Effects of...

Why does money affect output? (Basic Issues in Aggregate Supply)

Lucas Island Model

This began with the Lucas critique that individuals are rational enough to know when the money supply is being changed or their demand has actually risen. Intuition: when a producer observes a change in the price of his product, he knows it could be due to a change in relative prices (good) or a change in aggregate prices (bad). He'll attribute some of the increase to each, and thus nominal shocks can still affect output. Over time though, individuals will learn and monetary shocks will only matter if they are unexpected. He models that in equilibrium, output growth is determined by unexpected money shocks, however, the evidence does not support this. 

Imperfect Competition

After Lucas (1970s) exposed a theoretical crisis within Keynesian economics, that is, rational actors will adjust their behavior to monetary policy (parallel of Ricardian equivalence), new Keynesian literature became concerned with searching for rigorous models of wage/price stickiness based on maximizing behavior and rational expectations. To build up aggregate supply issues from microfoundations, these theories revolve around models of imperfect competition via monopolistically competitive firms (price makers). 
"New Keynesian economists inhabit a brave new theoretical world characterized by imperfect competition, incomplete markets, heterogeneous labor, and asymmetric information, and where agents are frequently concerned with fairness."
These affect empirical observations through nominal and real rigidities.
Assumptions:
No government, no investment, no net exports. Focus is on AS rather than AD. Optimal consumption is a decreasing function of the price and increasing in real income. Optimal labor supply increases with real wage. Firms are monopolistically competitive, thus they mark-up prices over marginal costs, and output is at a socially un-optimal low level. 
Key equations:
Household maximization problem:
\(\mathcal{L} = [\int \limits _{t =0} ^1 C_i ^{\frac{\eta - 1}{\eta}} di] ^{\frac{\eta}{\eta - 1}} + \lambda [S - \int \limits _{t = 0} ^1 P_i C_i di]\)
Firm's profits are:
\(\frac{P_i}{P} = \frac{\eta}{\eta - 1} \frac{1}{A_i} \frac{W}{P}\)
Implication is that firms markup price in accord with the elasticity of demand. The second two fractions are the marginal cost of the good. This model simply exhibits that consumers have a downward sloping demand curve and that AS is upward sloping. There isn't any implication that money has real effects  yet. 

Nominal Rigidities (Mankiw's Menu Costs) 

A nominal rigidity occurs if something prevents the nominal price/wage level from adjusting so as to precisely mimic nominal demand disturbances. =
Examples include: long-term wage contracts (firms and workers benefit from planning ahead and watching other firms' prices) and menu costs (hardly any profit incentive to change prices marginally, having large aggregate effects).
The conclusion is that this just causes coordination problems. However, firms will still adjust under reasonable price elasticities. Thus, we need real rigidities as well.   

Real Rigidities (Blanchard and Kiyotaki)

A real rigidity occurs if some factor prevents real prices/wages (as determined by firm's MC and MR) from adjusting or there is relative stickiness between prices/wages. These are usually triggered by nominal changes, thus magnifying the non-neutralities resulting from small nominal frictions.
Examples include: price mark-ups (prices less likely to respond to marginal cost/revenue increase/declines because of mark-ups added), customer markets (low frequency of search relative to frequency of purchase, so increased prices may create a bigger loss), complexity of the input-output table (gradual adjustment of prices is safest for firms operating in uncertain world where information is imperfect), capital market imperfections (due to asymmetric information that must be allayed, external finances is more costly to a firm than internal finance), and finally, efficiency wages.
Basic Efficiency Wages Model
Key Equations: