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?

A consumer has a budget of $3000 in a given period, and wishes to buy two goods, X and Y, so as to maximise his utility. The price of good X is $5 and the price of good Y is $2, and his MRS is given by the formula 2Y/X. How many units of good X will he buy in that period?

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: - \( 3 \)

There are two goods in the economy, wine and cheese. The price of wine is p1 dollars per litre of wine. The price of cheese is p2 per kilogram of cheese. The consumer has wealth M dollars to spend on wine and cheese. Let x1 denote the quantity of wine in litres she chooses to buy and let x2 denote the quantity of cheese in kilograms she chooses to buy. Suppose throughout that she can only purchase non-negative quantities of either good. Solve the following utility maximization problems when the preferences of a consumer are characterized by the following utility functions. For each part clearly specify what is the quantity of each good the consumer wishes to purchase given her budget constraint. (a) (10 points) U (x1; x2) = x1 x 4 2 [Hint: MU1 (x1; x2) = x 4 2 and MU2 (x1; x2) = 4x1x 3 2 .] (b) (15 points) U (x1; x2) =  x1 + 2x2 ￾ x 2 2 if x2  1 x1 + 1 if x2 > 1 [Hint: MU1 (x1; x2) = 1 and MU2 (x1; x2) = 2 (1 ￾ x2). First work out the solution for the case in which x 1 > 0 and x 2 > 0. Then show if p2 > 2p1 it is optimal for our consumer to spend her entire wealth on wine.] (c) (5 points) U (x1; x2) = maxfx1; x2g That is, U (x1; x2) =  x1 if x1  x2 x2 if x1 < x2 . Warning: In this case MU1 (x1; x2) and MU2 (x1; x2) do not exist! [Hint: Reason from Örst principles how the individual should allocate her wealth between the two goods if her aim is to maximize a utility function of this form. To do this, I recommend drawing a diagram to graph the budget set for di§erent price ratios. Can you Ögure out which 2 bundle or bundles maximise her utility for each particular budget set? To complete your answer you should now be able to work out for general prices and income levels which bundle or bundles maximize her utility.]

A consumer has an income of $10,000 to spend on two goods, X and Y, where the price ofgood X is $500 per unit, and the price of good Y is $1 per unit. The marginal rate ofsubstitution is given by the formula Y/X. How many units of good Y does this consumeroptimally c

Assume that product Alpha and product Beta are both priced at $1 per unit and that Ellie has $20 to spend on Alpha and Beta. She buys 8 units of Alpha and 12 units of Beta. The marginal utilities of the last unit of Alpha and Beta that she purchases are 40 utils and 20 utils, respectively. This indicates thatMultiple ChoiceEllie should make no change in consumption.given another dollar, Ellie should buy an additional unit of Beta.in order to maximize utility, Ellie should buy more Beta and less Alpha.in order to maximize utility, Ellie should buy more Alpha and less Beta.