# Microeconomics Question on Production Functions

BrumCo is a firm that produces heaters in a perfectly competitive market. The production of each model train requires plaster (P), and grams of white paint (W). Their production function is given by

f(P,W)= min{6sqrt(P), 3W-12}

(a)  Draw the isoquant corresponding to q = 48 heaters in a clearly labelled diagram where P is the horizontal axis and W is the vertical axis. Label two distinct input bundles (P, W ) in the diagram which give q = 48.

(b) Compute the marginal products of each input.

(c) Does this production function exhibit constant returns to scale? Using the marginal products you have computed in the previous part, explain your answer in no more than 25 words.

(d)  Suppose that BrumCo wants to produce q heaters when the price of plaster is $72/mL and the price of white paint is$18/g. Show that the minimum cost of such an undertaking is

c(q)=2q^2 +6q+72.

(e)  Suppose that the market price is p, and BrumCo will produce q units of heaters. Using the cost function you found in the previous part, find the supply function of BrumCo. Express it as a function of price.

(f)  Suppose that there are 60 identical firms like BrumCo who act as price-takers and the market demand for model trains is given by

QD =910−5p.

Show that the short-run market equilibrium price in this industry is p∗ = 50.

(g) In the long run, given no shocks to market demand, do you expect to see the number of firms in this industry increase, decrease or stay the same?

(h)  Assuming no shocks to market demand, what is the largest whole number of firms that this market can sustain in the long run equilibrium?