Suppose the consumer solves the following UMP: max (x1)^2 + (x2)^2 , s.t. p1x1 + p2x2 ≤ w where p1,p2 > 0. a) Plot the indifference curves b) Find the Marshallian demand functions. Show graphically the utility maximizing choices.
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Suppose the consumer solves the following UMP: max (x1)^2 + (x2)^2 , s.t. p1x1 + p2x2 ≤ w where p1,p2 > 0. a) Plot the indifference
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- Which of the following utility functions represent preferences which do not satisfy monotonicity? 1 U= - x+y 1 U= - 5ху Ou=xy O u= yFor each of the following utility functions draw the indifference curve that passes through (1,1). Label at least three points on each curve and indicate the direction of increased preference: a) u(x1, x2 ) = 3x1 b) u(x1, x2 ) = x1 + 2x2 c) u(x1, x2 ) = x1+ logx2 d) u(x1, x2 ) = min (2x1, x2) e) u(x1, х2) %3D max (1, x2)Individual that consumes two goods (X and Y) and has a CES Utility Function of the form: U = 100(X^(0.75) + Y^(0.75)). Income of 1000, the price of Good X is 10 and the Price of Good Y is 20 a) Find the Marginal Rate of Substitution as a function of the quantities consumed of Good X and Good Y. b) Write out the Lagrangian for this problem. c) Solve to find the demand for Good X, the demand for Good Y, and the highest level of utility for this individual. d) Now consider an increase in the price of Good X to 20. What is the demand for Good X and Good Y? What is the Utility of the consumer following the price change? e) Considering the change in demand for each good between parts c) and d), how much is due to the substitution effect and how much is due to the income effect? f) Show your answers on a graph.
- Kreese derives utility from two types of goods: cigars (good 1) and sleeveless shirts (good 2). A consumption bundle is a pair (r1, x2), where r, > 0 is the quantity of cigars and r2 2 0 is the quantity of sleeveless shirts (for simplicity, assume these goods can be measured in arbitrary-not just integer-quantities). (a) Suppose Kreese's preferences are monotone (more is better) and that he is willing to give up one sleeveless shirt in exchange for three cigars. Find a utility function representing his preferences. (b) Through trial and error, Kreese has discovered that there are no benefits to smoking more than 10 cigars per day. When smoking less than 10 cigars he is still willing to exchange one shirt for three cigars, but if he smokes more than ten cigars, there is no additional benefit and he only gains by owning more shirts. Find a utility function describing these preferences.For the utility function U = Qx0.28Q (1-0.28) and the budget 137 = 11Qx+6Qy find the CHANGE in optimal consumption of X if the price of X increases by a factor of 1.6. 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).Consider the following utility function: u(x1,x2) = x1 + x2. (A): i) Restate the consumer problem. ii) Form the Lagrangian function. iii) Find the first-order-conditions (FOC). iv) Divide Lx, by Lx,- v) From iv), can you determine the type of goods 1 and 2 are? (В): Draw the indifference curves for U = X1 + x2, where U = 1 and 2. What happens to the optimal demand for good 1, when: P1 P2 1. 3. Pi = P2
- Anna has an income of 1000 euros and utility function U(x,y)=x²y³. The prices of the two goods are P₁=5 and Py-20. a) Define the budget constraint and represent it graphically. (Suppose an individual has a utility function u (x1, x2) = 2*2. Present your mathmatical expressions below in the simplest form you can. a) Derive an expression for the marginal utility of good 1, and for the marginal utility of good 2. b) Using these, solve for an expression describing the slope of an indifference curve: MRS (11, 12). c) Sketch indifference curves for this consumer corresponding to u = 0,10,20. (Hint: z rı = k solves for m} = 11 (21) . Solve this expression and approximate it on a graph for the three values of k.)Find the marginal utility (MU) and marginal rate of substitution (MRS) of following utility functions. 1. U(x1,x2)=ln(x1)+x2,findMUx1,MUx2,andMRS(x1,x2) 2. U(x1,x2)=(2x21+x32)1/2,findMUx1,MUx2,andMRS(x1,x2)
- 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 xα3On what portion of the demand function does total revenue rise as price rises? The price of tea cakes goes up by 1%. As a result, the quantity purchased of tea falls by 1.5%. If income rises and consumers buy more of a particular good, that good is? Assume the budget constraint and the indifference curves are both linear. Assume the consumer is willing to tradeoff 1 of good X for 1 of good Y. If the relative price of one additional good X is giving up 1/2 of good Y, then the optimal bundle of the two goods is? What condition describes the optimal bundle of goods X and Y where utility is maximized [M=income, MU=marginal utility, P=price]Assume a consumer with the utility functionU(X,Y)=X²Y²and the typical budget constraintM= PxX+ PyY a. Derive the consumer’s demand for X and Y in terms of the parameters.b. If income of the consumer is 100, price of X is 2 and price of Y is 4, what the quantities of X and Y that gives the consumer maximum satisfaction?c. What is the consumer’s marginal rate of substitution between X and Y
