls of Tyre
that will consume her fortresses."

Why cite Bible?

Bible passages are quoted, not for religious reasons, but as partially historical record. This course does NOT support any organized churches or promote any religion.
outline of this section Production Possibility Frontier (Curve)
Equilibrium in a Closed Economy
Efficiency in Production
Efficiency in Consumption
Price determination in a closed economy
Equilibrium in an Open Economy

 

2. Production Possibility Frontier

 Assume

(i) PPF is negatively sloped:

To get more of one good, give up something else.

(ii) PPF is concave ⇒ increasing opportunity cost. The amount of y2 that an economy has to sacrifice to get an additional unit of y1 increases as y1 increases.

Example: If demand for oil increases, the amount of other goods to sacrifice to get an additional barrel of oil increases.

Technological efficiency  
 Technologically efficient production All Points along the PPF. 
 Allocatively efficient production Only one point, where MRT = MRS. 
  D: Technologically infeasible

B: Technologically efficient, any point on the PPF

C: Technologically feasible, but inefficient

A: Allocatively efficient. depends on price.

Allocative efficiency implies technological efficiency, but not conversely.

PPF: is the locus of maximum output combinations from given resources and technology.

 Shifts in PPF

When does the PPF shift?

(i)Technological improvement

(i) An increase in the supply of an input. discovery of new oil reserve.
 Unemployment

How does unemployment affect the PPF?

It does not.

Unemployment causes only a movement to a point inside PPF.  

3. Equilibrium in a Closed Economy

Problem

Choose x1, x2, y1, and y2 to maximize u(x1,x2)

subject to:

(i) F(y1,y2,L, K) = 0, (implicit form) or y2 = f(y1,L,K) (explicit form)

(ii) x1 = y1, x2 = y2 (Production = consumption for each good).

This problem is equivalent to: choose y1 and y2 to maximize u(y1,y2)

subject to the production possibility function,

F(y1,y2,K,L) = 0. (or y2 = f(y1,L,K) (explicit form)

 

The problem can be split into two smaller problems:

(i) efficient production,

(ii) efficient consumption,

Also, in a closed economy, production = consumption at the market prices. 
 Efficient production Cost minimization problem

for all y1, y2,

minimize Ci = wLi + rKi

subject to: yi = Fi(Li,Ki).

Equilibrium condition: MRTS = w/r. (wage-rent ratio)

 Isocost curve Isocost curve is defined as the locus of input combinations (L,K) along which production cost remains constant. That is, L and K satisfy:

C = wL + rK.

Solve the above equation for K:

K = C/r - (w/r)L,

where C/r is the vertical intercept and - (w/r) is the slope of the isocost curve. The negative sign indicates that an isocost curve is negatively sloped.

Of course, an isocost curve can be obtained for each product or industry. For industry 1, it is written as:

K1 = C1/r - (w/r)L1.

Likewise, for industry 2, it is

K2 = C2/r - (w/r)L2.

The slope of an isoquant is called marginal rate of technical substitution (MRTS). The above diagram shows that when production cost is minimized,

MRTS ≡ (defined as) MPL/MPK = w/r in equilibrium for each good j. 
Arbitrary Resource allocation between two industries  How does an economy allocate resources? MRTS must be the same (= w/r) for all goods using the two inputs, K and L.

Figure 3. An arbitrary allocation (E) of resources between two industries.
 Efficient resource allocation

MRTS = w/r for every industry using capital and labor.

This is shown in the Edgeworth box diagram.

 
 Two methods

(i) efficient output mix can be obtained by careful planning. The central planner chooses production and consumption, or

(ii) laissez-faire (nonintervention) policy: By letting the market maximize profits. Production and consumption decisions are decentralized.

Mathematically, the two methods yield an identical solution. In practice, the latter is more practical. (Communism vs. Free market)

 

 

3. Decentralized Production in a free market economy

   In a free market economy, production decisions are decentralized. Individual firms make production decisions.
Firm profits   Profit of firm 1 in industry i: π1 = piyi1 - (wLi1 + rKi1)

Profit of firm 2 in industry i: π2 = piyi2 - (wLi2 + rKi2)

Profit of firm 3 in industry i: π3 = piyi3 - (wLi3 + rKi3)

. . .

 From firm profits to industry profit Profit of firm n in industry i: πn = piyin - (wLin + rKin)

(Here, πj, in lower case, represents a firm's profit) Adding up all these profits, we get industry i's profit (in capital case):

Πi = piyi - wLi - rKi.

 Industry profits Specifically, profits of the two industries are:

Industry 1: Π1 = p1y1 - wL1 - rK1.

Industry 2: Π2 = p2y2 - wL2 - rK2.

The total profits of the economy is:

Π = p1y1 + p2y2 - w(L1 + L2) - r(K1 + K2).

Two assumptions   

There are two widely accepted assumptions in the general equilibrium model:

(i) perfect competion (PC)

(ii) full employment.

 Full employment means

Specifically, full employment means:

L1 + L2 = Lo, (aggregate demand for labor = fixed aggregate supply of labor)

K1 + K2 = Ko. (aggregate demand for capital = fixed aggregate supply of capital).

Thus, the total profits can be written as:

Π = p1y1 + p2y2 - (wLo + rKo).

 
Implications of undirected, unorchestrated decentralized decisions  Notice, however, that fixed supplies of inputs are fixed (by assumption). Factor prices, w and r, are endogenously determined. However, in equilibrium, these factor prices can be treated as fixed parameters, at least in the eyes of competitive firms.

Thus, undirected, unorchestrated, decentralized decisions of numerous producers who maximize their own profits (and disregarding the impacts of their decisions on other firms in the economy) unconsciously, unwittingly, unknowngly, unintentionally and impersonally and collectively, maximize the total revenue or GDP or NDP (no depreciation),

Π = p1y1 + p2y2


4. Planner's problem (Centralized production decision)

 

In a free market millions of undirected firms make production decisions, which collectively maximize the country's GDP.

We now consider an equivalent problem. A central planner makes production decisions.
 Planner's Problem The economy's problem is to choose L1, L2, K1, and K2 to:

Maximize Π = p1y1 + p2y2 - (wL1 + rK1) - (wL2 + rK2)

= p1y1 + p2y2 - (wLo + rKo)

(Note that Π= profit.)

subject to the PPF.

Recall that the total costs C = wLo + rKo can be treated as a fixed parameter. Hence, the problem reduces to:

maximize National income = p1y1 + p2y2

s.t. F(y1,y2,K,L) = 0. (A general expression of the PPF.)

Equivalent problem  This problem is equivalent to:

Maximize NDP = p1y1 + p2y2

s.t. y2 = f(y1,L,K).

 Equilibrium condition  Equilibrium condition:

MRT = p1/p2Isorevenue curve is defined in the output space, i.e., it is the locus of outputs along which revenue remains the same. Since prices are fixed, isorevenue curves are straight lines. The same locus when defined in the consumption space is called the budget line.

Figure 5, Isorevenue curve and PPF

 

5. Efficient Consumption

 Problem Choose the consumption bundle, x1 and x2, to Maximize u(x1,x2)

s.t. NDP = p1x1 + p2x2.

(Here, NDP is the income from the production decision problem.)

 Eq. condition

(There is a single consumer. Alternatively, all consumers have identical preferences or tastes)

MRS = p1/p2

   

 

6. Efficiency in Production and Consumption in a closed economy

Why consider it

Efficient output mix and efficient consumption bundles depend on prices.

In general, supply rises with the price, whereas market demand declines as price rises. (Law of demand).

If demand ≠ supply, then the market does not clear. Thus, we need to find the equilibrium price at which demand = supply for each good.

How does one find the equilibrium price, p1/p2 ?

In practice, free market impersonally determines the equilibrium price. Mathematically, it is the slope of the separating hyperplane that separates the highest indifference curve and PPF.

 
 Efficient production and consumption

Impose the condition: production = consumption for each good.

Maxmize u(y1,y2)

s.t. y2 = f(y1,L,K)

 Equilibrium condition MRT = MRS (must pay = willingness to pay)

(This determines pA = p1A/p2A = ...)

Figure 7, Equilibrium in a closed economy


 Equilibrium price?

Once equilibrium is established in a closed economy, it is possible to draw a separating hyperplane through the equilibrum point that separates the indifference curve and the PPF. In a two-good world, the separating hyperplane is a straight line.

In a three-good world, it will not be a line, but a three-dimensional plane.

In a many-good world where the number of goods exceeds three, one cannot employ a two- or three-dimensional graph, but algebra can be employed, and an algebraic expression can be obtained describing the equilibrium prices in a closed economy. Such expression defines a multi-dimensional hyperplane separting the PPF and indifference surface. At any rate, the separating hyperplane defines the autarky equilibrium prices in the domestic market.
 Properties of Eq point  

Given (p1/p2)A, the autarky point A maximizes

(i) NDP

(ii) utility

How to find it in practice Two approaches:

(i) Central planner: computes pA, directs each firm and person to engage in the production of certain commodities.

(ii) Market economy: laissez faire (leave people alone, according to Lao Zi)

Mathematically, the two methods are equivalent and yield the same answer.

In practice, market economy is better because it preserves individual incentives and freedom. The command economies of the former Soviet Union failed. This does not mean market economies are free of problems.
   

 

7. Equilibrium in an Open Economy

Statement of the Problem An open economy's problem is to choose x1, x2, y1 and y2

to maximize u(x1,x2)

subject to:

(i) PPF: F(y1,y2,K,L) = 0.

(Don't worry about zero in the above equation.

For instance, y12 + 4y22 = KL can be written as

y12 + 4y22 - KL = 0.)

(ii) p*1y1 + p*2y2 = p*1x1 + p*2x2. (balance of trade)

 

Undoubtedly, this is a complex problem with four unknown decision variables. However, we will break it up into two stages, so we deal with two decision variable problem in each stage.

First Problem

(Step 1)

  Maximize NDP = p*1y1 + p*2y2

subject to: y2 = f(y1,L,K).

Figure 8

Producers move from autarky point, A, to free trade production point, B.
Equilibrium condition MRT = p*1/p*2.

Remark: B (free trade production) maximizes NNP at p*1/p*2, but A does not

Second problem

(Stage 2)

 Choose x1 and x2 to maximize u(x1,x2)

Io = p*1x1 + p*2x2, where Io = p*1y1 + p*2y2 = maximized income.

Equilibrium condition

Eq condition for the optimal (best) consumption bundle:

MRS = p*1/p*2.

In other words, choose a bundle (F) on the budget constraint so that it is tangent to the highest indifference curve.

Remark: F (free trade consumtion) was not feasible before. Trade Triangle: BFC in Figure 10.
   
Numerical Example  

Copper ingot from Crete. Copper was alloyed with tin to produce bronze during the Bronze Age in Cyprus, and exported far and wide.