Consider an aggregate demand and aggregate supply model where, in the short run, aggregate capital is fixed at the level $\bar K$. The aggregate demand curve, aggregate output (Y) demanded as a function of aggregate price level (P), is given by a standard downward-sloping curve. The aggregate supply curve, aggregate output (Y) supplied as a function of aggregate price level (P), is not standard, and the question leads you to derive the aggregate supply curve. The aggregate production function is linear in capital and labour (L): $ Y = AL+ \bar K$, A > 0. The labour union is very powerful and dictates the minimum aggregate nominal wage rate as W. Each worker is endowed with one unit of labour which they supply inelastically if the producers offer the nominal wage $ W > \bar W$. A worker does not supply any labour if $W < \bar W$. At W = W, a worker is indifferent between supplying and not supplying her labour endowment. The number of workers available in the economy is fixed at $\bar L$.
(a) Derive, with a clear explanation, the aggregate labour supply $(L^S)$ in this economy as a function of the aggregate nominal wage rate, W.
(b) Note that the marginal product of labour is constant, A > 0.
(i) Derive, with a clear explanation, the aggregate labour demand $(L^D)$ in this economy as a function of the real wage rate, W/p.
(ii) Using your answer to part (i) above, derive the aggregate labour demand $(L^D)$ in this economy as a function of the aggregate nominal wage rate, W.
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