Question [3]: Just as for consumers, we can use the same trick for aggregation on the firm side. Each firm has discontinuous demand/supply, but in aggregate we will get (hopefully) a nice function.

Suppose there is a unit mass of firms (this language means that there are infinitely many firms who are all miniscule and together add to 1; firms are treated like a realization of a continuous random variable, in effect.) 1 Each firm can only hire zero or one unit of labour l, which has price w. Each firm has a technology indexed by which determines the level of the output good. The technology for a given firm is given by the production function f(l) = l. Suppose that the technology parameter for the firms is distributed on the interval [0, 4] according to

f() = ( 1 /2 - 1/ 4 if 2

( 1/ 2 + 1 /4 if 2

(It may be worthwhile to note that this means that there are no firms at the average productivity! Firms tend to be high or low productivity, but there are none in the middle.) Output y has price p. Firms are profit maximizers.

What is aggregate demand for labour?

What is aggregate supply of output?

(Careful, here the level of output varies for those firms that produce! So this is 'more complicated' than the previous part, where the level of labour was 1 for all firms that produce!)

Question [4]: Since we are at it and have done half the work already: Take the above question but suppose firms are actually bunched at the average. In other words, suppose the technology parameter is distributed on [0, 4] according to

f() = ( 1 /4 if 2

(1 1 /4 if 2

What is aggregate demand for labour?

What is aggregate supply of output?

Please answer question 4 ONLY. Information in question 3 may be needed to answer question 4.