Question: Standard Trade Model, Increasing Returns to Scale & Firm Heterogeneity

1. (Standard Trade Model) Bohemia is a country which produces two goods, wine and cheese, using capital and labor. The production process in both goods features constant returns to scale, but diminishing returns to each input factor.

 

1. 1. What is Bohemia’s consumption possibilities set in autarky? In other words, what combinations of wine and cheese can Bohemia afford to consume? Indicate it in a diagram. Denote the autarky production point by A.
   2. What is Bohemia’s consumption possibilities set if it can trade in world markets at some fixed relative price? Does the short run differ from the long run? Illustrate this in the same diagram.
   3. Show and discuss the sources of welfare gains from opening up to international trade in the short and long run.

 

2. (Increasing Returns to Scale) Consider a slightly deferent version of the general- equilibrium monopolistic competition model presented in class. Assume Home is a country populated by L individuals who are both consumers and workers. There is only one sector in this economy - the breakfast cereal sector.

Demand

Consumer preferences over different types of breakfast cereal (i) are given by:

 

where Q_i is the amount of cereal i that is consumed and N is the number of cereal varieties available to the consumer. These preferences are homothetic, so we can treat the demand side of this economy as if there were a representative consumer. The demand of this consumer for breakfast cereal j is given by

 

 

where P_j is the price of cereal j, Q is defined above, and

 

Supply

Suppose that a firm that produces breakfast cereal of type j has the following production function:

Q_j = L_j ! a;

where Q_j is the amount of breakfast cereal j produces, L_j is the total number of workers it hires, and a is the number of workers who are required before any production takes place (e.g. the CEO, accountant, etc). Therefore, L_j a is the number of production workers. This firm faces the following total cost function:

T C_j = W (Q_j + a) ;

where W is the wage paid to the workers it hires. You can think of W as the marginal cost (c) and aW as the fixed cost (F) from the model in class. Note that all firms have identical production technology.

Labor market

The wage in this economy adjusts such that the number of workers demanded by producers of breakfast cereals equals the total labor supply. The labor market clearing condition is:

 

1. 1. Price: Assume that each individual firm takes market demand Q and average price P as given. Rewrite the demand curve for firm j in the form P_j (Q_j), i.e. P_j as a function of Q_j. What are total revenue and marginal revenue in terms of Q_j? What is the profit maximizing condition for firm j? Given this condition, use the demand curve to write the firm’s desired price as a function of W. How is the optimal mark-up affected by b?
   2. Firm size: Assume that there is free entry of firms in the market for breakfast cereal. How much will firm j produce? Is the equilibrium firm size increasing or decreasing in the number of non-production workers (fixed costs)? What is the intuition for this result?
   3. Number of firms: Use your answer to (b) and the firm’s production function to solve for its labor demand, L_j. Then use the labor market clearing condition to solve for the equilibrium number of firms. Is the equilibrium number of firms increasing or decreasing in the number of non-production workers (fixed costs)? Is the equilibrium number of firms increasing or decreasing in population? What is the intuition for these results?
   4. Wages: Use the expression for the equilibrium price charged by firm j from (a) to derive an expression for real wages (W=P). How do real wages vary with country size?
   5. Welfare: Use what you have learnt about firm size and the number of firms, together with the utility function of the representative consumer, to write home country welfare as a function of L, a and b. Is welfare increasing or decreasing in country size? Why?