Demonstration microeconomics.-

6. (32 points) The aggregate production function in Solow (1957) was specified as: O = A(D)f(K, L). (1) (6 points) Derive the output elasticities of labor ( s, ) and capital (Ex ). (2) (10 points) Show that the proportional output growth can be composed into the weighted sum of input growth and rate of technical change: K L A + 6 * K (3) (10 points) Mathematically prove Euler's homogeneous function theorem which relates marginal measures to the value of a function as the following: (i) Show that for a homogeneous-of-degree-m function f(x,, x, ), the partial derivative of f (tx, , tx, ) with respect to f is: my " - f ( x , X 2 ) = x, f, ( tx , tx ) + x , f, (tx , tx ) (Note: fi(tx, , tx, ) = of (tx, , tx, )/ atx , f, (tx, , tx, ) = of (tx, , tx, )/ otx, ) (ii) Let #1, mf (x , x ) = x/ ( x , x ) + x / (x ,X2) (4) (6 points) According to the Euler's homogeneous function theorem, under the assumption of constant returns to scale, O=KMP, + LMP, . Show that when constant returns to scale is assumed and inputs are paid at their marginal productivity, output elasticity is equal to cost share.6. (32 points) The aggregate production function in Solow (1957) was specified as: Q = A(1)f(K, L). (1) (6 points) Derive the output elasticities of labor ( &, ) and capital ( Ex ). (2) (10 points) Show that the proportional output growth can be composed into the weighted sum of input growth and rate of technical change: O_A K O A + 61 L (3) (10 points) Mathematically prove Euler's homogeneous function theorem which relates marginal measures to the value of a function as the following: (i) Show that for a homogeneous-of-degree-m function f(x,, x, ) , the partial derivative of f(tx, , tx, ) with respect to f is: my "-' f ( x , x , ) = x f ( tx , tx ) + x , f, (tx , tx ) (Note: fi(tx, , tx, ) = of (tx, , tx )/ etx, , f, (tx, , tx, ) = of (tx, , tx, )/ atx, ) (ii ) Let # 1, mf ( x , x ] ) = x f ( x , x ) + x / 2 ( x ,*2) (4) (6 points) According to the Euler's homogeneous function theorem, under the assumption of constant returns to scale, O=KMP + LMP, . Show that when constant returns to scale is assumed and inputs are paid at their marginal productivity, output elasticity is equal to cost share