Tesla produces small cars in a perfectly competitive market using labour (L) and capital (K). Tesla's production function is given by:

f(L,K) = min {0.05L, K^1/2}

where q is the number of cars produced.

(a) Starting from L > 0, K > 0, suppose you double the amount of L and K. Is it possible for output (q) to more than double (i.e., increase from q to Aq where A > 2)?

(b) Find the minimum cost to produce q cars when the price of labour (w) is 400 and price of capital (r) is 10? [Hint: the answer would involve q.]

(c) Using the answer to part (b), we can show that Tesla's supply function is of the following form:

q = { a + bp for p > AVC_min and

0 for p less than or equal to AVC_min

where AVC stands for average variable cost. Find a, b, and AVCmin

(d) A new technology of producing cars has come to the market which only uses capital

g(K) = 0.5K^1/2

As in part (c), continue to assume w = 400 and r = 10.

Tesla maximises profits. It uses both technologies and sells q* cars in a perfectly competitive market where the price of small cars is p*. If 1/3 of cars are produced using new technology and 2/3 of the cars with old technology, then p* = ___________________; q* = _______________

How many cars do Tesla produce in total? What must be the price of cars (assuming that the market for cars is perfectly competitive)?