$1.$ Let y $=$ f $($x$1$; x$2)=$ f $($x$)$ be a production function, the price of y is p; and

the factor prices are given by wi for input xi for i $=1$; $2$: Assume the production

function is either decreasing or constant returns to scale.

a$.$ Write down the long$-$run pro$$t maximization problem.

b$.$ Now, write down the Short run pro$$t maximization problem assuming

factor $1$ is $$xed in the short$-$run at a strictly positive amount.

c$.$ Compute the envelope theorem for the short$-$run pro$$t in the $$xed factor

$($factor $1).$

d$.$ Now, maximize the Short run pro$$t choosing the $$xed factor, and give

the FOC. Explain now the relationship between your answer in $($a$)$ and this

answer answer.