Question:

Rolex is one of the most important brands in the production of watches. The quality of their products is very high, since each watch is subject to a unique production process that involves both precise machineries and expert human intervention. Indeed, in order to produce one watch 3 unit of labor (L) and 1 unit of capital (K) are used. Furthermore, suppose that the price of work, capital and the level of production are respectively \(\omega =1\), r = 4, and \(\bar{Q}\) = 10.

1. Write Rolex’ production function [Hint: notice that the two inputs are used in fixed proportions].
2. Represent graphically isoquants and isocost.
3. Find Rolex’ optimal input combination (L*,K*) in this scenario.
4. Derive the long-run factor demands as a function of \(\bar{Q}\), r and \(\omega \). [Hint: consider the optimality conditions, we ask you to find the optimal quantity of labor and capital demanded for different \(\bar{Q}\), r and \(\omega \) *].
5. Derive the long-run cost function, that is the long run optimal total costs as a function of \(\bar{Q}\), r and \(\omega \).
6. Looking at patent data, the American company Apple Inc. figures out that the market for watches is characterized by poor innovation. Hence, they decide to invest in such sector in order to come up with new and different products. After some years, smart watches are developed. This new product can be made in two different ways: either employing 10 units of labor (L) or 1 technological machinery (capital, K). The price of inputs is the same of before: \(\omega \) = 1, r = 4, and the firm wants to produce the same quantity of its competitor Rolex, \(\bar{Q}\) = 10. Write Apple’s production function [Hint: the two inputs are substitutes], and draw on a graph isoquant and isocost curves.
7. Derive Apple’s optimal input combination (L**,K**) in this scenario.
8. Derive the factor demands and the total cost function for Apple.

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