implicit_function_theorem

implicit_function_theorem - Spring 2005 Economics 425 John...

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Spring 2005 John Rust Economics 425 University of Maryland The Implicit Function Theorem and its use in Economics 1 Overview The implicit function theorem is an important result in calculus, since it tells us conditions under which certain variables in a relationship (i.e. the “dependent variables”) are “implicit functions” of other vari- ables (the “exogenous, or shifter variables”), and it provides a formula for showing how the endogenous variables change when the exogenous shifter variables change (i.e. it allows us to calculate the derivative of the endogenous variables with respect to a change in the exogenous shifter variables). 1.1 Comparative Statics Thus, the implicit function theorem is a basis for comparative static exercises in economics: in such exercises we want to Fnd out how certain “endogenous variables” change when the exogenous variables change. 1.2 Supply/Demand Example Before actually presenting the implicit function theorem, consider the following supply/demand exam- ple. Consider the price of wheat, p which is determined in competitive equilibrium by the intersection of the aggregate supply and demand curves for wheat. Let p * be the equilibrium price, and let r be the amount of rainfall. Let’s assume that rainfall affects the supply of wheat but not the demand for wheat, so we can write S ( p * , r ) = D ( p * ) (1) as the basic “supply=demand” equilibrium condition determining the equilibrium price of wheat, p * . Now in this example, the price of wheat, p * , is the “endogenous variable” (since the price is being determined to set supply equal to demand) and the amount of rainfall, r , is the “exogenous” or shifter variable. The amount of rainfall can change, but the amount of rainfall does not have to obey any supply/demand equilibrium condition. Rainfall is determined by the weather, and this is why we call it “exogenous”, meaning that this variable is determined “outside of the system”. However the price of wheat is “endogenous” because it is determined by our theory of economics, namely that prices adjust to set supply equal to demand. Thus p * is determined “inside the system” (in this case, the “system” is speciFed by the supply and demand equations), and this is why we refer to it as the endogenous variable. Now if rainfall changes, prices will adjust so that for each possible level of rainfall, there will be a new equilibrium price that will depend on the level of rainfall. Thus we can write p * ( r ) to emphasize that the equilibrium price of wheat is an implicit function of the level of rainfall. This implicit function must satisfy, for each possible level of rainfall r S ( p * ( r ) , r ) = D ( p * ( r )) (2) Now a common comparative static exercise is to ask, “does p * ( r ) increase or decrease as rainfall increases?” That is, we seek to Fnd out whether the derivative, p * ( r ) / r , is positive or negative (or 1
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zero). We would expect that if rainfall increases the quantity supplied, S ( p , r ) / r > 0, then the “supply curve shifts up” while the demand curve remains unchanged (since
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implicit_function_theorem - Spring 2005 Economics 425 John...

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