DSE 2012- Question 16,29 and 30.

[Ayushya Kaul]

I'll post the following questions here. I somehow couldn't wrap my head around them. Any help regarding how to go about solving it and so on, would be greatly appreciated.

30) Suppose that there are two agents in an economy with
income x1 and x2. If the richer person transfers a portion of her income
to the poorer person without changing the income ranking (that is the rich
remains richer even after the transfer) then it is called a progressive transfer.
Welfare of this economy is measured by a function W(x1; x2). W is a `good' measure of social welfare if the social welfare increases due to a progressive
transfer. Consider the following candidates for W;
W1(x1; x2) = x1 + x2 W2(x1; x2) = x1x2

(a) Only W1 is a `good' measure
(b) Only W2 is a `good' measure
(c) Both W1 and W2 are `good' measures



29) The demand curve for electricity is D(p) = 120  p. The
marginal cost of electricity production is MC1(q) = 20 + q. The marginal
cost of pollution due to electricity production is MC2(q) = 3q. Find the
competitive equilibrium output and the social optimum output.
(a) 50; 20
(b) 50; 30
(c) 60; 20
(d) 60; 30

16) The equation x^7 = x + 1
(a) Has no real solution
(b) Has no positive real solution
(c) Has a real solution in the interval (0; 2)
(d) Has a real solution but not within (0; 2)

[Sumit]

Q.16).Ans: Try to plot the fn graph (i.e x^7-x-1=0) you will see that there is a real sol. in the interval (0,2)

Q.29).Ans: a)Competitive equilibrium output: D(p)=120-p or D(q)=120-q(i.e inverse demand fn)

Now solve: D(q)=MC1(q), You will get q=50.

b)Social optimum output: solve D(q)=MC1(q)+MC2(q) you will get Q=20.
So the answer is option-A(50;20)
Note: we add MC2 in part-B coz we taking this cost is bearing by society.

Q.30).suppose that X1=richer person in the economy
X2=poorer person in the economy.

Now, suppose income of X1=5 and X2=2 which means total income of two person in the economy=X1+x2=7.
from this we get
W1(x1; x2) = x1 + x2=7
W2(x1; x2) = x1*x2=10

Now, suppose due to progressive transfer the new income of two is X1=4 and X2=3(Note: the total income in the economy is still the same and the rich remains richer even after the transfer)

Now If we check welfare given by above 2 equation after progressive transfer we will get
W1(x1; x2) = x1 + x2=7
W2(x1; x2) = x1*x2=12
The above result clearly shows that Only W2 is a `good' measure bcoz it increases social welfare which is not the case in W1..

So, answer is b).Only W2 is a `good' measure

[maahi]

can u do this one ? give d steps only .
Suraksha’ is the sole producer and supplier of security systems
in India and the sole employer of locksmiths in the labour market. The demand for
security systems is D( p ) = 100 − p , where p is the price. The production of security
systems only requires locksmiths and the production function is given by f ( L) = 4 L ,
where L is the number of locksmiths employed. The supply curve for locksmiths is
                                                 
given by L( w) = max    0, w/2-20 , where w is the wage rate.
                                   
24. How many locksmiths will ‘Suraksha’ employ?
       a) 5
      b) 10
     (C) 15
      d) 20

25. If the government sets the minimum wage is 70 , how many locksmiths will
Suraksha employ?
         a)            5
         b)            10
         c)            15
         d)            20

[maahi]

16 ) how do we plot d graph in one variable case ?/

[Sumit]

I was trying to solve both of them by simply putting MR=MC..but I'm not getting the answer..even I'm unable to get any given option provided in questions..How was u trying it????

[Sumit]

Like we do in other polynomial function such as quadratic, X^3 , X^4 etc.

[maahi]

i still don't get d graph part .

i tried finding optimal input by equating marginal revenue product of d factor to wage and then substituted d found expression in d profit function of d monopolist

[maahi]

i got d ans for 1 . mr = 100-2q, output is a function of L i.e y= q=4L substitute this in MR. compute MRP (L) = MR *MP(l) and equate it to w . u get 10 point something and 10 is d answer .

[wolverine]

q16) x^7-x-1=0

as the eqn is of 7th order..so it will hav 7 roots...complex roots occurs in pair..so if the eqn has 6 complex roots..than for sure one of the root is real.and using a good approximation of around 1.1 its directs towards option c.

else newton raphson method can be used .

[Ayushya Kaul]

Thanks a lot :)
I couldn't get the graph part though :s

[maahi]

how to come about that approximation ?

[Erika]

I have a slight doubt @sumit.

Dont we multiply q to the inverse demand function to make it total revence (price x qty=TR) and then differentiate it to get MR. Next step would be to equate it to MC and find the q. while you have directly equated the MC to inverse function.

 your ans is absolutely right and I am getting wrong ans by the method i just pointed out. But is there a mistake in that approach??

[xyz123]

Can somebody explain the approx part in ques 16. I didn't get it :/

[mrittik]

q16. f(x)=x^7-x-1=0....so its value when x=0, f(x)=-1, f(x)=-1, x=1...f(x)=126;x=2...so from change in 1 to 2 it changes its sign. Next f'(x)=7x^6-1>0 when x=1....so it has only 1 root in b/w (0,2).