Alex and Jessica own a coffee shop near campus. Suppose that their daily production function for workers, L, and capital K, is The wage rate is $100 daily and a unit of capital costs $50 daily. What is Alex and Jessica’s least-cost input combination to make 145 cups of coffee per day? What is their total cost?

To determine Alex and Jessica's least-cost input combination, we need to calculate the marginal product of labor (MPL) and the marginal product of capital (MPK) and compare the ratios to the input prices.

Given the production function is not provided, we cannot calculate the MPL and MPK accurately. However, we can assume that the relationship between labor (L) and capital (K) is linear and follow a Cobb-Douglas production function.

Let's assume the production function is given by:

Q = a * L^b * K^c

Where:
Q = quantity of cups of coffee per day (145 cups)
L = labor input
K = capital input
a, b, c = parameters (unknown)

We can assume that a = 1 (as it has no bearing on the input combination and cost).

The MPL and MPK are given by the partial derivatives of the production function with respect to labor and capital, respectively:

MPL = ∂Q/∂L = b * a * L^(b-1) * K^c
MPK = ∂Q/∂K = c * a * L^b * K^(c-1)

To find the least-cost input combination, we need to calculate the ratio:

MRPL/MRPK = MPL/MPK

Assuming the above production function, we have:

b * a * L^(b-1) * K^c / (c * a * L^b * K^(c-1))

Since a = 1 and we are trying to find the input combination for 145 cups of coffee per day, we can set the ratio equal to 145 and solve for L/K:

b * L^(b-1) * K^c / (c * L^b * K^(c-1)) = 145

Simplifying:

b * K / (c * L) = 145

This equation represents the relationship between labor and capital in the least-cost input combination. We can solve for L and K using this equation.

As for the total cost, it can be calculated by multiplying the quantity of each input by its corresponding price:

Total cost (TC) = (L * wage rate) + (K * capital cost)

Given that the wage rate is $100 daily and the capital cost is $50 daily, we can substitute the values and the obtained input combination for L and K into the equation to find the total cost.

To find Alex and Jessica's least-cost input combination, we need to determine the optimal quantities of workers (L) and capital (K) that minimize their cost while producing 145 cups of coffee per day.

The production function gives us the relationship between workers, capital, and the output of coffee cups. However, the specific form of the production function is not provided in the question. Without that information, we cannot explicitly determine the input quantities that yield exactly 145 cups of coffee. Instead, we will find the input combination that produces the closest possible number of cups to 145.

To do this, we will use the concept of isoquant, which represents different combinations of inputs that result in the same level of output. We will aim to find an isoquant that closely matches 145 cups.

Let's assume a certain combination of workers and capital (L1, K1) that yields an output close to 145 cups. For this combination, we need to calculate the total cost.

The total cost is the sum of the cost of labor and the cost of capital:
Total Cost = Cost of Labor + Cost of Capital

The cost of labor is given by:
Cost of Labor = Wage Rate * Number of Workers

The cost of capital is given by:
Cost of Capital = Cost of a Unit of Capital * Number of Units of Capital

Since the given wage rate is $100 per day and a unit of capital costs $50 per day, we can now compute the total cost.

Now, let's assume a certain number of workers (L1) and units of capital (K1), substitute these values in the total cost formula, and calculate the corresponding cost.

Note: The actual calculation will require the specific production function and the specific input quantities (L1, K1) chosen. Without that information, we can only provide a general explanation of the process.

If in case you have the specific production function and input quantities, please provide them so that we can assist you in the calculation.

You did not include the production function.