There are 2 firms in the yoghurt market, Dannon and Yoplait. They produce yoghurt using milk (M) and capital (K) according to the production function
q=2 (MK)^1/2
where q is the quantity of yoghurt produced. The price of milk is Pm = 4, and the price of capital is denoted r. The price of yoghurt is denoted py.
In the short run, both firms have a fixed plant size, so that capital is fixed. Dannon has plant size K ¯d = 100, while Yoplait has plant size K ¯y = 81.
1) Compute the short run cost functions of Dannon and Yoplait. (Hint: your short-run cost functions should depend on r.)
2) Deduce the equations of the short run supply curves of Dannon and Yoplait. Do the supply curves depend on the price of capital? the quantity of capital? Comment.
3) Derive the equation of the market short-run supply curve.
The market demand for yoghurt is D(py) = 261 − 40py, where py is the price of yoghurt.
4) Represent the short run market equilibrium on a graph with price on the vertical axis.
5) Derive the short run market equilibrium, that is, the equilibrium price pey, the total equilibrium quantity Qe and the quantities supplied by each firm qDe and qYe .
6) Which firm has the highest market share? Comment in light of the capital endowments.
7) Compute the quantity of milk used by each firm in equilibrium. Derive the marginal product of milk in each firm at the equilibrium. Comment. (Hint: Express marginal cost as a function of the marginal product of milk, as we did in lecture.)