Question

Kar-a-Tea supplies college students with boba using two inputs: black tea (b) and tapioca pearls (t). Kar-a-Tea can buy an ounce of black tea for the price pb = 4 and a pound of tapioca pearls for the price pt = 5. They are seeking an economist’s help in running their business. (a) Suppose they tell you that making boba is a proudly artisan business. Each drink is made in a perfect ratio between black tea and tapioca pearls, such that Kar-a-Tea cannot make extra drinks with excess of either input. Let f(b, t) = min{10b, 5t} represent their production function. (i) Solve for Kar-a-Tea’s cost function. (ii) If Kar-a-Tea wanted to make 50 drinks, what would be their cost? How much of each input would they use?

Answer 1

To find the cost function for Kar-a-Tea, we consider the price of inputs and the production function. The cost to make one drink is $1.40, so the cost for 50 drinks is $70, using 5 ounces of black tea and 10 pounds of tapioca pearls.

Step-by-step explanation:

To determine the cost function for Kar-a-Tea given the production function f(b, t) = min{10b, 5t}, we first understand that each drink requires inputs in such a way that neither input is in excess. To make one boba drink, Kar-a-Tea needs 1/10 of an ounce of black tea and 1/5 of a pound of tapioca pearls. Since the price per ounce of black tea is pb = $4 and the price per pound of tapioca pearls is pt = $5, we calculate the cost of inputs for one drink as (1/10)*$4 for tea + (1/5)*$5 for tapioca pearls.

The cost to make one drink, therefore, is $0.40 + $1.00 = $1.40. The cost function C(q) for q drinks is then $1.40q. To determine the cost to make 50 drinks, we apply the cost function: C(50) = $1.40 * 50 = $70. Thus, Kar-a-Tea would spend $70 to produce 50 drinks. As for the inputs, they would use 5 ounces of black tea (50/10) and 10 pounds of tapioca pearls (50/5) to make the 50 drinks.