Final answer:
To find the marginal productivities, calculate the partial derivatives of the production function with respect to x and y and substitute the given values. The marginal productivity of labor is 21 and the marginal productivity of capital is 37.
Step-by-step explanation:
To find the marginal productivities, we need to calculate the partial derivatives of the production function P(x,y) with respect to x and y. Given the production function P(x,y) = 0.2x2 + 2x + 3xy + 0.4y2 + 3y, we calculate:
Partial derivative with respect to x:
Px(x,y) = 0.4x + 2 + 3y
Partial derivative with respect to y:
Py(x,y) = 3x + 0.8y + 3
Now, substituting x = 10 and y = 5 into the partial derivatives, we get:
Px(10,5) = 0.4(10) + 2 + 3(5) = 4 + 2 + 15 = 21
Py(10,5) = 3(10) + 0.8(5) + 3 = 30 + 4 + 3 = 37
Interpreting the results, the marginal productivity of labor (x) when 10,000 work-hours per month are utilized and a capital investment of $5000/month is made is 21. This means that if one more thousand work-hours are utilized, the number of souvenir coffee mugs produced would increase by 21 hundreds. Similarly, the marginal productivity of capital (y) is 37. This means that if the capital investment is increased by $1000, the number of souvenir coffee mugs produced would increase by 37 hundreds.