# Increasing Costs

 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 an inward movement to a point inside the 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. How to allocate resources, K and L 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. 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 an AI (artificial ingelligence) planner orchestrated production and consumption, can such a totalitarian society (such as China) do better than the Free Market? A: It remains to be seen. Information asymmetry occurs between government and private agents, and AI cannot acquire full information. Data collection from private producers and consumers is difficult. In fact, data gathered in a totalitarian society is not reliable because they are often exaggerated. Thus, information assymetry can cause government failure. Wars are caused by information asymmetry, i.e., each side overestimates the probability that it will win. 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.

 4. 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) . . . There are about 30 million firms in the US (2012). 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 (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, Π = p1y1 + p2y2
 5. 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 GDP = 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. 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

 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, 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, 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 economic 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 GDP = p*1y1 + p*2y2 subject to: y2 = f(y1,L,K). Figure 8 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.