Indexation. (This problem follows Ball, 1988.) Suppose production at firm i is given by Yi = SLα

i , where S is a supply shock and 0 < α ≤ 1. Thus in logs, yi = s + αi . Prices are flexible; thus (setting the constant term to 0 for simplicity), pi = wi + (1 − α)i − s. Aggregating the output and price equations yields y= s + α and p = w + (1−α)−s. Wages are partially indexed to prices: w = θp, where 0 ≤ θ ≤ 1. Finally, aggregate demand is given by y = m − p. s and m are independent, mean-zero random variables with variances Vs and Vm.

(a) What are p, y, , and w as functions of m and s and the parameters α and θ?

How does indexation affect the response of employment to monetary shocks?

How does it affect the response to supply shocks?

(b) What value of θ minimizes the variance of employment?

(c) Suppose the demand for a single firm’s output is yi = y−η(pi − p). Suppose all firms other than firm i index their wages by w = θp as before, but that firm i indexes its wage by wi = θi p. Firm i continues to set its price as pi = wi +
(1 − α)i − s. The production function and the pricing equation then imply that yi = y − φ(wi − w), where φ ≡ αη/[α + (1 − α)η].
(i) What is employment at firm i , i , as a function of m ,s, α, η , θ, and θi ?
(ii) What value of θi minimizes the variance of i ?
(iii) Find the Nash equilibrium value of θ. That is, find the value of θ such that if aggregate indexation is given by θ, the representative firm minimizes the variance of i by setting θi = θ. Compare this value with the value found in part (b).