Can anyone please explain why the answer to this econ problem is such?

Question

Hans has $27 which he decides to spend on x and y. Commodity x costs $16 per unit and commodity y costs $10 per unit. He has the utility function U(x, y) = 5x^2 + 2y^2 and he can purchase fractional units of x and y. Hans will choose

>> only y<<

why only y?

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He also restricts his consumption to 6,500 calories per day. There are 1,500 calories in a bag of Doritos and 500 calories in a seafood salad. If he spends his entire money budget each day and consumes no more calories than his calorie limit, he can consume up to

>>> 3 bags of Doritos per day but no more.<<

why?

thanks in advance

Answer 1

The Utility function defines a circle around the center point (0,0). So if you change U((x,y) by 2², 3².....,z² you´ll find sooner or later a point that maximizes his utility subject to the restriction  16x+10y=27 which is a line  for x&gt;0 and y&gt;0. You´re turning the circle bigger till it crosses the line. At that time you have to get the crossing points which are possibly (01),(1,0) or (a,b).

As we don´t know the exact points we apply Lagrange

L(x,y,µ)=5x²+2y²+µ(16x+10y-27)

dL/dx=10x+µ16=0

dL/dy=4y+µ10=0

dL/µ=16x+10y-27=0

From one and two equations we get

25x=16y

x=0.64y

Operating with the third equation

16 0.64y+10y-27=0

and simplifying we get y

20.24y=27

y=27/20.24=1,3 units

16x+10y=27

16x+13-27=0

16x=14

x=14/16=0.8 units that do not reach one unit

If he can buy fractions of x and y he will buy 1.3 and 0.8 respectively. If he cannot buy fractions of those items he will only buy 1 unit of  x.

Answer 2

A. First of all note that the utility function is not convex, so the standard utility maximization technique is not applicable! ! ! Further, intuitively, if you calculate the utility level when only y is consumed, and compare it to that of only consuming x, you&#039;ll see that under only y the utility is higher. Besides, your utility function has an ellipse-shaped curvature, so each point on that curve corresponds to some utility level, i.e. the higher the point the higher the utility, i.e. you are looking for a corner solution.

B. Are you sure you have provided all the necessary info in the setup?