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 p5 = 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 KararTea cannot make extra drinks with excess of either input. Let f (b, t) = min{10b, 5t} represent their production function. (i) Solve for KaraTea's cost function. (ii) If KaraTea wanted to make 50 drinks, what would be their cost? How much of each input would they use? (b) Now suppose they eventually let on that their business is a lot more complicated. They do not really adhere to any specic recipe, instead changing it as they face price uctuations in the tea and tapioca markets. Kar-a-Tea tells you that their production function is really f(b,t) = 4[b% + $12. (i) Calculate the marginal products of black tea (b) and tapioca (t). (ii) Solve for the cost function. (While this function looks a bit different than ones we have looked at so far, you can still solve for the optimal inputs by using the tangency condition.) (iii) If KaraTea wanted to make 50 drinks, what would be their cost? How much of each input would they use? (Non-integer answers are OK.) (iv) Find the average cost and marginal cost functions. What do these tell you about returns to scale