Consider the following budget constraint. \[ \mathrm{I}=\mathrm{PB}^{*} \mathrm{~B}^{*}+\mathrm{PT}^{*}{ }^{*} \mathrm{~T} \] and Utility Function \[ U(x, y)=\left(6 T^{5}+4 B^{5}\right)^{1 / 5} \] Where

[ U=q_{\frac{1}{2}}^{\frac{1}{2}} q_{v}^{\frac{1}{2}} . \] We showed earlier in the course that maximizing this utility subject to a budget con-straint yields the result \[ \frac{q_{y}}{q_{x}}=\frac{P_{x}}{P_{y}} \] You may use this result below. This world also has a production possibilities frontier (PPF) whose equation is \[ 800=q_{x}^{2}+4 q_{v}^{2} \] (a) Calculate the Marginal Rate of Transformation (MRT). (b) Give the equation relating MRT and prices. (c) Calculateq z​ ,q y​ andP y​ P z​ ​ . (d) Draw the PPF, labelling three points with numerical values. Arithmetic hint:800​ =28.3, 200​ =14.1.) (e) Show on the graph how the price ratioP y​ P z​ ​ is determined. Please add appropriate labels.Skip questionStart SolvingExitExitQnA

To solve the utility maximization problem given the utility function \( u(x, y) = 5xy \), the budget constraint \( 5x + y = 30 \), and non-negative consumption of goods \( x \) and \( y \), follow these steps: 1. **Set up the Lagrangian**: \[ \mathcal{L}(x, y, \lambda) = 5xy + \lambda (30 - 5x - y) \] 2. **Find the partial derivatives and set them to zero**: \[ \frac{\partial \mathcal{L}}{\partial x} = 5y - 5\lambda = 0 \quad \Rightarrow \quad y = \lambda \] \[ \frac{\partial \mathcal{L}}{\partial y} = 5x - \lambda = 0 \quad \Rightarrow \quad \lambda = 5x \] \[ \frac{\partial \mathcal{L}}{\partial \lambda} = 30 - 5x - y = 0 \quad \Rightarrow \quad y = 30 - 5x \] 3. **Substitute \( \lambda \) from the first equation into the second equation**: \[ y = 5x \] 4. **Substitute \( y = 5x \) into the budget constraint**: \[ 5x + 5x = 30 \] \[ 10x = 30 \] \[ x = 3 \] 5. **Find the corresponding \( y \)**: \[ y = 5x = 5 \times 3 = 15 \] So, the x-coordinate of the point that solves this individual's utility maximization problem is \( x = 3 \). The correct answer is: - \( 3 \)

A consumer has income equal to $1,000, to spend on two goods, X and Y, where price ofgood X is $20 per unit, and price of good Y is $50 per unit. The X-intercept of the consumer'sbudget constraint has __ units and the Y-intercept of the consumer's budget constraint has__ units.

U(x, y) = (3x + 2y)2 .The price of x is px = $10 per unit and his income is $200.(a) Obtain the equation of Johnathan’s indifference curve for the utility level U = 100. Drawthis indifference curve. (2 marks)(b) The price of y is py = $8 per unit. Obtain the marginal rate of substitution (MRS) and theequation of the budget line. Using a graph, find Johnathan’s optimal consumption bundle.In this graph, show the budget line, the optimal bundle, and the corresponding indifferencecurve. Make sure to label carefully all the curves. (3 marks)(c) Suppose that the price of y drops to py = $6 per unit (the price of x remains the same,at px = $10 per unit, and the income remains the same). Obtain the equation of the newbudget line. Using a new graph, find Johnathan’s optimal bundle with this new price. Inthis graph, show Johnathan’s new budget line, new optimal bundle, and the correspondingindifference curve. Make sure to label carefully all the curves. (3 marks)(d) Now suppose that the price of y is py = $8 per unit if Johnathan buys less than 10 units ofthis product, and py = $6 per unit if he buys 10 units of y or more (as an example, 20 unitsof y would cost $120). Assume that the price of x remains the same, at px = $10 per unit.Derive the equation of the budget line and draw it in a separate graph. (3 marks)(e) Using a new graph, find the optimal bundle(s) for the problem in part (d). In this new graph,show the budget line, the optimal bundle and corresponding indifference curve. Make sureto label carefully all the curves. (3 marks)

Consider the following utility functions (in a world with Good X andGood Y ):I. U (x, y) = √xyII. U (x, y) = 4x + 3yIII. U (x, y) = min{x, y} (That is, U (4, 3) = 3, U (1, 1) = 1, U (2, 3) = 2, etc.)IV. U (x, y) = √x + y(a) Fill out the last 3 columns of Table 1. “MRS” stands for Marginal Rate of Substi-tution here. (You would need to copy this table into your answer.)(b) Fill out the the first 4 columns of Table 1 with a Yes/No entry for each cell. Justifyyour answers.(c) Do all utility functions display diminishing MRS? Justify your answers.Note: For the utility function U (x, y) = min{x, y}, calculus cannot be used. How-ever, think about the concept of MRS as how much of y you are willing to give upfor a bit more of x, as represented by the steepness of the indifference curve at apoint. Can you find some value for MRS in this case?(d) Sketch an indifference curve for each of the above utility functions.Table 1. Question 1: Properties of Common Utility FunctionsMonotone Strongly Monotone Convex Strictly Convex MU x MU y MRSI. No 12 x 12 y − 12II. Yes −4/3III. YesIV. Yes Yes1