Economics question

3. Consider the function f(x, y) = (2 - }) defined on {(x, y) : 1, y > 0}. (a) Graph {(r, y) : x, y > 0, f(x, y) = -1}. (b) Find an equation of the hyperplane tangent to the graph of f(r, y) = = at the point (I, y, z) = (1, 1,0). (c) Decide whether or not f(.) is homogeneous. If f(.) is homogeneous, then determine its degree of homogeneity and explicitly verify Euler's Theorem. (d) Find the directional derivative of f in the direction w = (1, 1) at the point (r, y) = (.5, 1). (e) Let g(u, v) = (u' +v? + 1, u+ 1). Use the chain rule to compute all partial derivatives of fog when u = v = 0.6. An agent has utility function u(r) where r is income. The agent has initial wealth w. With probability p the agent suffers a loss of / dollars, reducing her wealth to w - l. The agent's expected utility is pu(w -1) + (1-p)u(w). The agent's certainty equivalent C is implicitly defined by the equation u(C) = pu(w - 1) + (1 - p)u(w). (1) (a) Show that if u is continuous, there exists a certainty equivalent. That is, there is a solution to equation (1). (b) Show that if u is strictly increasing, then there exists a unique certainty equivalent. That is, there is one and only one C that solves equation (1). (c) Suppose that for given values (po, wo, lo) equation (1) has a solution Co. State con- ditions on u and its derivatives under which you can locally solve equation (1) for C as a differentiable function C = g(p, w, /) in a neighborhood of (po, wo, ) with g(po, wo, lo) = Co. Write down a formula for Dy(po, wo. lo)- (d) State economically plausible conditions under which g is decreasing in p.Consider the initial-value problem y = y -x, y(0) = 3 Use Euler's method with (a) h = 0.1 and (b) h = 0.05 to obtain an approximation to y (1). Given that the exact solution to the initial-value problem is y ( x) = *+1-7ex. compare the errors in the two approximations to y (1).Apply the fourth-order Runge-Kutta method with h = 0.1 to determine an approximation to the solution to the initial-value problem below at x = 1: V = V - X, y(0) =\fExample 3 A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K . Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by de -2.2067 x10-12(0* -81x10*) e(0) = 1200K where o is in K and / in seconds. Find the temperature at / = 480 seconds using Euler's method. Assume a step size of h = 240 seconds.9.6. Consider the initial-value problem dy with y(O) = 13 dx - = (y - 1)2 10 This is the problem discussed in section 9.3 in the illustration of a "catastrophic failure" of Euler's method. a. Find the exact solution to this initial-value problem using methods developed in earlier chapters. What, in particular, is the exact value of y (4) ? b. Using your program/worksheet from exercise 9.3 a, find the numerical solution to the above initial-value problem with *max = 4 and step size Ax = /2. (Also, confirm that this numerical solution has been properly plotted in figure 9.3 on page 198.) c. Find the approximation to y(4) generated by Euler's method with each of the follow- ing step sizes (use your answer to the previous part or your program/worksheet from exercise 9.3 a). Also, compute the magnitude of the error in using this approximation for the exact value found in the first part of this exercise. i. Ax = 1 ii. Ax = 7 iii. Ax = 7 iv. Ax = 7 9.7. Consider the following initial-value problem dy dx = -4y with y(0) = 3 The following will illustrate the importance of choosing appropriate step sizes. a. Find the numerical solution using Euler's method with Ax = /2 and N being any large integer (this will be more easily done by hand than using calculator!). Then do the following: i. There will be a pattern to the ya's. What is that pattern? What happens as k -+ co ? ii. Plot the piecewise straight approximation corresponding to your numerical solution along with a slope field for the above differential equation. Using these plots, decide whether your numerical solution accurately describes the true solution, especially as x gets large. 210 Euler's Numerical Method iii. Solve the above initial-value problem exactly using methods developed in earlier chapters. What happens to y(x) as x - co ? Compare this behavior to that of your numerical solution. In particular, what is the approximate error in using y for y(xx) when Xx is large? b. Now find the numerical solution to the above initial-value problem using Euler's method with Ax = /10 and N being any large integer (do this by hand, looking for patterns in the yx's )). Then do the following: i. Find a relatively simple formula describing the pattern in the ya's. ii. Plot the piecewise straight approximation corresponding to this numerical solution along with a slope field for the above differential equation. Does this numerical solution appear to be significantly better (more accurate) than the one found in part 9.7 a? 9.8. In this problem we'll see one danger of blindly applying a numerical method to solve an initial-value problem. The initial-value problem is with y(0) = 0 a. Find the numerical solution to this using Euler's method with step size Ax = 1/2 and Xmax = 5. (Use your program/worksheet from exercise 9.3 a). b. Sketch the piecewise straight approximation corresponding to the numerical solution just found c. Sketch the slope field for this differential equation, and find the exact solution the above initial-value problem by simple integration. d. What happens in the true solution as x - /3 ? e. What can be said about the approximations to y(xx) obtained in the first part when * * > /3 ? 9.9. What goes wrong with attempting to find a numerical solution to (y - 1)43 dy dx - = 1 with y(0) =0 using Euler's method with, say, step size Ax = h