Q$4(25$ points$)$ Firm X has the following production function $(,)=(+$#$).$

$(1)(5$ points$)$ Find the marginal product of labor and capital and the marginal rate of technical substitution, and state if the returns to capital and labor are increasing, decreasing or constant. Explain what is $$weird$$ about this MRTS$.$

$(2)(5$ points$)$ Graph the isoquants of the production function and state whether the

production function exhibits constant, increasing or decreasing returns to scale.

$(3)(15$ points$)$ Now suppose that the production function is $(,)="$

$!$$$($&$\text{'}($#$),$where t measures years.

$($a$)(5$ points$)$ How does this affect the marginal products of labor and capital, the returns to scale and the MRTS$?$

$($b$)(10$ points$)$ What is the rate of productivity increase over time? Graph the isoquants for q $=100$ for t $=1,2.$ Explain.