min √(x) s.t. μ 20 with variable μ Rm. In particular, show that an optimal solution μ* to this program must satisfy T n t=1 j=1 T n P(Rt(*)¹Ã¡) Ai‚j ≤ xi, Vi, ΣΣP (Rt‚j ≥ (µ*)'Aj) (µ*)¹A; = (µ*)¹x. t=1 j=1 (1) (2)
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Derive the Karush-Kuhn-Tucker conditions for this Bid-price policy program (also shown in the image for clarity),
min J˜µT(x)
s.t. µ ≥ 0
with variable µ ∈ ℝ^m. In particular, show that an optimal solution µ* to this program must satisfy the constraints in the image below:
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- Question 3 The Black-Scholes model is a time series (X₂) that follows the dynamics X₁ = X₁-1*exp(μ+0€) where μ, >0 and () is a time series of independent and identically distributed standard Gaussian random variables. Note that E(exp(μ+σt)) = "¹+0² (i) Show that the log-returns of this process X, form a white noise process (ii) Suppose E(X) = 0. Compute E(X₂). Determine if the variance of X, is falling, remains constant or is growing in t.Consider the following utility function: u = 400 (1 + 1)−1. xy a. Is this utility function homogenous? Briefly explain. b. Use implicit differentiation to find the MRCS. c. Write down an expression for the indifference curve if u = 20.1 Question You are given: • Px+t = 0.9 for 0 ≤ t ≤ 3. • i = 8%. Calculate 2:31.
- Find the optimal solution for the following problem. Note: Round your answers to 3 decimal places. Maximize C = 13x + 9y subject to and 6x + 11y ≤ 18 16x + 21y ≤ 41 x ≥ 0, y ≥ 0. a. What is the optimal value of x? X b. What is the optimal value of y? c. What is the maximum value of the objective function?A:IV > partial derivative + V - L "dxdy Recall that for a general function fx), the integral feldx representa the difference of the area below the curve y = f(x) but above the r-axis when f(x) 2 0, and the area above the 104 CHAPTER 3. MULTIPLE INTEGRALS eurve but below the r-axis when f(x)s0. Similarly, the double integral of any continuous funetion f(x, y) represents the difference of the volume below the surface z=f(x, y) but above the xy-plane when f(x,y) 20, and the valume above the surface but below the xy-plane when flz,y) s0. Thus, our method of double integration by means of iterated integrals can be used to evaluate the double integral of any continuous funetion over a rectangle, regardless of whether fix, y)20 or not. Exumple 3.3. Evaluate [ sintx+ y)dxdy. Solution: Note that f(x,y)= sin(r+y) is both positive and negative over the rectangle [0,) (0,2n1 We can still evaluate the double integral: sin(x + yldzdy = (-conlx + y)dy (-cos(y+)+ cos y)dy - - sin(y+n)+ sin y. --sin3r…IRR is the rate at which; a.NPV is zero b.NPV is negative c.IRR = NPV d.NPV is positive
- If f'(x) = x5, what is f(x)? (Remember the constant of integration.) f(x) =Please show all work ง 5. Evaluate the vector surface integral JJ (+2², x + 2y + 32², 5x − 2y + 2³) . Ndo. S Here, S is the sphere Sy²+2 1. = Since S is a closed surface, the outside of S is considered as the positive side on S. Hint: use the divergence theorem.In order to find a local minimum or maximum of a function, how do we manipulate the fırst derivative? Make it equal to zero O Make it positive O Make it negative Make it equal to f(x)