Solow-Growth-Model - The Solow Growth Model The Solow...

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The Solow Growth Model The Solow Growth Model is a model of capital accumulation in a pure production economy: there are no prices because we are strictly interested in output = real income. Everyone works all the time, so there is no labor/leisure choice. In fact, there is no choice at all: the consumer always saves a fixed portion of income, always works, and owns the firm so collects all “wage” income and profit in the form of all output. We will not need to model the “consumer”. We assume all people work all the time, and we assume they save, hence invest, a fixed portion of their income. There is no government, hence no taxation nor subsidies; this is a closed economy , so there is no trade. Since there are no prices there is no need for money: there are no financial markets, etc. This model, then, is a model that captures the pure impact savings = investment has on the long run standard of living = per capita income. Since we allow for population growth, this model may be called the Blue Lagoon Model (i.e. as opposed to Robinson Crusoe, two people can reproduce). Ingredients : Consumers and Firms. All consumers own the firms, so consumers receive all output, and therefore all profit and rent. Aggregates : Output = Real Income = Y t in period t . Capital Stock = K t ; Population Size = N t = Labor Supply (since everyone works all the time). Consumption = C t ; Savings = S t ; Investment = I. t . Per Capita : Output = Real Income = y t = Y t /N t in period t . Capital Stock = k t = K t /N t ; Consumption = c t = C t /N t ; Savings = s t = S t /N t ; Investment = i. t = I t /N t ; THE POPULATION = THE LABOR SUPPLY The population grows at a constant rate n : t t t t t t N n N N N N N n ) 1 ( % 1 1 1 ! " # $ " % " ! ! ! THE FIRM The firm produces according to a Cobb-Douglas production function: & & & & & & & & & & & & & & & & & t t t t t t t t t t t t t t t t t t t t t t t t Ak y Ak N K A N K A N N N K A N N N K A y N N K A N Y N AK Y " " ( ) * * + , " ( ) * * + , " " " " " $ $ $ $ $ $ : Capita Per : Aggregate 1 1 1 1 1 1
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THE CONSUMER : CONSUMPTION AND SAVINGS The consumer saves the fraction s of income: t t t t t t t S C Y Y s C sY S ! " $ " " hence ) 1 ( and Notice this reflects the fact that there is no government (no taxes), and no imports/exports (no trade). EQUILIBRIUM GROWTH 1. POPULATION If the population size starts at N 0 , then 0 2 1 ) 1 ( ) 1 )( 1 ( ) 1 ( N n N n n N n N t t t t ! " " ! ! " ! " $ $ ! We need only know the initial N 0 and the growth rate n to find the population size in any period t . 2. SAVINGS AND INVESTMENT Since there is no government (no taxes), there are no imports/exports (no trade), consumers receive everything from the firms, and there are no financial markets, savings is simply investment (the only place consumers can put their “money”, which is actually output, is simply back into the firm): t t S I " 3. CAPITAL ACCUMULATION Aggregate capital grows according to the following law of motion : t t t I K d K ! $ " !
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