8. Slutsky Equation Exercises PDF

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8. INCOME AND SUBSTITUTION EFFECTS

Exercise 1. Slutsky (Cobb-Douglas) The utility function is u = x1x2, and the budget constraint is m = p1x1 + p2x2. a) Derive the optimal demand curve for good 1, x1(m,p1), and good 2, x2(m, p2). b) Assume m=160, p1=8 and p2=1. Based on your answer in part a, what is the optimal consumption bundle (x1,x2)? c) Suppose price of good 1 falls to p1’=2 (while m=160 and p2=1 still). Using your answer in part a, find the new optimal consumption bundle (x1M,x2M). In a graph with p1 in the vertical axis and x1 in the horizontal axis, plot your results from parts b and c. Draw the Marshallian demand curve for good 1. d) Consider your answer in part b as your endowment of good 1 and good 2. With new prices (p1’=2, p2=1), how much is this endowment worth? Define this amount as mS, and the change in income, Δm=m-mS, as the income “compensation” to keep purchasing power unchanged despite the change in prices. e) Using the new prices (p1’=2, p2=1), and the compensated income, mS, find the new optimal consumption bundle (x1S,x2S): that is x1S(mS,p1’) and x2S(mS,p2). In a graph with p1 in the vertical axis and x1 in the horizontal axis, plot your results from parts b and e. Draw the Slutsky demand curve for good 1. f) How does the Marshallian demand from part c compares to the Slutsky demand in part e? What is the substitution effect and the income effect for both good 1 and good 2? Exercise 2. Hicks (Cobb-Douglass) The utility function is u = x1x2, and the budget constraint is m = p1x1 + p2x2. a) Derive the optimal demand curve for good 1, x1(m,p1), and good 2, x2(m, p2). For m=160, p1=8 and p2=1, what is the optimal consumption bundle (x1,x2)? b) Changing only price of good 1 to p1’=2, find the new optimal consumption bundle (x1M,x2M). Plot the initial x1 you found in part a and the new x1M. Draw the Marshallian demand curve for good 1.

c) Define as uH the utility amount you get from consumption bundle in part a? Find an alternative consumption bundle (x1H,x2H) that gives you the same utility (uH=x1x2), but that satisfies the optimal condition MRS=p1’/p2 with the new prices (p1’=2 instead of p1=8). d) In a graph with p1 in the vertical axis and x1 in the horizontal axis, plot your result from part a and c. Draw the Hicks demand curve for good 1. e) How does the Marshallian demand from part b compares to the Hicks demand in part d? What is the substitution effect and the income effect for both good 1 and good 2?

Exercise 3. Slutsky (Quasilinear) The utility function is u = x1½ + x2, and the budget constraint is m = p1x1 + p2x2. a) Derive the optimal demand curve for good 1, x1(p1, p2), and good 2, x2(m, p1, p2). b) Looking at the cross price effects (∂x1/∂p2 and ∂x2/∂p1) are goods x1 and x2 substitutes or complements? Looking at income effects (∂x1/∂m and ∂x2/∂m) are goods x1 and x2 inferior, normal or neither? c) Assume m=100, p1=0.5 and p2=1. Using the demand function you derived in part a, what is the consumption of x1 and of x2? d) Consider a price drop for only good 1 to p1’=0.25. Calculate the new demands for good 1 and good 2 and label them x1M and x2M. Plot in a graph with p1 on the vertical axis the x1 you found in part c, and x 1M. Graph the resulting Marshallian demand curve. e) Assume you are endowed with the amounts x1 and x2 you found in part c. What is their total worth, given the new prices? Define this as mS = p1’x1 + p2x2. If mS was your original budget constraint and you had to maximize your utility, what would be the resulting optimal consumption x1S(p1’,p2) and x2S(mS,p1’,p2)? f) Plot the original x1 from part c and x1S from part e in a graph with p1 on the vertical axis. Draw the Slutsky demand curve. How does it differ from the Marshallian demand curve? What is the substitution effect and the income effect of this price change?

Exercise 4. Hicks (Quasilinear) The utility function is u = x1½ + x2, and the budget constraint is m = p1x1 + p2x2.

a) Derive the optimal demand curve for good 1, x1(p1, p2), and good 2, x2(m, p1, p2). When m=100, p1=0.5 and p2=1, what is the consumption of x1 and of x2? b) Consider a price drop for only good 1 to p1’=0.25. Calculate the new demands for good 1 and good 2 and label them x1M and x2M. Plot in a graph with p1 on the vertical axis the x1 you found in part a, and x 1M. Graph the resulting Marshallian demand curve. c) Define as uH the utility amount you get from consumption bundle in part a? Find an alternative consumption bundle (x1H,x2H) that gives you the same utility (uH=x1½+x2), but that satisfies the optimal condition MRS=p1’/p2 with the new prices (p1’=0.25 instead of p1=0.5). d) Plot the original x1 from part a, and x1H from part c in a graph with p1 on the vertical axis. Draw the Hicks demand curve. How does it differ from the Marshallian demand curve? What is the substitution effect and the income effect of this price change?

Exercise 5. Slutsky (Perfect Complements) The utility function is u = min(0.5x1,x2), and the budget constraint is m = p1x1 + p2x2. a) What are the demand functions x1(m,p1,p2) and x1(m,p1,p2)? For m=90, p1=4 and p2=1, what are the consumption amounts x1 and x2? b) Assume only p1 changes to p1’=1, define the new consumption values as x1M and x2M. c) Assume you are endowed with the amounts x1 and x2 you found in part a. What is their total worth, given the new prices? Define this as mS = p1’x1 + p2x2. If mS was your original budget constraint and you had to maximize your utility, what would be the resulting optimal consumption x1S(m,p1’,p2) and x2S(mS,p1’,p2)? a) Compare x1M from part b and x1S from part c. Do they differ? What is the substitution effect and the income effect of this price change? Exercise 6. Hicks (Perfect Complements) The utility function is u = min(0.5x1,x2), and the budget constraint is m = p1x1 + p2x2. a) What are the demand functions x1(m,p1,p2) and x1(m,p1,p2)? For m=90, p1=4 and p2=1, what are the consumption amounts x1 and x2?

b) Assume only p1 changes to p1’=1, define the new consumption values as x1M and x2M. c) Define as uH the utility amount you get from consumption bundle in part a. Find the consumption bundle (x1H,x2H) that gives you the same utility (uH=min(x1,x2)), but that satisfies the new budget line slope p1’/p2 with the new prices (p1’=1 instead of p1=4). d) Compare x1M from part b and x1H from part c. Do they differ? What is the substitution effect and the income effect of this price change?

Exercise 7. Slutsky (Perfect Substitutes) The utility function is u = 3x1 + x2, and the budget constraint is m = p1x1 + p2x2. a) What are the demand functions x1(m,p1,p2) and x1(m,p1,p2)? For m=100, p1=4 and p2=1, what are the consumption amounts x1 and x2? b) Assume only p1 changes to p1’=2, define the new consumption values as x1M and x2M. c) Assume you are endowed with the amounts x1 and x2 you found in part a. What is their total worth, given the new prices? Define this as mS = p1’x1 + p2x2. If mS was your original budget constraint and you had to maximize your utility, what would be the resulting optimal consumption x1S(m,p1’,p2) and x2S(mS,p1’,p2)? d) Compare x1M from part b and x1S from part c. Do they differ? What is the substitution effect and the income effect of this price change?

Exercise 8. Hicks (Perfect Substitutes) The utility function is u = 3x1 + x2, and the budget constraint is m = p1x1 + p2x2. a) What are the demand functions x1(m,p1,p2) and x1(m,p1,p2)? For m=100, p1=4 and p2=1, what are the consumption amounts x1 and x2? b) Assume only p1 changes to p1’=2, define the new consumption values as x1M and x2M. c) Define as uH the utility amount you get from consumption bundle in part a. Find the consumption bundle (x1H,x2H) that gives you the same utility (uH = 3x1 + x2), but that satisfies the new budget line slope p1’/p2 with the new prices (p1’=2 instead of p1=4). d) Compare x1M from part b and x1H from part c. Do they differ? What is the substitution effect and the income effect of this price change?

Exercise 9. Members Only Discounts (Slutsky vs. Hicks) Your store sells groceries, x1, and your typical customer has preferences between your groceries and all other goods, x2, according to the following utility: u = x1x2. Prices of both types of goods are normalized to one (p1=1 and p2=1). Your typical customer’s budget constraint is m=p1x1+p2x2, where available income is m=100. a) What share of income does your customer spend on your groceries (α=x1p1/m)? What is the optimal amount of x1O and x2O the customer will purchase? What utility level, uO, will be attained? What is your company’s revenue, RO=p1x1O? b) You are considering introducing a 10% price discount rate (d=0.1), so that the new budget constraint faced by your customer is m=p1(1-d)x1+p2x2 (or 100=0.9x1+x2). How would your answers in part a change? Label the new amounts with “M” and find the new optimal quantities x1M, x2M, the new utility uM, and your company’s new revenue RM=p1(1-d)x1M. c) You are considering introducing a lump-sum membership fee, S, as mandatory to access discounts. The budget constraint faced by your customer is then m-S= p1(1-d)x1+p2x2 (or 100-S=0.9x1+x2). You choose S so that you don’t alter your customer’s original purchasing power from part a: m-S=p1(1-d)x1O+p2x2O (or 100-S=0.9x1O+x2O). How will your customer react? Label the new amounts with “S” and find the new optimal quantities x 1S, x2S, the new utility uS, and your company’s new revenue, RS=S+p1(1-d)x1S. d) Consider an alternative membership fee, H, so that m-H= p1(1-d)x1+p2x2 (or 100H=0.9x1+x2). This time you choose H so that you don’t alter your customer’s original utility, uO, you found in part a. Find the new optimal quantities, x1H and x2H that satisfy MRS=p1(1-d)/p2, but that are constrained by uO=x1Hx2H. What is the new membership fee H? What is your company’s new revenue, RH=H+p1(1-d)x1H. e) Which scheme is better for your revenues? Choose between RO, RM, RS, and RH. f) What about your other customers? They all have the same utility (u = x 1x2), but some are richer (m>100) and some are poorer (m...