1. Caty consumes only goods X and Y . Her utility function is: U(X; Y ) = min{X;

Question

1. Caty consumes only goods X and Y . Her utility function is: U(X; Y ) = min{X; Y}. We

are given that PX = 3; PY = 6, and Caty's income is 18.

Calculate Caty's optimal consumption bundle, (X, Y). (Hint: Since Caty's indif-

ference curves are not smooth and curvy", we cannot use MRS = MRT to solve

for the optimal bundle. Draw a diagram to see where Caty's optimal bundle must be

on her IC. How do you characterize this bundle mathematically?)

2. John consumes only goods X and Y . His utility function is: U(X, Y ) = X + 2Y . We are

given that PX = 3; PY = 3, and John's income is 30.

(a) Calculate the slopes of John's budget constraint and his indierence curves, as viewed

with Y as the vertical axis and X as the horizontal axis

(b) Calculate John's optimal consumption bundle, (X, Y). (Hint: Since John's indif-

ference curves are not smooth and curvy", we cannot use MRS = MRT to solve for

the optimal bundle. Draw a diagram to see where the John's optimal bundle must

be on his IC. How do you characterize this bundle mathematically?)

Explanation / Answer

a)

All two dimensional budget constraints are generalized into the equation:

P_x x+P_y y=m

Where:

m= money income allocated to consumption (after saving and borrowing)

P_x= the price of a specific good

P_y= the price of all other goods

x= amount purchased of a specific good

y= amount purchased of all other goods

The equation can be rearranged to represent the shape of the curve on a graph:

y= (m/P_y)-(P_x/P_y) x, where (m/P_y) is the y-intercept and (-P_x/P_y) is the slope,

representing a downward sloping budget line.

The budget constraint equation can be rearranged to represent the shape of the curve on a graph:

y=18/6 - 3/6*x

=>y = 3-x/2

x=0 y=3

y=0 x=6

An individual consumer should choose to consume goods at the point where the most preferred available

indifference curve on their preference map is tangent to their budget constraint.

Linear utility-

If the utility function is of the form U(x,y)=x+y then the marginal utility of x, is U_1(x,y)=alpha and

the marginal utility of y, is U_2(x,y)=beta. The slope of the indifference curve is, therefore,

frac{dx}{dy}=-frac{beta}{alpha}.

Observe that the slope does not depend on x or y: the indifference curves are straight lines.

when x<y U(x,y) = x dy/dx = -infinity

x>y U(x,y) = y dy/dx = 0

In this case, Indifference curve will be L shaped i.e X and Y are perfect complement.

then our budget constraint curve will intersect on (x,y) = (2,2).

2.)----------------------

a)

The indifference curves are straight line, given U(X, Y ) = X + 2Y

tan theta = -1/2. theta = approx 153 degree, goods are perfect substitute.

b)

Given : P_X = 3; P_Y = 3, and John's income is 30

Then Budget constraint equation will be as equation formula in part(1):

y=30/3 - 3/3 x

=>y=10-x

x=0 y=10

y=0 x=10

The intersection of the Budget equation can be found at (x,y) = (0,10).

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