Decisions and Consumer Behavior

1. Let $$u(x_{1},x_{2})=x_{1}+x_{2}$$ u ( x 1 , x 2 ) = x 1 + x 2 be a utility function. There exists no preference relation which is represented by this utility function. 2. Let $$x_{1}\succ x_{2}$$ x 1 ≻ x 2 and $$x_{2}\succ x_{3}$$ x 2 ≻ x 3 . Then, the assumption of transitivity implies that $$x_{1}\succ x_{3}$$ x 1 ≻ x 3 . 3. If $$u(x_{1},x_{2})=x_{1}\cdot(x_{2})^{5}$$ u ( x 1 , x 2 ) = x 1 ⋅ ( x 2 ) 5 is a utility representation of a preference ordering, then $$v(x_{1},x_{2})=\frac{1}{5}\ln x_{1}+\ln x_{2}$$ v ( x 1 , x 2 ) = 1 5 ln⁡ x 1 + ln⁡ x 2 , too, is a utility representation of the same preference ordering. 4. Preferences that fulfill the principle of monotonicity are always convex. Assume an individual has income b > 0 at his disposal, which he can spend on two goods of quantities x 1 and x 2. 1. A consumer’s preference relation is represented by the utility function $$u(x_{1},x_{2})=x_{1}\cdot x_{2}$$ u ( x 1 , x 2 ) = x 1 ⋅ x 2 . Let x 1 be marked on the x-axis and x 2 on the y-axis. If so, the price-consumption path for all $$p_{1}> 0,p_{2}> 0$$ p 1 > 0 , p 2 > 0 is a straight line through the origin with a slope of $$\frac{p_{1}}{p_{2}}$$ p 1 p 2 . 2. For an individual, two goods are perfect complements. If so, the cross-price elasticity of the Marshallian demand always equals zero. 3. The individual’s demand will decrease if the price of good 1 decreases, provided that x 1 is an inferior good. 4. For an individual, two goods are perfect substitutes. In such case, at the optimum, the demand for one good will always be zero.