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Examining a two-period OLG model

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Consider a two-period OLG model, where each generation consists of n people. There is no production in this model and each person receives constant income Y. Young persons can buy government bonds when young and sell them when old. Let 1 denote the first period (young) and 2 the second period (old). For simplicity, assume constant interest rate.
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This solution examines a two-period OLG model and calculates its aggregate consumption to check if the Ricardian equation holds in this situation.

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