Figure 1.3. Cost Minimization with Leontief Technology From Figure 1.3, we see that the minimum point is(4, 4). Therefore, the cost function Answer 1.5. If the demand system is a solution of a cost minimization problem, then it must satisfy the properties listed in Proposition 1.6. Property(1)in the proposition is obviously satisfied. Property(2)requires symmetric cross-price effects, that is 2 '12 Therefore, b12= b21. With 012= b21, the substitution matrix is 5w12w2y w1 a w a w2 (3u12u22y-2uiu2'y/ We have <0 b2 =1m(1-听)=0 Thus, the substitution matrix is negative semi-definite. Finally, property(4)is implied by the fact that the substitution matrix is negative semi-definite. Therefore, to be consistent with cost minimization, we need and only need condition: b12= b21 5x x 1 2 ax =bx 1 2 _y b _y a y=f(x) Figure 1.3. Cost Minimization with Leontief Technology From Figure 1.3, we see that the minimum point is ( y a , y b ). Therefore, the cost function is: c(w1, w2, y) = w1 a + w2 b  y. Answer 1.5. If the demand system is a solution of a cost minimization problem, then it must satisfy the properties listed in Proposition 1.6. Property (1) in the proposition is obviously satisfied. Property (2) requires symmetric cross-price effects, that is, ∂x1 ∂w2 = ∂x2 ∂w1 or 1 2 b12 y √w1w2 = 1 2 b21 y √w1w2 . Therefore, b12 = b21. With b12 = b21, the substitution matrix is ⎛ ⎜⎝ ∂x1 ∂ w1 ∂x1 ∂ w2 ∂x2 ∂ w1 ∂x2 ∂ w2 ⎞ ⎟⎠ = b12 ⎛ ⎜⎝ −1 2w− 3 2 1 w 1 2 2 y 1 2w−1 2 1 w− 1 2 2 y 1 2w− 1 2 1 w−1 2 2 y −1 2w 1 2 1 w− 3 2 2 y ⎞ ⎟⎠ . We have ∂x1 ∂ w1 < 0, and        ∂x1 ∂ w1 ∂x1 ∂ w2 ∂x2 ∂ w1 ∂x2 ∂ w2        = b2 12        −1 2w−3 2 1 w 1 2 2 y 1 2w− 1 2 1 w−1 2 2 y 1 2w− 1 2 1 w− 1 2 2 y −1 2w 1 2 1 w− 3 2 2 y        = 1 4 b2 12y2 w−1 2 w−1 1 − w−1 1 w−1 2  = 0. Thus, the substitution matrix is negative semi-definite. Finally, property (4) is implied by the fact that the substitution matrix is negative semi-definite. Therefore, to be consistent with cost minimization, we need and only need condition: b12 = b21. 5