The table below represents the production function for Caia's Sweaters, a small company specializing in custom

sweater designs. The number in the cells represents the number of sweaters that can be produced with various

combinations of labor $(L)$ and capital $(K).$

a$.$ In what way is this production function long run?

b$.$ A short run production function is part of the long run production function.

Describe a short run production function from the table when $K$ is fixed.

Describe a short run production function from the table when $L$ is fixed.

c$.$ Suppose that Caia's Sweaters employs $3$ units of capital and $4$ workers. How many sweaters will be produced?

d$.$ Suppose that Caia's Sweaters employs $3$ units of capital and $4$ workers but the owner, Caia Itadori, is considering

adding another worker to the company. What will be the marginal product of the new worker?

e$.$ Suppose Caia's Sweaters employs $3$ units of capital. Does the short run production function exhibit diminishing

returns with respect to labor? If so$,$ when do the diminishing returns "set in$"?$ Show your work!