microeconomics. question 2

2. - Consider the following decision problem: max pg wa: (2.9)6113, subject to 4ln(:c + 1) = y. a. On the :cy plane, given any so 2 0, the slope of the graph of the constraint 4 ln(a:+ 1) = y is equal to __________ (the answer should be an expression of 3;). b. Suppose that p = 10 and w = 6. i. iv. On the 33; plane, given any a: 2 0, the slope of the level surface of the objective function 103; 6:1: is equal to ____________ . The optimum (3;, y) is: :1: = ___________ and y = ___________ . At this optimum, the gradient of the objective 10y 6:1: is equal to ____________ At this optimum, which of the following vectors is aligned with the gradient of the constraint function 41n(ar; + 1) y? M B.[_j0[ c-[fo] D. [ 6/10 [ 1 E. none of the above c. Suppose that 10 remains to be 10 while to varies. i. ii. The optimum is uniquely (0, 0) if and only if m is ____________ (higher than, lower than, equal to [ll in the blank with one of the three items]) ____________ [provide a number] The equation system produced by the Lagrange method with equality constraints is valid when w is ____________ (higher than, lower than [ll in the blank with one of the two items]) ___________ [provide a number]