3. Equilibrium in a Closed Economy  
Problem  Choose x_{1}, x_{2}, y_{1}, and y_{2} to maximize u(x_{1},x_{2}) subject to: (i) F(y_{1},y_{2},L, K) = 0, (implicit form) or y_{2} = f(y_{1},L,K) (explicit form) (ii) x_{1} = y_{1}, x_{2} = y_{2} (Production = consumption for each good). This problem is equivalent to: choose y_{1} and y_{2} to maximize u(y_{1},y_{2}) subject to the production possibility function, F(y_{1},y_{2},K,L) = 0. (or y_{2} = f(y_{1},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 y_{1}, y_{2}, minimize C_{i} = wL_{i} + rK_{i} subject to: y_{i} = F^{i}(L_{i},K_{i}). Equilibrium condition: MRTS = w/r. (wagerent 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: K_{1} = C_{1}/r  (w/r)L_{1}. Likewise, for industry 2, it is K_{2} = C_{2}/r  (w/r)L_{2}. 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) MP_{L}/MP_{K} = 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. 
Two methods  (i) efficient output mix can be obtained by careful planning. The central planner chooses production and consumption, or (ii) laissezfaire (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) 
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: π^{1}
= p_{i}y_{i}^{1}  (wL_{i}^{1} +
rK_{i}^{1})
Profit of firm 2 in industry i: π^{2} = p_{i}y_{i}^{2}  (wL_{i}^{2} + rK_{i}^{2}) Profit of firm 3 in industry i: π^{3} = p_{i}y_{i}^{3}  (wL_{i}^{3} + rK_{i}^{3}) . . . There are about 30 million firms in the US (2012). 
From firm profits to industry profit  Profit of firm n in industry i: π^{n} = p_{i}y_{i}^{n}
 (wL_{i}^{n} + rK_{i}^{n}) (Here, π^{j}, in lower case, represents a firm's profit) Adding up all these profits, we get industry i's profit (in capital case): Π_{i} = p_{i}y_{i}  wL_{i}  rK_{i}.

Industry profits  Specifically, profits of the two industries are:
Industry 1: Π_{1} = p_{1}y_{1}  wL_{1}  rK_{1}. Industry 2: Π_{2} = p_{2}y_{2}  wL_{2}  rK_{2}. The total profits of the economy is: Π = p_{1}y_{1} + p_{2}y_{2}  w(L_{1} + L_{2})  r(K_{1} + K_{2}).

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: L_{1} + L_{2} = L^{o}, (aggregate demand for labor = fixed aggregate supply of labor) K_{1} + K_{2} = K^{o}. (aggregate demand for capital = fixed aggregate supply of capital). Thus, the total profits can be written as: Π = p_{1}y_{1} + p_{2}y_{2}  (wL^{o} + rK^{o}). 
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 or NDP (no depreciation), 
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 L_{1}, L_{2},
K_{1}, and K_{2} to: Maximize Π = p_{1}y_{1} + p_{2}y_{2}  (wL_{1} + rK_{1})  (wL_{2} + rK_{2}) = p_{1}y_{1} + p_{2}y_{2}  (wL^{o} + rK^{o}) (Note that Π= profit.) subject to the PPF. Recall that the total costs C = wL^{o} + rK^{o} can be treated as a fixed parameter. Hence, the problem reduces to: maximize National income = p_{1}y_{1} + p_{2}y_{2} s.t. F(y_{1},y_{2},K,L) = 0. (A general expression of the PPF.) 
Equivalent problem  This problem is equivalent to: Maximize GDP = p_{1}y_{1} + p_{2}y_{2} s.t. y_{2} = f(y_{1},L,K). 
Equilibrium condition  Equilibrium condition: 
5. Efficient Consumption  
Problem  Choose the consumption bundle, x_{1}
and x_{2}, to Maximize u(x_{1},x_{2})
s.t. GDP = p_{1}x_{1} + p_{2}x_{2}. (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 = p_{1}/p_{2} 
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, p_{1}/p_{2} ? 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(y_{1},y_{2}) s.t. y_{2} = f(y_{1},L,K)

Equilibrium condition  MRT = MRS (must pay = willingness to pay) (This determines p^{A} = p_{1}^{A}/p_{2}^{A} = ...) 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 twogood world, the separating hyperplane is a straight line. In a threegood world, it will not be a line, but a threedimensional plane. In a manygood world where the number of goods exceeds three, one cannot employ a two or threedimensional 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 multidimensional 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 (p_{1}/p_{2})^{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 p^{A}, 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 x_{1},
x_{2}, y_{1} and y_{2} to maximize u(x_{1},x_{2}) subject to: (i) PPF: F(y_{1},y_{2},K,L) = 0. (Don't worry about zero in the above equation. For instance, y_{1}^{2} + 4y_{2}^{2} = KL can be written as y_{1}^{2} + 4y_{2}^{2}  KL = 0.) (ii) p*_{1}y_{1} + p*_{2}y_{2} = p*_{1}x_{1} + p*_{2}x_{2}. (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*_{1}y_{1}
+ p*_{2}y_{2}
subject to: y_{2} = f(y_{1},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 x_{1} and x_{2} to maximize u(x_{1},x_{2})
I^{o} = p*_{1}x_{1} + p*_{2}x_{2}, where I^{o} = p*_{1}y_{1} + p*_{2}y_{2} = 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. 
