Suppose that there are two goods (X and Y). The price of X is $2 per unitu, and

Question

Suppose that there are two goods (X and Y). The price of X is $2 per unitu, and the price of Y is $1 per unit. There are two consumers ( A and B). The utility functions for the consumers are: for consumer A: U (X,Y)= X^.5Y^.5 and for consumer B: U(X,Y)=X^.8Y^.2

Consumer A has an income of $100, and Consumer B has an income of $300.

1. Derive the marginal utlities of goods X and Y for consumer A

2. Derive the marginal utilities of goods X and Y for consumer B

3. Set consumer A's MRS equal to the price ratio. Then use the budget constraint to solve for her optimal bundle

4. Set consumer B's MRS equal to the price ratio. Then use the budget contraint to solve for his optimal consumption bundle.

5. Calculate the MRS for each consumer at their optimal consumption bundles.

6. Suppose that there is another conusmer (let's call her C). You don't know anything about her utility function or her income. All you know is that she consumes both goods. What do you know about C's MRS at her optimal consumption bundle? Why?

Explanation / Answer

1

for A

MUx = dU/dx = .5(X^-.5)*(Y^.5)

MUy = dU/dy = .5(X^.5)*(Y^-.5)

2

for B

MUx = dU/dx = .8*(X^-.2)*(Y^.2)

MUy = dU/dy = .2*(X^.8)*(Y^-.8)

3. for A

MRS = delU/delx/delU/dely = Px/Py = .5(X^-.5)*(Y^.5)/.5(X^.5)*(Y^-.5) = y/x = Px/Py = 2

Y = 2x

100 = 2x+y

y = 50, x = 25 (optimal bundle)

4. For B

MRS = delU/delx/delU/dely = Px/Py = .8*(X^-.2)*(Y^.2)/.2*(X^.8)*(Y^-.8) = Px/Py = 2

soving above eqn

y = x/2

300= 2x+y

x = 300/2.5 = 120

y = 60

x= 120, y = 60, (optimal bundle)

5.

MRS for A = y/x = 2

MRS for B = 4y/x = 2

6.

for C at optimum consumption bundle

MRS = Px/Py = 2

because at optimum consumption level,

I = 2x+y

if we differentiate

0 = 2dx/dy + 1

MRS = -dy/dx = 2 = px/py

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