DSE 2009

[s]

Can someone please explain how to go about this sum.

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


for the first part i maximised the monopolist's problem  and derived output supply function as a function of w then provided he'll produce this y* labor emplyment is L* = Y*/4 that is how i got the labor demand function of firm then at market equilibrium i equated LD with LS ..but my ans L*=6 and the correct ans is 10..please help

[anon_econ]

For part 1 if nothing else is striking u just calculate her profits at all the 4 levels of employment n pick the maximum one (since there is no 'none of the above' option). Plz tell me the correct answer for the second 1. I'm getting 10 for that 1 as well..if it's correct i'll tell u how i did it.

[s]

here suraksha is monopolist in output mkt n monopsonist in factr mkt so frm profit max problem

p(y(L)). y(L) - w(L). L   ......i got L* = 10...for the second part what i did was put w=70 in MRPL or labor dd fn of monopsonist (w= 400 - 32L) n got L*= 330/32...which is 10.31...i dunno if that is the right approach

the correct answer is 10...

hey Vasudha what was your approach?

[anon_econ]

Ok now i'm not sure if there r 2 ppl with the same username or what :S
I did it the same way and got the same answers, except that even for part 1 i was getting 10 point something..not 10 exactly

[s]

lol i posted the question, later figured out the way to solve :)

[ritu]

how do we do this:
i tried hard but couldn't think of anything regarding this question...
isi 2010:
30.for all x,y belonging to (0,infinity),a function f: (o,infinity)-R satisfies the inequality If(x)-f(y)I<= Ix-yI^3...then f is
a.an increasing function
b.a decreasing function
c.constant function
d.none of these


q23.let f be a real valued continuous function on [0,3]suppose that f(x) takes only rational values and f(1)=1.then f(2) =
a.2
b.4
c.8
d.none of these

[s]

for 1st question, consider an increasing fn f(x)=x let x=1/2, y=1...so |f(1)-f(2)|=1/2 and |1/2-1|^3= 1/8...1/2 is not less than or equal to 1/8 so this result dint hold for an increasing fn..because if it did it must hold for all increasing functions.

consider a decreasing fn f(x)=1/x let x=1,y=2 now |f(1)-f(2)|=1/2 and |x-y|^3=1/8 and inequality doesnt hold again so its not true for decreasing functions.

consider constant fn f(x)=1 , let x=2 y=3, |f(x)-f(y)|=0 and |x-y|^3=1 which satisfies inequality and always will for any constant fn as the LHS will always be 0...

so (c)

for 2nd question f takes only rational values,if f was incr or decr given its continuous in the interval it can take irrational values for some x...only for a constant fn its guranteed that f will take a particular rational value..so f is constant thus f(2)=1 ...so (d)

[ritu]

for second question....what if f(x)=x ^2...then it satisfies f(1)=1 and also the condition that it takes rational values only....then option giving 4 as answer is also applicable....????
pls reply

[anon_econ]

Ritu how is it taking only rational values?

[ritu]

like if say x=root 2,f(x)=2...similarly for any irrational number...it will give a squared value which will be rational....only borderline case i can think of is root pi and root e....????

[ritu]

a utility maximizing consumer with a given money income consumes two commodities X & Y.he is a price taker in the market for X.for Y there are two alternatives ...
a.he purchases  Y from the market being a price taker or
b.the govt supplies a fixed quantity of it through ration shops free of cost.is the consumer necessarily better off in case b???explain with respect to following cases:
1.indiff curves are strictly convex to origin
2.X & Y are perfect substitutes
3.X & Y are perfect complements

[anon_econ]

there r infinitely many irrational numbers b/w any 2 rational numbers. if a function is continuous it has to take on both rational and irrational values unless it takes on only one rational value (as in this case). in ur example x=(2)^(1/4) will give an irrational value of y. hope u get the point.

[anon_econ]

and for the micro question i'm not sure but i think he will not necessarily be better off in the 3rd case. See in the 2nd scheme he can consume y* and M/px where y* is the quantity given to him by the ration shop. now obviously if y* is gr8er than what he was consuming in the 1st scenario, he would b better off. but if y* is less than that quantity u can see graphically that (M/px,y*) would not be on a higher IC in the case of perfect complements..

[ritu]

yes vasudha thanx a lot...i got the point...:)

[ritu]

but i think in case of perfect substitutes he can be better of....because of additive utility.....he will spend all his income on x plus he gets wot govt gives him of y....how do we incorporate price considerations..if px<py...then utility is x+y*
if px>py then utility is m/py+y* but wot if px=py?..wot do u think abt it?????

[anon_econ]

Yeah he would be better off in the case of convex preferences as well as perfect substitutes..i had said that he could be worse off only in the case of perfect complements. in case of PS he would either be consuming at the x-intercept or at the y-intercept or he would be indifferent b/w all points on his budget constraint. In any case he would be better off. u can see it diagrammatically.

[ritu]

dse
1.consider the function f mapping points of the plane into the plane,defined by f(x,y)=(x-y,x+y)the range of this function is....
a.the 45 degree line
b.a ray through origin but not the 45 degree line
c.the entire plane
d.the first and third quadrant

i think its c...but i dnt kno if wot i am thinking is correct....x-y and x+y are independent vectors....when we plot f(x,Y) in 2-D space these two will span the entire vector space...
i might be horribly wrong so pls tell me the correct answer...



2.suppose we are interested in estimating the mean height of MA students in delhi university...an average height estimated from a random sample size 30 is better than that estimated from a random sample of size 20 coz....
a.the sample of size 20 is likely to be more biased because its less representative
b.the sample of size 30 is likely to yield more precise estimates of average height than sample size of 20.
c.the central limit theorem prescribes a minimum sample size of 30
d.both a and b..

b and d can be rejected....it may be a or c....wot is the right answer????



3.suppose f:[0,1] to [0,1] is a continuous nondecresing function with f(0)=0 and f(1)=1.define g:[0,1] to [0,1] by g(y)=min{x belongs to [0,1] such that f(x)>=y}.then
a.g is non decreasing
b.if g is continuous,then f is strictly increasing
c.neither a nor b is true
d both a nd b are true
[dse 2010]


kindly help

[anon_econ]

1. i'm also getting c by elimination of the other options.
2. umm i think it would be b
3. i'm getting d but the solutions say it's a so i don't know.

[ritu]

can u please explain that how u arrived at ur answer in case of question 3???????

[anon_econ]

hey i just realized i had made a mistake in that micro question. even in the case of strictly convex preferences he may not be better off. u can see it graphically as well as intuitively. if he is given a very small quantity of y, he might not be better off even tho now he can get M/px of x. so the final answer is that only in the case of perfect substitutes would he NECESSARILY be better off.
in that maths question u can plot such a function n c what f(y) would be for each y..for eg if u hv f(x)=x then g(1/2) would be the min value from among all those x's for which f(x)>=1/2. this would give u g(1/2)=1/2. g has to be non decreasing (just think of what a decreasing g would mean). and according to me if f is not STRICTLY increasing then g is not continuous..i dunno what my mistake is.. :/