2. [a] Let f '. RELi R+ he a a rm's differentiable production function satisfying the usual assuInptions. Suppose that :I.' E E1 is a. vector of inputs and that w E R1 are the factor prioes. Let p E R... denote the output price. The prot of the rm is given by: Tr[p,a=} = 111:}: [pf[1'}- arr]. Derive and explain Shephard's lemma: arr E = .'Ifi [19, w} (h) Now let f : Eir JR... be dened by: minimi- Suppose this rm is a price taker in the output market and in the factor market. The output price is p and the factor prices are is = {1, 1}. Derive the rm's supply metion. [e] Suppose now that 3:1 is xed at E1 = 1. Derive the short run supply metion. Which is more elastic? the shortrun supply or the longrun supply? For what output level is 3:1 = 1 the optimum plant size':ll [d] Suppose that p = 4 and that the government imposes a lump sum tax T = 2.50 on the rm if it produoes positive output. What will the rm's output he i] in the longrun and ii) in the shortrun where 31 is xed at 1'? {A lump-sum tax is deducted from pretax prots, at {y} T}