Question

Interpret the utility function: U(x1 , x2,...,xn )= β1x1 + β2x2 + ··· + βnxn ; Similarly, One study reports a utility function that had the form : U(TW,TT,C)= −0.147TW − 0.0411TT − 2.24C, where TW = total walking time to and from bus or car; TT = total time of trip in minutes, C = total cost of trip in dollars; rigorously interpret the meaning of that utility function. 

Answer 1

Utility function $U(x1,x2,...,xn)$ shows the measure of satisfaction that consumers receive for choosing of some set of goods. The utility function $U(TW,TT,C)=-0.147TW-0.0411TT-2.24C$ is always less than zero (an individual does not like any trips). This function shows the pindividual utility as a preferences over a set of the following actions during a trip: trip from some start point to some finish point. Individual can chose within 2 options: walking (denoted by TW) or using a bus/car during a period that is equal to $TT-TW$ (incliding payment for the traveling costs denoted by C).

The first order derivatives of the U with respect to TW, TT, and C can relate a change in U to a change in arguments TW, TT, and C).

$dU/dTW=-0.147<0$ (1)

$dU/dTT=-0.0411<0$ (2)

$dU/dC=-2.24<0$ (3)

The inequations (1) and (2) shows that individual does not like to go from a home to bus/car rather than to use car/bus. Hovewer. individual does not like to pay.

Let's assume that the TW is equal to zero and analyze the indifference curve based on the U1 and U2: $U1(TT1,C1)$ = $U2(TT2,C2).$

Hence, $-0.0411TT1-2.24C1=-0.0411TT2-2.24C2$

Then, $-0.0411(TT1-TT2)=-2.24(C2-C1)$

Hence, if the total time of trip increases, then cost of trip should be decresed to provide the constant utility.

Conclusions:

1) An individual does not like any trips

2) An individual does not like to go from a home to bus/car rather than to use car/bus

3) Positive total walking time to and from bus or car is result of inequation (3) that shows that utility decreases when the cost for trip increases.