Consider an economy which is divided into different sectors, each producing a differentiated product. Workers in each sector are organized in a trade union which monopolizes the supply of labour to all firms in the sector. Because of its monopoly position, the trade union in each sector may dictate the nominal wage rate  to be paid by employers in that sector. However, employers are free to choose the level of employment. For simplicity, assume that the number of working hours for the individual worker is fixed, so total labour input is proportional to the number of workers employed. Workers in sector  are educated and trained to work in that particular sector, so they cannot move to another sector to look for a job. If a worker fails to find a job in his sector, he therefore becomes unemployed. His real income will then be equal to the real rate of unemployment benefit

• Consider an economy which is divided into different sectors, each producing a differentiated product. Workers in each sector are organized in a trade union which monopolizes the supply of labour to all firms in the sector. Because of its monopoly position, the trade union in each sector may dictate the nominal wage rate  to be paid by employers in that sector. However, employers are free to choose the level of employment. For simplicity, assume that the number of working hours for the individual worker is fixed, so total labour input is proportional to the number of workers employed. Workers in sector  are educated and trained to work in that particular sector, so they cannot move to another sector to look for a job. If a worker fails to find a job in his sector, he therefore becomes unemployed. His real income will then be equal to the real rate of unemployment benefit . An employed worker in sector  earns the real wage  where  is an index of the general price level, so his net income gain from being employed is . The trade union for sector  cares about this real income gain for its employed members, but it also cares about the total number of jobs  secured for the membership. Assume that the union sets the nominal wage with the purpose of maximizing a utility function  of the form:

                                                                    subject to

The representative employer in sector  uses a technology described by the production function:

where  is the volume of real output produced and sold in sector , and  is a productivity parameter. The employer representing industry  produces a differentiated product and therefore has some monopoly power, and hence faces a downward-sloping demand curve of the form:

where  is the aggregate output from all the  sectors, hence  is the market share captured by each industry.

• Derive the price-setting equation and show that the firm always sets its prices as a constant mark-up over marginal cost. Provide an intuitive account.

• Derive the labour demand function for sector . What is the relationship between numerical real wage elasticity of labour demand at the sectoral level and price elasticity for sectoral output demand? Give an intuition account behind this relationship.

NOTE: Some functions did not appear in the above, i have attached images of the questions which has all the materials.