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: - \( 3 \)

To find the Lagrange multiplier associated with the equality constraint for the given utility maximization problem, 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 \( y \)**: \[ y = 5x = 5 \times 3 = 15 \] 6. **Find the Lagrange multiplier \( \lambda \)**: From the equation \( \lambda = 5x \): \[ \lambda = 5 \times 3 = 15 \] So, the Lagrange multiplier associated with the equality constraint is \( \lambda = 15 \). The correct answer is: - \( 15 \)

A consumer spends an amount 𝑚𝑚 to buy 𝑥𝑥 units of one good at the price of 6 per unit and 𝑥𝑥 units of a different good at the price of 10 per unit. Here 𝑚𝑚 is positive and suppose that 8 < 𝑚𝑚 < 40. The consumer’s utility function is 𝑈(𝑥, y) = 𝑥y + y^2+ 2𝑥 + 2y, so that her problem is: Maximize 𝑥y+y^2 +2x+2y subject to 6𝑥+10𝑥y=𝑚 (a) Find the optimal quantities 𝑥∗ and y∗ and the Lagrange multiplier, all of them as functions of 𝑚. (b) Write down the maximum value of the utility function as a function of m. (c) Find the derivative of the above utility function for m=20 (d) What are the solutions for 𝑥∗ and y∗ if (i) 𝑚 ≤ 8? (ii) 𝑚 ≥ 40?

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(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)

1. Lagrange multiplier as shadow price of income. Consider a consumer withutility function u (x1; x2) = x1x2. She has income M and faces prices p1; p2:Using Lagrange method derive her optimal consumption bundle (x1; x2) andthe level of utility at the optimum u(x1; x2): This is a function of (p1; p2; M ) :Show that du(x1; x2)=dM = ; the value of the Lagrange multiplier.2. Exercise 2.2 from Lengwiler.3. Exercise 2.3 from Lengwiler.