Functions of several variables and partial differentiation, Unconstrained Optimization with two independent variables, Stationary points, Sufficient conditions for optimisation and saddle point Question 2: A firm has the production function Q=f(L,K)=100L0.5K0.5, where L is labour and K is capital. The wage rate is $5 per hour and the rental price of capital is $4 per hour and the price of the good is $1. 1. Find the marginal product (MP) of labour and capital (that is, fL,fK ) 2. Show that there are diminishing marginal productivities, that is, fLL<0,fKK<0 3. Find the cost function. 4. Find the amount of labour and capital the firm should hire to maximize profits. [Hint: write the profit function in terms of L and K and then take partial derivatives. Use the concepts involved in Question 1 above to determine the profit is maximised.] (Question 1: Let f(x,y)=2x2+x2y+y3+12y2. Find the stationary points and for each stationary point determine whether it is a maximum, minimum, saddle point or can’t tell.)
Functions of several variables and partial differentiation, Unconstrained Optimization with two independent variables, Stationary points, Sufficient conditions for optimisation and saddle point
Question 2: A firm has the production function Q=f(L,K)=100L0.5K0.5, where L is labour and K is capital. The wage rate is $5 per hour and the rental price of capital is $4 per hour and the price of the good is $1.
1. Find the marginal product (MP) of labour and capital (that is, fL,fK )
2. Show that there are diminishing marginal productivities, that is, fLL<0,fKK<0
3. Find the cost function.
4. Find the amount of labour and capital the firm should hire to maximize profits. [Hint: write the profit function in terms of L and K and then take partial derivatives. Use the concepts involved in Question 1 above to determine the profit is maximised.] (Question 1: Let f(x,y)=2x2+x2y+y3+12y2. Find the stationary points and for each stationary point determine whether it is a maximum, minimum, saddle point or can’t tell.)
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