Question

TUTORIAL2 Chapter 7 Part 1 Key Concepts and Equations: Production Isoquant: shows all combinations of input quantities that yield the same level of output. Higher isoquant: higher level of output Marginal Rate of Technical Substitution: MRTS is the slope of the isoquant at any input combination. It tells us the rate at which we must increase the qty of input 2 per unit decrease in qty of input 1. MRTS diminishes as we move down the isoquant from left to right MRTS = MP(Z1)/ MP(zz) A linear isoquant means MRTS is constant- a linear isoquant inputs are perfect substitutes. An L-shaped isoquant means MRTS cannot be defined at the kink -inputs are perfect compliments 3 types of returns to scale: increasing- on downward sloping portion of the LRAC constant on the flat portion of the LRAC and decreasing- on the upward sloping portion of the LRAC Conditional input demand functions: the demand for each input as a function of y, W1 and w2. There are no z's in these equations The slope of the isocost is w/W2 Two principles of cost minimization The cost minimizing input bundle is on the isoquant y F(z1 72). The MRTS is equal to w/w2 at the cost minimizing bundle slope of the isoquant slope of the isocost line MP(z1) / MP(Z2) - W,/W2 If the price of one input changes, the optimal combination of inputs will change. Total cost will change If goods are perfect substitutes, the optimal input mix will be a corner solution. When isoquants are convex, we get an interior solution

Answer 1

Production function y = z_1 . z_2^1/2, Input prices = (w_1, w_2) then Now Conditional input dd function. at profit max, MRTS z_1, z_2 = w_1/w_2 MRTS_12 = 2y/2z_1/sy/2z_2 rightdoublearrow squareroot z_2/z_1 middot 1/2 (z_2)^-1/2 = 2z_2/z_1 thus at equation 2z_2/z_1 = w_1/w_2, thus 2z_2 w_2/w_1 = z_1 so, from production function: - q = 2z_2 middot w_2/w_1 . (z_2)^1/2 thus w_1 q/2w_2 = (z_2)^3/2 then, thus z_2^cz = (w_1q/2w_2)^2/3 z_1^cx = 2w_2/w_1 middot z_2^cx = 2w_2/w_1 middot (w_1 q/2w_2)^2/3 = (q)^2/3 (2w_2/w_1)^1 - 2/3 Now 2z_2/2w_2 = (w_1 q/2) (- 2/3) (w_2)^-5/3 z_1^cx = (q)^2/3 (2w_2/w_1)^1/3 \r\n so, both dd function slopes downwards. 2z_1/2w_1 < 0, 1(b) w_1 = 5, w_2 = 10 Now q bar = 27. So, C^x = w_1 z_1^cx + w_2 z_2^cx = 5 middot (q)^2/3 (2w_2/w_1)^1/3 + 10 middot q^2/3(w_1/2w_2)^2/3 = (27)^2/3 [5 middot (20/5)^1/3 + 10 middot (5/20)^2/3] = 9[5 (4)^1/3 + 10 middot (1/4)^2/3] = 107.15