Problem $1.$ Determine the returns to scale for the following production functions. Give a mathematical proof of your answer.

$($A$)F(A,K,L)=A{K}^{\frac{1}{6}}{L}^{\frac{1}{6}}$

$($C$)F(A,K,L)=\sqrt[\phantom{2}]{K}+\sqrt[\phantom{2}]{AL}$

$($B$)F(A,K,L)=AK+L$

$($D$)F(A,K,L)=A(K^{\frac{1}{2}}+L^{\frac{1}{2}}{)}^2$

Problem $2.$ Consider the production functions from Problem $1.$

$($A$)$ Find the marginal products of capital and labor.

$($B$)$ Establish whether the marginal products are diminishing. Explain your answer.

$($C$)$ Check whether there is a positive effect of technology on marginal products. Explain your answer.

$($D$)$ Check whether there is factor complementarity. Explain your answer.

Problem $3.$ Production Functions. Ignore technology A$.$ Give an example of the following production functions, and prove that your examples are valid. Here $K$ is capital and $L$ is labor. If you think such an example does not exist, prove it$.$

$($A$)$ A production function $F(K,L)$ with diminishing marginal products and constant returns to scale.

$($B$)$ A non$-$Cobb$-$Douglas production function $F(K,L)$ with constant returns to scale.

$($C$)$ A production function $F(K,L)$ with rising marginal products.

$($D$)$ A production function $F(K,L)$ without factor complementarity.

$($E$)$ A production function with three factors of production, $F(K,L,M),$ and constant returns to scale.

$($F$)$ A production function with three factors of production, $F(K,L,M),$ and factor complemenarity between $K$ and $L,L$ and $M,$ and between $K$ and $M.$

$($G$)$ A production function with three factors of production, $F(K,L,M),$ and factor complemenarity between $K$ and $L,L$ and $M,$ but no complementarity between $K$ and $M.$