1. Consider utility function Ux1,x2=x,0.5x20.5 for x1<0 and x2<0.Show that the function satisfies (i) twice differentiability, (ii) non-satiation, (iii) strict convexity.
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- 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 xα32. Supose that a consumer has utility function U with U (₁,22) = x1426, where ₁ and ₂ are demand of commodity 1 and 2. Suppose that the price of commodity 1 and 2 is $3 and $5, respectively. This consumer has money m = 12 to spend and wants to maximize utility with constraint the money. (a) Write the Lagrangian function. (b) Find the critical point using first order condition. (c) Find the Hessian matrix to check whether the critical point is maximiser or minimiser. (d) Find the maximum utility.4. Consider utility function Ux1,x2=x10.5x20.5 for x1<0 and x2<0 .Show that the function satisfies (i) twice differentiability, (ii) non-satiation, (iii) strict convexity.
- What is the logarithmic transform of the utility function U=xα1xβ2xγ3 given the budget constraint px1x1+px2x2+px3x3=M . Select one: a. ln U=ln xα1+ln xβ2+ln xγ3 b. ln U=ln xγ1+ln xβ2+ln xα3 c. ln U=α ln x1−β ln x2−γ ln x3 d. ln U=α ln x1+β ln x2+γ ln x3given this utility function 95x10.3x20.2x30.25 first the first, second and cross order patial derivative with respect to each of the derivatives x1 x2 x3 what is the total utility derive if x1=24 x2=15 x3=5ASAP 1. Consider a consumer with utility u(x1,x2) = .5lnx1 + .5lnx2 (b) Now, consider an equivalent representation of the above utility functionx51 x52 Does this utility function have the "increasing difference property"? Howabout the "strict increasing difference" property?
- 1. Joan is on summer break and spends most of her time either playing video games or browsing the internet. Her utility function is U (q1, 42) 399.24$, where qi represents the 0.2,0.8 %3D hours spent browsing the internet and q2 hours playing video games. Joan does not have a budget constraint but she has 15 hours to spend on both activities. (a) Write the equation for one of Joan's indifference curves when the level of utility is equal to 100. (b) Write Joan's time constraint. (c) Find the amount of hours that Joan will spend on each activity that maximizes her utility and satisfies her time constraint. Use either the substitution or Lagrangian method. (d) Can Joan afford any bundle that yields a utility of 50? Explain. (e) How does Joan's optimum amount of time spent on each activity change if her utility function is U (q1, q2) = 5ln(3) + In(q1) + 4ln(q2)?2. Consider a consumer whose preference relation over the consumption set X = R can be repre- sented by the utility function: u (x1, x2, 13) = A (x1 – T1)ª (x2 – ã2)° (x3 – ã3)" - where A, a, B and y are all strictly positive. (a) Why can you assume A = 1 and a + B+y= 1 without loss of generality? Do so for the rest of the question. (b) Solve the UMP and derive the consumer's Walrasian demand. (c) Find the indirect utility function, the minimum expenditure function, and the Hicksian demand function.Graphical and Intuitive Optimization Consider a consumer who must allocate their available income between the consumption of two goods, Beer and Pizza. Their preferences over Beer and Pizza adhere to the usual assumptions we have made about preferences. The indifference curve map below reflects their preferences and the budget constraint reflects current income and prices. In the simplest terms possible, define or explain the term "Marginal Utility of Beer" (MUB). In the simplest terms possible, define or explain the term "Marginal Rate of Substitution of beer for pizza" (MRSBP). In the simplest terms possible, intuitively explain the meaning of MUB/PB, where PB represents the price of one unit of Beer.
- 1. For each of the three utility functions below, find the substitution effect, the income effect, and the total effect that result when prices change from p = (2,1) to p' = (2,4). Assume the consumer has income I = 20. (a) Before doing any calculation, make an educated guess about the relative magnitude of the three substitution effects and the three income effects to be found below. (b) u(x1, x2) = x1 + x2. (c) u(x1, x2) = x1x2. (d) u(x1,x2) = min {x1,x2). (e) Rank the substitution effects and the income effects found above by their magnitude. To what extent do they conform to your guess?5. Consider a consumer that seeks to minimize his expenditure E to achieve a given 1/41/4 level of utilityŪ. Assume that E = p₁x₁ + p₂x²; Ū=x^x^; and p₁ and p₂ are given. a) Set up the Lagrangian. b) Show the first-order conditions or minimization. c) Derive the expressions for the optimal levels of x, and x₂. d) Using the second-order conditions, verify if the solution generates a minimum value for E (Use H₂ to verify).[Indifference Curves] Consider the utility function U(C1, C2) and B 1 In(C1)BIn(C2) 1. Using a program of your choice (say excel or matlab) plot indifference curves in the space (C1, C2) for U =-0.5, -1, and -1.5. Consider values of Cin the interval [0.1, 1]. Set the range of the vertical axis to [0, 2]. 2. Find an analytical expression for the slope of the indifference curve 3. Find the slope of indifference curve when C1 is 0.2 and 0.8. Which one is steeper? Explain