Question

Suppose a consumer's preferences are given by U(X,Y) = X²*Y. Thus, the marginal utility of X, MUx = 2XY and the marginal utility of Y, MUy = X². Suppose the consumer has $180 to spend and the price of good Y is $1. Sketch the price-consumption curve for the prices of

• Px = $1
• Px = $2
• Px = $3

To do this, carefully draw the budget constraints associated with each of the prices for good X, and indicate the bundle that the consumer chooses in each case. Also, be sure to label your graph accurately.

Answer 1

There will 3 cases of budget line

Case1

i.e. p_x=1 and p_y=1

Budget line is given by

1*X+1*Y=180

X+Y=180

Case2

i.e. p_x=2 and p_y=1

Budget line is given by

2*X+1*Y=180

2X+Y=180

Case3

i.e. p_x=2 and p_y=1

Budget line is given by

3*X+1*Y=180

3X+Y=180

Case 1

In case of utility maximization, we know that

x=2Y

Put X=2Y in budget line

X+Y=180

2Y+Y=180

3Y=180/3=60

X=2*60=120

Optimal Bundle (X=120, Y=60)

Utility=X²Y=(120)²60=864000

^{We make following schedule to draw budget and utility curves.}

Budget line ,X+Y=180		Utility=864000
X	Y	X	Y
0	180
20	160
40	140
60	120	60	240.00
80	100	80	135.00
100	80	100	86.40
120	60	120	60.00
140	40	140	44.08
160	20	160	33.75
180	0	180	26.67

Optimal consumption bundle is shown on the graph. It is the point where budget line is tangential to utility curve.

Case 2

In case of utility maximization, we know that

x=Y

Put X=Y in budget line

2X+Y=180

3Y=180/3=60

X=Y=60

Optimal Bundle (X=60, Y=60)

Utility=X²Y=(60)²60=216000

^{We make following schedule to draw budget and utility curves.}

Budget line, 2X+Y=180		Utility=216000
X	Y	X	Y
0	180
10	160
20	140
30	120	30	240.00
40	100	40	135.00
50	80	50	86.40
60	60	60	60.00
70	40	70	44.08
80	20	80	33.75
90	0	90	26.67

Optimal consumption bundle is shown on the graph. It is the point where budget line is tangential to utility curve.

Case 3

In case of utility maximization, we know that

2Y=3X

X=(2/3)Y

Put X=(2/3)Y in budget line

3X+Y=180

3*(2/3Y)Y+Y=180

Y=180/3=60

X=2/3*60=40

Optimal Bundle (X=40, Y=60)

Utility=X²Y=(40)²60=96000

^{We make following schedule to draw budget and utility curves.}

Budget line,3X+Y=180		Utility=96000
X	Y	X	Y
0	180
10	150
20	120	20	240.00
30	90	30	106.67
40	60	40	60.00
50	30	50	38.40
60	0	60	26.67

Optimal consumption bundle is shown on the graph. It is the point where budget line is tangential to utility curve.