math_details - Long-run Inflation: In long-run equilibrium:...

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Unformatted text preview: Long-run Inflation: In long-run equilibrium: ( ) / ( ) ( , ( )) o M t P t L i Y t = , where the M(t) indicates that the variable is changing continuously over time. The subscript zero denotes a variable that is constant at a given long-run equilibrium. Taking the derivative of the above with respect to time one gets 2 ( ) ( ) ( ) ( ) ( ) dM t dP t M t dL dY dt P t dt P t dY dt- = Now multiply the LHS by P/M and the right hand side by 1/L , since these two quantities are equal in equilibrium: 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) dM t dP t M t P t dL dY dt P t dt P t M t dY dt L dM t dP t dL dY dL dY Y M t dt P t dt LdY dt LdY dt Y dL dM t dP t YdL dY dY L dY M t dt P t dt LdY Ydt Ydt Y - = - = = - = = Now all of the variables of the form dX/X are growth rates, and the quotient of two...
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math_details - Long-run Inflation: In long-run equilibrium:...

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