Question

An ice cream vendor has the following production function:

Q = LM

Where, Q = the number of ice cream cones produced per day, L = the number of workers hired per day, and M = the number of ice cream machines rented per day. Associated with this production function are the following marginal product relationships:

MPL = M and MPM = L.

Plot the isoquant for 80 ice cream cones per day.

Find the optimal combination of inputs, if the wage rate per day = $30, the ice cream machine rental per day = $150, and 80 ice cream cones are to be produced.

Answer 1

The optimal combination of inputs for the ice cream vendor to produce 80 cones per day at minimum cost is hiring 4 workers and renting 20 machines, resulting in a total cost of $3,330.

Step-by-step explanation:

To plot the isoquant for 80 ice cream cones per day, we use the production function Q = LM. This means that to produce 80 cones, the product of the number of workers (L) and the number of machines (M) must equal 80.

The isoquant curve will show all the combinations of L and M that result in the production of 80 ice cream cones.

Given the marginal products MPL = M and MPM = L, and the respective cost of hiring a worker (W=$30) and renting a machine (R=$150), we find the optimal combination of inputs by setting the ratio of the marginal products equal to the ratio of input costs,

i.e., MPL/W = MPM/R. This results in the equation M/$30 = L/$150, which simplifies to L = 5M.

For the production of 80 ice cream cones, Q = LM becomes 80 = L*5L, which gives us L2 = 16, so L = 4. This means M = 80/L which is 20. Thus, hiring 4 workers and renting 20 machines would be the cost-minimizing input combination to produce 80 cones, with a total cost of (4*$30) + (20*$150) = $3,330.