(due October 13) 1. Assume that x and y are related by G(x,y) = x3 + y3 - y2 - 3
ID: 1189604 • Letter: #
Question
(due October 13)
1. Assume that x and y are related by G(x,y) = x3 + y3 - y2 - 3yx - x = 0
(a) What values of x solve the equation G(x,1)=0? What values of y solve G(0,y)=0? What values of
(x,y) solve G(x,y)=0 subject to the constraint y=-x?
(b) All of the points that solve the equation G(x,y) = 0 form one or more curves in (x,y)-space. Your
answers to (a) identified which six specific points on those curves?
(c) Using the method explained in class, calculate dy/dx at each of the points that you listed in (c).
2. Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and y represent the
quantities of two goods, X and Y. For parts (a)-(g), assume that good X costs pX=$3 per unit and
good Y costs pY=$4 per unit.
(a) With good X on the horizontal axis and good Y on the vertical axis, draw the consumer’s budget line
if she has W=$31 to spend, Show the exact coordinates of the intercepts.
(b) On your budget line, show the optimal point (which you calculated in Homework #6). How much
utility does that point produce? Draw the indifference curve that passes through that optimal point,
showing the exact coordinates of at least three points on the curve.
(c) On the same diagram, draw three new budget lines, representing the consumer’s budget constraint
for other possible values of her wealth: W=$7, W=$55, and W=$79. For each budget line, show the
exact coordinates of its intercepts.
(d) If the consumer’s wealth is W=$7, then calculate the optimal point and show it on the appropriate
budget line. Do the same for W=$55 and W=$79. Your diagram should now show four possible
budget lines and four optimal points.
(e) Calculate the marginal rate of substitution of X for Y, at each of the four optimal points
(f) Connect the four optimal points to show the income expansion path. Are X and Y normal or inferior
goods?
For parts (g) and (h), W is an unknown exogenous variable.
(g) Derive the optimal consumption of X, as a function of W. Also derive the optimal consumption of
Y, as a separate function of W. Are X and Y normal or inferior goods?
(h) Using your answer to part (f), what fraction of the consumer’s income is spent on good X? Does this
fraction increase or decrease as W increases? What happens to this fraction in the limit as wealth
increases (i.e., diverges to infinity)?
Now assume that the consumer’s wealth is fixed at W=$31, and the price of Y is fixed at py=4.
(i) Assuming that the optimal point satisfies te first-order condition, derive the consumer’s demand for
X, as a function of pX. Under what conditions (i.e., for which values of pX) should the consumer
choose a boundary point?
(j) Derive the consumer’s demand for Y, as function of pX. Based on this demand function, are X and
Y complements or substitutes?
(k) Using your answers to (i) and (j), calculate the consumer’s optimal consumption of X and Y, for each
of the following five prices: pX=1, 3, 5, 7, 15. (Most of the answers are not integers, so you should
report them to precision 0.1.)
(l) Draw a diagram showing the five budget lines for the five prices of part (h). On the five budget lines,
show the five optimal points from part (k). Connect the points to show the price expansion path, for
changes in the price of X.
(m) As pX rises, does the substitution effect tend to increase or decrease the consumer’s consumption of
X and Y? As pX rises, does the income effect tend to increase or decrease the consumer’s
consumption of X and Y? For which of these goods, if any, do the income and substitution effects
work in the opposite directions? If they do work in opposite directions, then which effect is stronger
in this case?
(n) Calculate the elasticity of the consumer’s demand, as a function of the quantity demanded.
*Note: for (2-b), the question like mentioned is based on previous homework #6 that you guys already work on. it is on my last post in my account. the problem about utilities function and budget lines. Please let me know if there is anything confusing about this problem and thank you for the help you provide to me.
Explanation / Answer
ans 1.
a)
G(x,y)= x3+y3- y2-3yx-x =0
values of x for G(x,1)=0
x3- (1)3-1 -3 (1) x - x =0
x3 -4x=0
x(x2-4) = 0
x=0, x=2 and x=-2
points are (0,1) , (2,1) and (-2,1)
values of y when G(0,y) =0
0+ y3-y2 -3y(0) -0=0
y3- y2= 0
y2 ( y-1) =0
y=0, y=1
points are (0.0) and (0,1)
solve G(x,y)=0 subject to the constraint y=-x
x3+y3- y2-3yx-x =0
x3+ (-x)3 - (-x)2 - 3(-x) x -x= 0
2x2- x= 0
x=0, x=1/2
points are ( 0,0) and (1/2, -1/2)
b) points are mentioned above which we got from part a)
points are (0,1) , (2,1) and (-2,1) ,(0.0) and (0,1) ,( 0,0) and (1/2, -1/2) we can plot these x-y space
c)
G(x,y)= x3+y3- y2-3yx-x =0 is implicit function
therefore dy/dx = - G'x/G'y = - (3x2-3y-1) /(3y2-2y-3x)
at (0, 1)
dy/dx = - ( 0- 3-1)/ (3-2-0) =4
at (2,1)
dy/dx= - ( 12-3-1)/ (3-2-6) = 8/5
at (-2,1)
dy/dx = - (12- 3-1) /( 3-2+6)= - 8/7
at (0,0)
dy/dx = -(-1)/ 0 = not defined
at (0,1)
dy/dx = - (0-3-1) /(3-2-0) =4
at (0,0)
dy/dx = -(-1)/ 0 = not defined
at (1/2, -1/2)
dy/dx= - (3/4 + 3/2 -1)/ (3/4 +1-3/2) = -5
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