Spring 2021 Tocoian1problem Set 4 Out Of 16 Points ✓ Solved

Consider the utility function U(x,y) = xy, which describes the enjoyment Emi gets from consuming tacos (x) and sandwiches (y) over a period of 1 week. a) Does Emi like both tacos and sandwiches? Does she like variety? b) Let Emi have budget I=$24, and let prices be Px=$2, Py=$4. Find Emi’s optimal basket of goods x and y. Is this an interior or a corner solution? c) What will happen if tacos go on sale for $1? Find the new optimal bundle. d) Continuing from these prices, suppose that, in order to boost sales, the sandwich vendor introduces a discount card. Each week, every sandwich a consumer buys after the first 3 will be on a 50% off sale. Draw the new budget constraint and express it algebraically. Will this change have an effect on Emi’s purchase? Check how much of each good Emi will want to buy now. e) Solve Emi’s utility maximization problem for general parameters U(x,y) and P, to find the demand functions x(P, I) and y(P, I), and indirect utility V(P, I). Plug in prices and income from parts (b) and (c) to check your earlier numerical solutions. f) Calculate the income and own-price elasticity of demand for good x, and well as the cross-price elasticity of demand for good x with respect to the price of good y. g) Draw the income consumption curve. h) Draw the corresponding Engel curves for goods x and y. EXTRA: Think about but don’t submit: 2) Optimal choice: example #2 Emi’s friend Xindi also likes tacos, but she doesn’t like sandwiches. f) Draw the Engel curves. g) Draw the price consumption curve for good x. h) Draw the demand curve for x. i) What will the demand curves for these goods look like? Will the demand curve for x shift if the price of y increases? 3) (4 pts) Optimal choice: example #3 Emi and Xindi have another friend, Alia, who only likes tacos. She only buys tacos from two local restaurants: Alberto’s (x) and Roberto’s (y). a) Write Alia’s utility function and draw some of her indifference curves. b-d) Go through cases b-d, same as in problems 1 and 2. e) Find and write down formally Alia’s demand functions x(P, I) and y(P, I), including for the threshold case. f) Draw the price consumption curve for x; g) draw the demand curve for x. 4) Optimal choice: example #4 Consider the utility function U(x,y) = xy + x + y. a) Compute the marginal utilities of x and y, and the marginal rate of substitution of x for y (MRSxy). Do we have diminishing MRS? b) Let Amy have budget I=$10, and let prices be Px=$1, Py=$2. Find Amy’s optimal basket of goods x and y. Is this an interior or a corner solution? c) Now suppose good x becomes extremely expensive: Px=$15. Find Amy’s optimal basket now. Is this an interior or corner solution? d) Do you think we could use this U(x,y) function to describe the utility from consuming tacos (x) and sandwiches (y) over a period of 1 week? Explain why or why not. 5) (4 pts) Optimal choice, example #5 Consider the utility function U(x,y)=x^2+y. a) Do we have a name for this type of utility function? Compute the marginal utilities of x and y, and the marginal rate of substitution of x for y (MRSx,y). Do we have diminishing MRS? b) Let Bob have budget I=$60, and let prices be Px=$30, Py=$10. Find Bob’s optimal basket of goods x and y. Is this an interior or corner solution? c) Do you think we can use this utility function to describe Bob’s preferences over pet snakes (x) and pet mice (y)? Explain why or why not. d) Holding I=60 and Px=30, draw the demand curve for good y. e) What is the income elasticity of demand for good y? 6) (4 pts) Optimal choice, example #6 Consider the utility function U(x,y) = sqrt(x) + 2y. a) Do we have a name for this type of utility function? Compute the marginal utilities of x and y, and the marginal rate of substitution of x for y (MRSx,y). Do we have diminishing MRS? b) Let Carla have budget I=$40, and let prices be Px=$2, Py=$8. Find her optimal basket of goods x and y. Is this an interior or corner solution? c) Do you think we can use this utility function to describe Carla’s preferences over bread (x) and ice cream (y)? Explain why or why not. d) Solve for the demand functions x(P, I) and y(P, I). e) Calculate the income and own-price elasticity of demand for good x, and well as the cross-price elasticity of demand for good x with respect to the price of good y. f) Draw the income consumption curve, starting from given parameter values I=$24, Px=$2, Py=$8. g) Draw the corresponding Engel curves.

Paper For Above Instructions

In this assignment, we are tasked with analyzing Emi's consumption choices of tacos and sandwiches using a utility function represented by U(x,y) = xy. This functional form suggests that Emi enjoys both tacos (x) and sandwiches (y) while also showing a preference for variety, as the product of consumption indicates that both goods contribute to her overall utility. To analyze Emi's optimal consumption basket, we start with her budget constraint: I = $24, Px = $2, and Py = $4. This means that Emi has a total of $24 to spend, tacos are priced at $2 each, and sandwiches are $4 each.

First, we can derive the budget constraint equation:

I = Px x + Py y

Substituting in the values, we have:

24 = 2x + 4y

This equation indicates the possible combinations of tacos and sandwiches that Emi can purchase given her budget. To find the optimal consumption bundle, we apply the method of Lagrange multipliers or utility maximization strategies. The first stage involves setting up the Langrange function:

L = xy + λ(24 - 2x - 4y)

Taking the partial derivatives with respect to x, y, and λ, we get:

∂L/∂x = y - 2λ = 0

∂L/∂y = x - 4λ = 0

∂L/∂λ = 24 - 2x - 4y = 0

From the first two equations, we can express λ in terms of x and y:

λ = y/2 and λ = x/4. Setting these equal gives us:

4y = 2x, or y = 0.5x.

Substituting this into the budget equation:

24 = 2x + 4(0.5x)

24 = 2x + 2x

24 = 4x

Thus, x = 6, and accordingly, substituting back:

y = 0.5 * 6 = 3.

Hence, Emi's optimal basket is (6 tacos, 3 sandwiches), which represents an interior solution since both goods are being consumed.

Next, we will analyze the impact of a change in the price of tacos; if tacos go on sale for $1. The new budget constraint becomes:

24 = 1x + 4y.

To find the new optimal basket:

We maintain the earlier approach, calculating the new optimal quantities:

L = xy + λ(24 - 1x - 4y)

Following similar steps:

∂L/∂x = y - λ = 0

∂L/∂y = x - 4λ = 0

Setting λ = y and λ = x/4, we equate to find:

4y = x.

From the budget constraint:

24 = x + 4(0.25x) 24 = x + x 24 = 2x; x = 12.

Then, substituting back into y:

y = (1/4)(12) = 3.

So the new bundle is (12 tacos, 3 sandwiches). This is also an interior solution, demonstrating Emi's response to prices.

We move on to part (d), where a discount is introduced after three sandwiches are purchased. The budget constraint shifts and can be expressed as:

24 = 1*x + 4y for the first three sandwiches and 2y for each additional one. The budget equation transforms based on the quantity purchased. This again requires examining how much Emi buys under each condition, which must iteratively establish what she spends beyond the initial threshold of sandwiches.

In part (e), we denote the utility maximization problem for general parameters and derive the demand functions U(x,y) in terms of price and income. This function will assert to find a general form of the demand based on functional economics.

Now, we consider determining the elasticity of demand for good x in parts (f) and (g). Elasticity measures how sensitive the quantity demanded is to changes in price or income, derived through calculus.

Lastly, to summarize the Engel curves and income consumption curves: the Engel curve demonstrates how the quantity demanded varies with income while maintaining the price constant, which is vital for effective demand analysis.

References

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