Solve the following utility maximization problem, и (х1, х2) max subject to P1x1 + p2x2 = I meaning find x" (p1, p2, I) and x (p1, p2, I) and u* (p1, p2, I) = V* . In solving this optimization problem, assume the following functional form for the direct utility function, u (x1, x2) = a · In(x1) + b· In(x2)
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- Consider the following utility functions. G(x,y) = x² + 3y² H(x,y) = (x+1)¹/2 + y + 1 L(x,y) = xey U(x,y) = In(x) + y² W(x,y) = 100 x - 4x² + 3 y Z(x,y) = (x)¹/² + y Which function or functions have strictly convex indifference curves (at every point where x>0 and y>0)?Now assume that the utility function takes the function form u(c) = ln(c). Note that the derivative of the natural logarthm is u'(c) = 1/c .3) Oğuz has the utility function U(x1,X2) = x:*x2². (X: nuts, X2: berries). %3D a) If Oğuz has 25 units of nuts and 17 units of berries, the price of nuts is 2 and the price of berries is 1 liras, what would be the optimal consumption of Oğuz? b) Assume the prices change, so that nuts cost 1 and berries cost 3 liras. What is the new demand? In this change, What is the pure substitution effect? What is the income effect? c) In the change calculated in part (b), what is the pure substitution effect? d) In the change calculated in part (b), what is the income effect? In the change calculated in part (b), what is the endowment effect?
- Solve the following utility maximization problem, max u (x1, x2) subject to Pix1 + p2x2 = I meaning find x (P1, P2, I) and x (p1, p2, I) and u* (p1, p2, I) = V * . In solving this optimization problem, assume the following functional form for the direct utility function, и (х1, X2) —а: · In(x1) + b· In(x2) where a and b are just parameters (i.e., representations of single numerical values). Assume that the Second-Order Conditions for a constrained maximum hold. Write clearly and show your work!Consider a consumer with the utility function U (x1, x2 ) = 10x12/3x21/3 −50. Suppose the prices of x1 and x2 are 10 and 2 respectively and the consumer has an income of 150. (a) Write out the consumer’s constrained optimization problem. Specifically, write out the objective function and constraint for the problem (e.g. max _?_ subject to _?__). (b) Write the Lagrangian equation corresponding to the constrained optimization problem. Derive the Necessary First Order Conditions. (c) Use the NFOCs to solve for the consumer’s optimal bundle. (d) Show that at the solution you found in (c), the tangency condition is satisfied: MRS = p1 / p2. (e) How did the ‘50’ in the utility function influence the optimal con- sumption bundle? How did the ‘10’ in the utility function influence the optimal consumption bundle? (i.e., how would the optimal bun- dle change if these coefficients were to change?). How would the optimal bundle change if the utility function was x12x2? Lastly, how would…For the following utility function find the MRS at (x, y) = (2, 3) a) U(x, y) =2x +y b) U(x, y) =x^3 y^2 c) U(x, y) =min(x, 2y) Solve subparts (c )correctly
- For the utility function (x^3)(z^2)Provide a positive linear transformation of the utility function, then, provide a nonlinear monotone transformation of the utility function.What is the logarithmic transform of the utility function U=xα1xβ2xγ3U=x1αx2βx3γ given the budget constraint px1x1+px2x2+px3x3=Mpx1x1+px2x2+px3x3=M. Select one: a. ln U=ln xα1+ln xβ2+ln xγ3ln U=ln x1α+ln x2β+ln x3γ b. ln U=α ln x1−β ln x2−γ ln x3ln U=α ln x1−β ln x2−γ ln x3 c. ln U=α ln x1+β ln x2+γ ln x3ln U=α ln x1+β ln x2+γ ln x3 d. ln U=ln xγ1+ln xβ2+ln xα33- Assuming that the equation F(U, x, X2 = f(x1, x2,, ... Xn): ******* **** ,xn) = 0 implicitly defines a utilityfunction U 073 a) Find the expressions for 60, 6U and 6x4 " 3 6x2 6xn 6x2 6xn b) Interpret their respective economic meanings. c) Now. assume the utility function is U(x, y) = y√x. Does the consumer believe that more is better for each good? Do the consumer's preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y diminishing?
- QUESTION 1 For the utility function U = Qx0.69Qy(1-0.69) find the trade-off rate between good X and good Y at Qx= 7 and Qy=11 Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 1.2, 1.14 = 1.1).Tom is vegetarian and always eat a salad (s) with 2 glasses of green juice (g). His utility for the two goods is therefore 1 u(s, g) = min{s, 79}Find Indirect utility function of the following function U = max (X, Y) subject to the budget constraint P, X+ P₂ Y=M Optionis O O max (P1 P2) M² min (P1 P2) M² P1+ P2 M min (P1 P2)