Math, microeconomics

Question

Minimize c = 2x+4y subject to

0.8x+0.8y≥8

x+2y≥14

x ≥ 0, y ≥ 0.

Minimum value c =?, occurs at the endpoints (x, y) =? ,(smaller x-value) and (x, y) =? (larger x-value) and? " Choose the correct word "

•the region is empty

•no other points

•the function is unbounded

•the line connecting them

Answer 1

.8x +.8y > 8

multiply by 10/8

x + y > 10

it's endpoints are (0,10) and (10,0)

x+2y > 14

this constraint is bounded below by a line from endpoint (0,7) to endpoint (14,0)

the two bounday lines intersect at a point which is possibly a cost minimizing point

the 2nd constraint line boundary is the same as C=2x+4y =28

divide by 2 to get x+2y =14,

x+y=10 and x+2y= 14 intersect

subtract 1st equation from 2nd to get

2y-y = 14-10

y=4, and x=6, the intersection point is (6,4)

c(x) = c(6)= 2x+4y = 2(6)+4(4) =12 +16= 28

endpoints

have c(14)=2(14)+0= 28

cost is minimized at any point on the line segment from (6,4) to (14,0)

an infinite number of points including (6,4), (8,3), (10,2), (12,1), (14,0)

all have cost = 28= minimum cost