1. Closed Economy  
purpose  The idea here is to compare the best outcome in a closed economy with that in an open economy. This problem is similar to one's occupational choice problem. There are thousands of different jobs, and most people usually choose at most one or two jobs, instead of personally producing everything they need. 
Autarky 
(auto = self, arkein = rule, defend in Greek) = Self sufficiency Autarky imposes a stringent condition on production: x_{i} = y_{i} 
maximize u(x_{1},x_{2}) ( x_{i} = consumption of good i, y_{i} = production of
good i) Figure 2. Equilibrium in a closed economy How does one solve the utility maximization problem of a closed economy? As the above diagram shows, the maximum welfare is achieved at the tangency point between the highest indifference curve and the PPF. Thus, when a maximum is attained, the two curves must have the same slopes. Marginal cost of good 1, measured in terms of the resources used up or the other good sacrificed is represented by the slope of the PPF, and called the marginal rate of transformation (MRT), i.e. MRT ≡ a_{L1}/a_{L2 }(opportunity cost or relative price of 1 in autarky) 

Consumers' willingness to pay for a good is often represented by the slope of the indifference curve is called the marginal rate of substitution (MRS), MRS≡ MU_{1}/MU_{2} (MU_{i} = marginal utility of good i.) 

Equilibrium condition  Thus, the consumption point which satisfies the following equilibruim is the solution to the problem (willingness to pay for good 1 = its opportunity cost), MRS = a_{L1}/a_{L2 } 