Question

Suppose firm j’s output is given by yj = n 1−α j , where 0 < α < 1 (α is a parameter). Suppose the firm must pay a fixed cost b < α if it wants to operate. That is, the firm’s profits are given by π (nj ) = 0 , if nj = 0 and π (nj ) = n 1−α j − wnj − b , if nj > 0 where w is the wage.

(a) Under what condition on w (in terms of the parameters α and b) would firm j be willing to hire a positive number of workers (i.e., choose nj > 0)? Let w¯ denote the level of w for which this condition is just met, and n¯ the amount of labour the would the firm hire if w = ¯w and the firm chose to operate. Solve for n¯ in terms of α and b. (HINT: Assuming the firm does operate, what’s the optimal level of n? When is this better than choosing n = 0?)

(b) Explain intuitively the reason for the condition on w you found in part (a). In what direction does w¯ change if the fixed cost b increases? Explain your answer.

(c) Give a mathematical statement of firm j’s labour demand function nj as a function of the wage w.

(d) Give a mathematical statement of average labour demand n (as a function of w) across all firm’s. For any cases where an individual firm would be indifferent between operating and not operating, assume that a fraction of them choose to operate and the remainder don’t, where any such fraction between 0 and 1 (inclusive) is a possible outcome. Draw a diagram showing this (average) labour demand curve.

(e) Assume there is no government or public goods, i.e., g = τ = 0, and that the household’s utility function is U (c, l) = log (c + φl) where φ > 0 is a parameter. The household maximizes this utility function subject to the budget constraint c = w (1 − l) + π and the NNCs c ≥ 0 and 0 ≤ l ≤ 1. Find the household’s optimal choice of its labour supply NS = 1 − l given the wage w. Draw a diagram showing this labour supply curve. Will this labor supply curve shift if π changes? (HINT: Depending on the value of w, one or more of the NNCs could bind, or the household might be indifferent between multiple bundles that are equally optimal. You may find it helpful to calculate the MRS, and then think about what this says about the shape of the household’s indifference curves.)

(f) An equilibrium in the labour market is a combination of n ∗ , w∗ such that n (w ∗ ) = NS (w ∗ ) = n ∗ , where n (w) is the (average) labour demand function you found in part (c), and NS (w) is the labour supply function you found in part (d). We say the equilibrium is unique if there is only one such combination of n ∗ , w∗ . We say there are multiple equilibria if there are multiple such combinations. We say there are no equilibria if there are no such combinations. Based on your answers from (c) and (d), for each of the four following possible cases, draw a labour market supply-and-demand diagram illustrating it. Be sure to point out where any equilibria are located (either in the diagram, or in words). i. A case where there are multiple equilibria, all having n ∗ = 0. ii. A case where the equilibrium is unique and has n ∗ = 1. iii. A case where the equilibrium is unique and has 0 < n < 1. iv. A case where there are multiple equilibria, and at least one has n ∗ > 0

Answer 1

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