Question

U(q1,q2)= q1^0.5 + q2^0.5, which ones are correct (can be more than 1)

1. Q2 in as inferior good

2. The indifference curves are downward sloping

3. Preference satisy “more is better”

4. Demand function for q2 is downward sloping

5. Engel curve for q1 is downward slopping

6. Income-consumption curve is a straight line

Answer 1

We need to derive demand function of each good ( well both have same symmetrical demand)

Equilibrium condition,

Mu1=0.5 /q1^0.5. Mu2=0.5/q2^0.5

Mu1/p1=mu2/p2

0.5/q1^0.5*p1=0.5q2^0.5*p2

Q2^0.5*p2=q1^0.5*p1

Q2=q1*(p1/p2)^2

M=p1q1+q1*(p1/p2)^2

M=q1(p1+p1^2/p2^2)

Q1=M/(p1*p2^2+p1^2)/p2=M*p2^2/(p1*p2^2+p1^2)

So by symmetry,.

Q2=M*p1^2/(p2*p1^2+p2^2)

1) as Q2 is positively proportionate to income(M), so Increase in income leads to increase in Q2 ,so Q2 is normal good . statement is flase.

2)U=Q1^0.5+Q2^0.5

Let U=4, q1=4. ,q2=4

Let Increase q1=9. ,to remain at Same U=9, q2=?

4=3+q2^0.5

1^2=q2

Q2=1

So as q1 Increases ,so to remain at same utility level, q2 decreases .so indifference curve is downward sloping, showing negative relationship between q1 and q2.

Statement is right

3)U is Increasing function of q1 and q2 ,so more q1 and q2 leads to higher utility level.

The consumer want to maximize its utility level,so yes preference satisy "more is better"

Statement is right

4)Q2=M*p1^2/(p2*p1^2+p2^2)

As you can see p2( price of q2) is in denominator ,as p2 Increase q2 decreases and vice versa,so yes demand curve is downward sloping. Statement is right

5) Engel curve shows relationship between QUANTITY demanded and CONSUMER income.

Q1=M*p2^2/(p1*p2^2+p1^2)

As M Increases Q1 Increases means there is a positive relationship between QUANTITY demanded and income.

So Engel curve is upward sloping curve.

Statement is wrong

6)q1=M*p2^2/(p1*p2^2+p1^2)

1$ change in income leads to p2^2/(p1*p2^2+p1^2) change in demand of q1. Because prices are exogenous(fixed).so every 1$ Increase in income Increases q1 by same factor.

And same for q2.

So yes income -consumption curve is straight line.

Statement is right