Task.2.

2) If the price of automobiles were to increase substantially, the demand curve for gasoline would most likely

A) shift leftward. B) shift rightward.

C) become flatter. D) become steeper.

3) If the price of automobiles were to decrease substantially, the demand curve for automobiles would most likely

A) shift rightward. B) shift leftward.

C) remain unchanged. D) become steeper.

4) Suppose a market were currently at equilibrium. A rightward shift of the demand curve would cause

A) an increase in price but a decrease in quantity.

B) a decrease in price but an increase in quantity.

C) an increase in both price and quantity.

D) a decrease in both price and quantity.

5. Draw out examples of each of the following indifference curves: imperfect substitutes, perfect substitutes, and perfect complements.

6) If two bundles are on the same indifference curve, then

A) the consumer derives the same level of utility from each.

B) the consumer derives the same level of ordinal utility from each but not the same level of cardinal utility.

C) no comparison can be made between the two bundles since utility cannot really be measured.

D) the MRS between the two bundles equals one.

7) The consumer is in equilibrium when

A) MRT = MRS.

B) Px/Py = MUx/MUy.

C) the budget line is tangent to the indifference curve at the bundle chosen.

D) All of the above.

8. Jody enjoys having exactly 1 teaspoon of sugar with every cup of coffee she has. What does this say about her indifference curves between the two goods? What happens to her utility level when she is given 5 teaspoons of sugar with one coffee? (Just an explanation)

9) Jay's Utility function is given by U(x,z) = 3x10.2 x20.8 and P1=$2 and P2=$4 and his budget is $200.

Write out the Lagrange but don't solve it

Find the utility maximizing values of x1 and x2

10)What does the substitution effect cause a consumer to do if the price of a good increases?

11) What does the income effect cause a consumer to do if the price of a good increases? What else is needed here?

12) What can we say about the substitution and income effects on a decrease in wages? What is an Engel curve and how does it relate here? If leisure is viewed as an inferior good, how would the labor demand curve look?

13) Suppose m=120, p1= 5, and p2=6.

What is the budget equation?

What is the slope of the budget line?

How does the budget set change with the following:

a 20% increase in m

a 2$ decrease in p2.

Task.3.

Directions. Find all the problems. Homework is due Monday, February 2,2021 at 5pm.1.Budget sets.Say we have 2 goods, and that the absolute price of good 1 is10, and of good 2 is 20 (so the absolute price vector isP= (P1;P2) = (10;20),and incomemis 100).a. Dene the consumption set, and then plot the budget set at thisP:b. In class, I discussed the "set inclusion" ordering on the subsets of the con-sumption setC=Rn+:Show in the above setting the budget sets get "smaller"under set inclusion assuming either component ofPincreases, ormdecreases.c. Show that the imposition of positive sale tax of good 1 (not good 2) hasthe same impact as a rise in theP1:d. Say the price of good 1 increases from 10 to 20 whenever more than 1unit of good 1 is purchases. Draw the new budget set, and show its convex. Isthe new budge set strictly convex? Explain.2.Preferences. Letdenote the consumers preference relation onC=R2+:Answer the following:a. Sayis reexive, complete, but not transitive. Show that the consumerspreferences could "cycle" (i.e., if forj= 1;2;3;:::;n;and consumption bundlesxnwe could havexjxj1andx0xn:b. Sayis reexive, complete, and transitive.(i) Can indierence curves "cross"?(ii) If so, what additional assumption on preferences rules this out. Also,provide a detailed argument as to why this assumption indeed does rule outcrossing indierence curves.(iii) Show the consumer cannot "cycle" (i.a., part (a) cannot happen in thiscase).(iv) Show that under "strictly monotonic" preferences, indierence curvescannot be "thick".3.Convex Preferences and optimal solutions. Letdenote the consumerspreference relation onC=R2+:We say a preference relationis convex (re-spectively, strictly convex) if for any two bundlesxandysuch thatx~y(i.e.,xandyindierent), then for any2[0;1](respectively,2(0;1)), andz=x+ (1)y; zx~y(respectively,zx~y):We say a preference relationis continuous if the two sets: weakly less preferred:WLP(x) =fy2Cjx1

y;x2Cgand weakly preferred:WP(x) =fy2Cjyx; x2Cgare "closed"(i.e., contain their boundaries. See discussion in class.Answer the following questions. Let the consumption set beC=R2+:(a) Show ifis convex,WP(x)convex.(b) Show ifis strictly convex,WP(x)is strictly convex.(c) IsWLP(x)convex?Consider a consumer facing a budget setB(p;m) =fx2Cjpxmgforp >>0:Dene the best choice setX(p:m) =fx2Cjxxfor allx2B(p;m)g(d) Show ifis convex,X(p;m)might have many elements (i.e., manyoptimal demand choices).(e) Show ifis strictly convex,X(p;m)is a unique for each price-incomepair.4. Say we have a utility functionu(x) =x1x12for2(0;1):(a) Construct the Marginal rate of substitution.(b) Discuss how the Marginal rate of substitution is related to the slope ofan indierence curve at a pointx >>0(i.e., each component ofxis strictlypositive).5. Answer the following:(i) Why is a utility function considered to be an "ordinal" concept?(ii) Show that ifu(x)represents a consumers preference relation, any^u(x) =10u(x)represents that same utility function.(iii) In question 4, of show that ifu(x1;x2)=x1x(1)2for2(0;1);theMRS between the two goods for^u(x)is not impacted by this strictly increasingtransformation.(iv) Actually, show ifh(y) :R!Ris a strictly increasing continuouslydierentiable transformation,^u(x) =h(u(x))represents that same preferencesasu(x).(v) Show in part (iv) that the MRS at the same for both^u(x)andu(x)assumingx >>0(all components ofxare strictly positive). Can you showactually for allx0;the preferences are the same for^u(x)andu(x)? (hint:answer is yes, but do not use the MRS).(vi) Leth(y) = lny:Forx >>0;show the MRS is the same for both^u(x)andu(x)ifu(x1;x2)=x1x(1)2for2(0;1):.

Task.4.

1. Budget sets. Say we have 2 goods, and that the absolute price of good 1 is 10, and of good 2 is 20 (say P = (P1;P2) = (10;20), and income m is 100. a. Dene the consumption set, and then plot the budget set at this P: b. Show the budget set gets "smaller" under set inclusion assuming either component of P increases, or m decreases. c. Show that the imposition of positive sale tax of good 1 (not good 2) has the same impact as a rise in the P1: d. Say the price of good 1 increases from 10 to 20 whenever more than 1 unit of good 1 is purchases. Draw the new budget set, and show its convex. Is it strictly convex? Explain.

2. Preferences. Let denote the consumers preference relation on C =Rn +: Answer the following: a. Sayis reexive, complete, but not transitive. Show that the consumers preferences could "cycle" (i.e., if for j = 1;2;3;:::;n; and consumption bundles xn we could have xj xj1 and x0 xn: b. Say is reexive, complete, and transitive. (i) Can indierence curves "cross"? (ii) what additional assumption rules this out. Show also that this assumption indeed does rule out crossing indierence curves. (iii) Show the consumer cannot "cycle" (i.a., part (a) cannot happen). (iv) Show that under "strictly monotonic" preferences, indierence curves cannot be "thick".

3. Convex Preferences and optimal solutions. Let denote the consumers preference relation on C =Rn +: We say a preference relation is convex (re-spectively, strictly convex) if for any two bundles x and y such that x~y (i.e., x and y indierent), then for any 2 [0;1] (respectively, 2 (0;1)), and z = x+(1)y; z x~y (respectively, z x~y): We say a preference relation is continuous if the two sets: weakly less preferred: WLP(x) = fy 2 Cjx y;x 2 Cg and weakly preferred: WP(x) = fy 2 Cjy x; x 2 Cg are "closed" (i.e., contain their boundaries. See discussion in class. Answer the following questions. Let the consumption set be C =Rn +: (a) Show if is convex, WP(x) convex. (b) Show if is strictly convex, WP(x) is strictly convex. (c) Is WLP(x) convex?

Consider a consumer facing a budget set B(p;m) = fx 2 Cjp x mgfor p >> 0: Dene the best choice set X(p:m) = fx 2 Cjx x for allx 2 B(p;m)g (d) Show if is convex, X(p;m) might have many elements (i.e., many optimal choices. (e) Show if is strictly convex, X(p;m) is a unique element.

4. Say we have a utility function u(x) = x 1 x1 2 for 2 (0;1):(a) Construct the Marginal rate of substitution. (b) Discuss how the Marginal rate of substitute in related to the slope of an indierence curve at a point x >> 0 (i.e., each component of x is strictly positive).

5. Answer the following: (i) Why is a utility function considered to be an "ordinal" concept? (ii) Show that if u(x) represents a consumers preference relation, any ^ u(x) = 10u(x) represents that same utility function. (iii) In question 4, of show that if u(x1;x2)=x 1 x(1) 2 for 2 (0;1); theMRS between the two goods for ^ u(x) is not impacted by this strictly increasing transformation. (iv) Actually, show if h(y) : R ! R is a strictly increasing continuously dierentiable transformation, ^ u(x) = h(u(x)) represents that same preferences as u(x). (v) Show in part (iv) that the MRS at the same for both ^ u(x) and u(x): (vi) Show in part (v) that the MRS is the same if (vii) let h(y) = lny: Show the MRS is the same for both ^ u(x) and u(x) if u(x1;x2)=x 1 x(1) 2 for 2 (0;1):