Question

2. There are two industries, hatching and laying, and two goods, eggs and chick- ens. The price of an egg is l; the price of a chicken is p. The hatching industry produces y chickens using x eggs, which it buys from the laying industry. Its production function is yi-4(x1)2. The laying industry produces x2 eggs using chickens, which it buys from the hatching industry. Its production function is x,-(y2)1/2. (We assume away any consideration of time lags, non negativity constraints, the indivisibility of live chickens or eggs, and claims to precedence by all parties.) Express in terms of p the profit-maximizing quantities of chickens and eggs produced or used in each industry. Assuming that all eggs produced in the laying industry are used as inputs by the hatching industry, what is the value h the net output p fhc indus of p? What, then, is the net output of chickens available for consumption?

Answer 1

P_egg = 1, p_chicken = p, y_1 = 4 squareroot x_1 rightarrow Hatching Industry laying p_x = 1, p_y = p x_2 = squareroot y_2 so for Hatching, at eqn pi Max, pi = py_1 - p_x x_1 pi = 4p squareroot x_1 - x_1 partial differential pi/partial differential x_1 implies 4p/2 squareroot x_1 - 1 = 0 implies 2p/squareroot x_1 = 1 80 2p = squareroot x_1 thus x_1 = 4p^2 & y_1 = 4 squareroot x_1 y_1 = 8p for laying rightarrow pi = x_2 = py_2 = squareroot y_2 - py_2 so partial differential pi/partial differential y_2 implies 1/2 squareroot y_2 - p = 0, so 1/2p = squareroot y_2 thus y^x_2 = 1/4p^2, x^x_2 = 1/2p] \r\n Now total egg produced in laying = 1/2p so inputs used by Hatching implies 1/2p = x^x_1 so 1/2p = 4p^2 implies 1/8 = p^3 hence p = 1/2 Ans Net o/p of chickens available for consn = y^x_1 - y^x_2 = 8p - 1/4p^2 = 8/2 - 1/4(1/2)^2 = 4 - 4/4 = 3