31301two%20Inputs.Rtex

# 31301two%20Inputs.Rtex - RTS [email protected][email protected] 1 [email protected][email protected] 2 ² F 1 =F 2...

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Notes on Production with Two Inputs On Production with Two Inputs Consider a production function with two inputs : I shall call the inputs 1 and 2 [think of "capital" and "labor"]. Suppose that the production function is given by q = F ( x 1 ; x 2 ) = x 1 = 2 1 x 1 = 3 2 where q , x 1 and x 2 are non-negaitive quantities. Example: q = x 1 = 2 1 x 1 = 3 2 [here x 1 & 0 and x 2 0] : Fix q = 4 : Observe that the input combination ( x 1 ; x 2 ) = (4 ; 8) produces q = 4 : But (16 ; 1) ; and (1 ; 64) are also producing q = 4 : q = 4 as the set of all input-combinations that produce q = 4 : we write: I ( q 0 ) = f ( x 1 ; x 2 ) 0 : F ( x 1 ; x 2 ) = q 0 g : Then for I (4) contains (4,8), (16,1), (1,64) and many, many other combi- nations. The slope of the isoquant is dx 2 =dx 1 = ± [( 1 ) 2 )] The absolute value of this slope is the Rate of Technical Substitution ( RTS ) between the two inouts along an isquant:
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Unformatted text preview: RTS = [( @[email protected] 1 ) = ( @[email protected] 2 )] ² F 1 =F 2 : In the case where q = x & 1 x ± 2 , RTS = ( &=± )( x 2 =x 1 ) The ±convex" case: write U ( q ) = f ( x 1 ; x 2 ) & 0 : F ( x 1 ; x 2 ) & q g : We say that the technology of the &rm described by the production function is convex if for any two points in U ( q ) the line joining the two points is entirely in U ( q ) : More formally; let (& x 1 ; & x 2 ) and (~ x 1 ; ~ x 2 ) are in U ( q ) : Then; for any ² in (0,1), ( ² & x 1 + (1 ± ² )~ x 1 ; ² & x 2 + (1 ± ² ) ; ~ x 2 ) is in U ( q ) : ** Indi/erence curves cannot interesect*** 1...
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