Answer True or False a)optimal consumption bundle of a utility-maximizing consum

Question

Answer True or False

a)optimal consumption bundle of a utility-maximizing consumer is always on the budget line

b)utility function of a consumer is u(x1, x2) = x1 + x2. Prices are p1 = 4, p2 = 6. Income is positive. Consumer will demand both goods in the optimum

c)I marginal rate of substitution between good #1 and good #2 is larger than the price ratio p1/p2, personl with utility function u(x1, x2) = x1x2 should consume less of good 1 and more of good 2 to achieve his optimum (x*1, x*2)

d) substitutional production function f(K, L) = K + L shows constant returns to scale

e) price-taking company's fixed costs do not matter for short-run profit maximization

Explanation / Answer

Answer
a)
True
The utility is maximum at MUx/MUy=Px/Py, this condition is possible only at the IC touching budget line so the optimum consumption bundle is on budget line but the point is decided by IC curve.

b)
False,
The utility function is for substitutes, and the MRS is
MUx1=1 and MUx2=1
so the MRS=1
which means the MU is 1 by purchasing any one of them.
the Price of x1 is lower so the per unit utility will be higher than good two at all point so the consumer can not purchase good 2.

c)
False
The optimum condition is
MUx/MUy=Px/Py
here
MUx/MUy>Px/Py
whereas we cannot change Px/Py, we can only change MUx/Muy
we have to decrease the ratio and that is possible by reducing the consumption of good2 because reduce in consumption increases marginal utility and that decreases the ratio.

d)
True
f(tK,tL)=tK=tL=t(K+L)=t*f(K,L)
so it shows constant return to scale

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