1. Why consider a 2 × 2 × 2 model? | |
Why a general model | (i) Although it provides a rough and ready explanation as to why Portuguese might export wine and Colombians export coffee, the Ricardian model 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 natural resources that are used in the production process. The emergence of capitalists and landowners is the source of the so-called "class struggle." (ii) The US imports cars even though we produce them. The Ricardian model does not explain this.
(iii) The specific factors model is a short run model, since each industry has a fixed factor. In the long run, capital is a variable input. Likewise, land or natural resources are a variable input in the long run. (iv) Other countries learn the production technology (Italy, India, Thailand eventually produce silk.) Although it is possible to consider a general model with many factors of production and many outputs, at least for the purpose of this course, it would be useful to consider a model with two factors and two goods. We also assume that countries have identical production technologies. |
Gains from Trade (mathematical portion) |
The purpose of this chapter is to demonstrate that there are gains from trade even when a country produces many goods. In other words, a country benefits from trade when it is cheaper to import a product than to produce it in the home market using domestic resources. (The country has the technology to produce a good, but it is cheaper to import it.)
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First maritime traders |
Phoenicia, (Canaanites, Philistines)
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Phoenicians |
(i) 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 could easily land on shores. (Phoenician's trade was based on comparative advantages. They excelled in purple dye and construction. Their outposts were Tyre, Sidon and Carthago.) Hiram, the King of Tyre had supplied master craftsmen as well as the cedar and cypress timber to King Solomon to build the Temple.) (ii) alphabet, glass, slave trade, crucifixion (iii) Tyrian purple . It costs about €2000 or $2500 per gram or $70,000 per ounce today. It is 60 times more expensive than gold ($1200/oz). and 40 grams are required to dye a handkerchief ($100,000) A purple toga would have cost $50 million today. How Theodora rescued Justinian's throne |
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A Phoenician mask. Phoenicians and Philistines 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. (its habitat was the coast of Lebanon and Syria now.) They also traded slaves captured on the lands they traveled. Phoenicians were the descendants of the original Canaanite inhabitants. As other groups (e.g., Israelites and Philistines) moved in, they were ousted to the edge and migrated to coastal cities. Phoenicians built coastal city of Tyre around 2750 BC. 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. Romans wore woolen clothes (togas and tunics), |
outline of this section | 2. Production Possibility
Frontier (Curve) |
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. |
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How to allocate resources, K and L, between two industries | |
Efficient resource allocation | 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) This condition must hold for every firm using capital and labor, regardless of the output they produce. |
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.
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Efficient resource allocation | MRTS = w/r for every industry using capital and labor. This is shown in the Edgeworth box diagram. Along the contract curve OABO', the output of industry 1 rises while that of industry 2 declines, which yields the production possibility frontier. |
Equivalence of Central Planning and Laissez-faire (Theory) | (i) An 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, and relegated to numerous producers and consumers. Mathematically, the two methods yield an identical solution. (with the exception of two market failures) In practice, the latter is more practical. (Communism vs. Free market) If central planning were superior, the Soviet Union would have surpassed the US.) If the central planner were an omnicient (all-knowing) deity directing all aspects of production and consumption, this equivalence can be restored. Q: If a central planner aided by AI (artificial ingelligence) orchestrated production and consumption, can such a command economy (such as China) do better than the Free Market? A: Information asymmetry occurs between government and private agents, and AI cannot acquire full information. Data collection from private producers and consumers is difficult . |
Nonequivalence (Empirical Reality) | The communist experiment of the former Soviet Union for the period 1917 - 1991 clearly demonstrated that central planning in practice cannot do better than the laissez-faire policy. China still insists on central planning. |
5. Central 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. |
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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 = 0 in long run equilibrium) 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 GDP = p1y1 + p2y2 s.t. y2 = f(y1,L,K). |
Equilibrium condition | Equilibrium condition: |
6. Efficient Consumption | |
Problem | Choose the consumption bundle, x1
and x2, to Maximize u(x1,x2)
s.t. GDP = p1x1 + p2x2. (Here, GDP 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 |
7. 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)
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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, at point A the market clears, i.e., supply = demand for each good. Moreover, the autarky point A maximizes (i) GDP (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) Mathematically, the two methods are equivalent and yield the same answer. In practice, market economy is better because it preserves individual incentives and economic freedom. The command economies of the former Soviet Union failed. This does not mean market economies are free of problems. |
8. 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 GDP = p*1y1
+ p*2y2
subject to: y2 = f(y1,L,K). Producers move from the autarky point, A, to the free trade production point, B. |
Equilibrium condition | MRT = p*1/p*2. Remark: B (free trade production) maximizes GDP 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 consumption) 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. Archeological Museum, Crete. |
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