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**Unformatted text preview: **The Ohio State University Department of Economics Econ 501A&Prof. James Peck Equations for the Final The utility maximization problem is max u ( x;y ) subject to : p x x + p y y = M The Lagrangean approach transforms a constrained optimization prob- lem into an unconstrained problem of choosing x, y, and the Lagrange multiplier, &; to maximize L = u ( x;y ) + & [ M & p x x & p y y ] : By di/erentiating L with respect to x, y, and & , and setting the derivatives equal to zero, the resulting ¡rst order conditions are: @u @x & &p x = 0 ; @u @y & &p y = 0 ; and M & p x x & p y y = 0 : x & ( p x ;p y ;M ) and y & ( p x ;p y ;M ) are known as the generalized demand func- tions . The production function speci¡es the most output that can be produced with a given combination of inputs, based on the technology available to the ¡rm. x = f ( K;L ) Technological e¢ ciency occurs if the ¡rm is on its ¢production frontier.£ That is, it is impossible to achieve more output with the same inputs. We typically assume that marginal products are positive @f ( K;L ) @K > and @f ( K;L ) @L > : A production isoquant is a curve describing the set of capital-labor combinations yielding the same output, according to the production function. The marginal rate of technical substitution is de¡ned to be the negative of the slope of the isoquant. The MRTS is the rate at which the ¡rm would be willing to give up capital in exchange for labor. We assume diminishing MRTS. In the long run , all inputs are variable, so the ¡rm can choose any combi- nation of capital and labor. In the short run , at least one input is ¡xed and cannot be varied. 1 Returns to Scale For long run decisions, we may be interested in what happens as we vary all of the inputs simultaneously. The production function exhibits decreasing returns to scale if, for & > 1 , we have f ( &K;&L ) < &f ( K;L ) : The production function exhibits constant returns to scale if, for & > 1 , we have f ( &K;&L ) = &f ( K;L ) : The production function exhibits increasing returns to scale if, for & > 1 , we have f ( &K;&L ) > &f ( K;L ) : Hold all but one of the inputs &xed (say, &x K = K ). The total product of labor is given by the function, x = f ( L ; K ) . The average product of labor is de&ned as AP L = f ( L ; K ) L : The marginal product of labor is de&ned as MP L = @f ( L ; K ) @L : Cobb-Douglas example: x = K & L ¡ AP L = K & L ¡ L = K & L ¡ & 1 MP L = @K & L ¡ @L = ¡K & L ¡ & 1 AP K = K & L ¡ K = K & & 1 L ¡ MP K = @K & L ¡ @K = ¢K & & 1 L ¡ MRTS = MP L MP K : Diminishing marginal returns (to labor) occur when the marginal product (of labor) eventually falls as L increases. @MP L @L < 2 Cost minimization subject to an output constraint: min wL + rK subject to f ( K;L ) = x (1) Set up the Lagrangean, Lagr: = wL + rK + & [ x & f ( K;L )] The &rst order conditions are @Lagr: @L = 0 = w & & @f @L @Lagr: @K = 0 = r & & @f @K @Lagr: @& = 0 = x & f ( K;L ) Solving, we have & = w MP L = r MP K : (2) This is the condition that the MRTS equals the input price ratio. It is alsoThis is the condition that the MRTS equals the input price ratio....

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