Sam has been spending a lot of time with Pat, Inspired to live a new lifestyle, Pat has also gotten seriously into gardening. Suppose Pat's production function and marginal products for producing backyard lettuce are given by:

Y = M^(1/2)L^(1/2) MPM = L^(1/2)/2M^(1/2)MP_L = M^(1/2)/2L^(1/2)

Y is the quantity of lettuce produced, L is the number of hours Pat works the garden, and M is the amount of an indexed unit of all other factors. M would include raw materials (seeds, dirt, fertilizer etc.), capital (hose, sprinkler system, chicken wire), and land (backyard space) used in production.

Pat uses 100 hours of their time and 4 units of M in their lettuce garden, making 20 heads of lettuce at the end of summer. Each unit of M cost $5 and the price of lettuce at the grocery store is $5 per head.

v.Suppose Pat works a job with flexible hours at the local hemp store that pays $20 an hour. If they valued all inputs in the production function at their opportunity cost, what is their economic profit from growing lettuce?

w.Sam decides to advise Pat on how to minimize their costs of lettuce production. What functions might Sam derive to determine Pat's cost minimizing combination of inputs?

x.Derive these functions. Given the input prices above, calculate the optimal amount of M and L to produce Y= 20.

y. Suppose the Pat was so impressed by the economic cost savings from this exercise and planned to start a business selling backyard lettuce. Given the input and product prices above, would Sam invest in this business?