Question

Steve consumes earplugs and other goods. His utility function ux,y is increasing in both his consumption of earplugs x and other goods y . The price of other goods is py and Steve has a sufficiently large wealth w . Suppose ux,y=v(x)+ay and that Steve can buy n-1 units of earplugs at total cost cn-1 .

Write out Steve’s utility of consuming n-1 units of earplugs and spending the rest on other goods.
Suppose that the nth earplug costs pn , write out Steve’s utility of consuming n units of earplugs and spending the rest on other goods.
Show that the reservation price for the nth earplug rn is pya×vn-vn-1 .
(Hint: look at the definition of the reservation price.)
Note that this implies that if earplugs were perfectly divisible, the reservation price function rn=pyav'(n) .
Solve Steve’s utility maximisation problem (subject to prices px,py and wealth w) to get his inverse demand function for earplugs x . Is it the same as the reservation price function?
In the lecture’s example, a=1 and py=1 such that the reservation price curve is px=v'(x) . What do you think the additional term pya here captures?

Answer 1

Steve's utility of consuming n-1 units of earplugs and spending the rest on other goods can be calculated by substituting n-1 for x in the utility function. Similarly, his utility of consuming n units of earplugs can be calculated by substituting n for x in the utility function. The reservation price for the nth earplug is given by the difference in utility between consuming n and n-1 units of earplugs, multiplied by pya.

Step-by-step explanation:

To find Steve's utility of consuming n-1 units of earplugs and spending the rest on other goods, we substitute n-1 for x in the utility function: u(n-1, y) = v(n-1) + ay. To find Steve's utility of consuming n units of earplugs and spending the rest on other goods, we substitute n for x in the utility function: u(n, y) = v(n) + ay. The reservation price for the nth earplug, rn, is given by the difference in utility between consuming n units and consuming n-1 units, multiplied by pya: rn = (v(n) + ay) - (v(n-1) + ay) = pya * (v(n) - v(n-1)). The additional term pya captures the marginal utility of consuming the nth earplug.

The inverse demand function for earplugs, x, can be found by solving Steve's utility maximization problem, which involves maximizing ux, y subject to prices px, py, and wealth w. The inverse demand function represents the quantity of earplugs that Steve is willing to buy at a given price. The reservation price function, rn, represents the maximum price Steve is willing to pay for an additional earplug. In the lecture's example, where a=1 and py=1, the reservation price curve is given by px = v'(x). The additional term pya captures the effect of the price of other goods and Steve's preference for consuming earplugs and other goods.