ORMAT A detailed explanation of Kruman (1980) trade model with impeded trade.
Notation
- consumption of variety i
- production of variety i
- fixed labor requirement
- variable labor requirement
- labor endowment of the home and foreign country
- the portion of the shipped good that arrives to destination
- consumption of the foreign variety in the foreign country
- consumption of the foreign variety in the home country
- output of the foreign variety
- preference parameter such that is the elasticity of substitution
- factory price of the domestic varieties in the home country
- delivered price of the domestic varieties in the foreign country
- factory price of the foreign varieties in the foreign country
- delivered price of the foreign varieties in the home country
Setup
Consumers maximize
Notice that in this utility formulation marginal utility of each product is independent of the consumption of other products. In a more general case the summation sign is usually raised to power .
is restricted to fall between 0 and 1. to satisfy diminishing marginal utility. And to allow for zero consumption of some varieties. These restrictions imply that the elasticity of substitution is greater than 1:
Some useful transformations:, , .
Every producer i produces a unique product that requires the following amount of labor
The shipped good is used up in the transportation, so that only the portion of the shipped good arrives at the destination. With transportation cost there is a difference between the factory and the delivered price because the buyer pays for the delivered good and for the melted part. Denote delivered price with a hat to get the price of the domestic product in the foreign country and the price of the foreign product at home
and
From the first order conditions, the relative demand for foreign varieties at home is:
Quick reminder: By consumer first order condition the ratio of marginal utilities equals prices
or
Which gives
Introducing transportation cost yields
Using definition of prices we have the expression for relative demands in terms of relative wage
Demands
Set income equal to expenditures on all goods
Express value of every good relative to good 0
Rearranging the above expression gives
It is useful to rewrite the above expression as
And then simplify it to get the expression for the share of income spent on each product
The term in the denominator on the right hand side is the CES price index
or .
So the CES demand is a function of the product price relative to the overall price level. The higher is the overall price level the higher is the demand. Several properties of the price index are of note:
(1) even when the prices are the same, the number of varieties makes the price index smaller if :
(2) the elasticity of substitution governs how much the number of goods affects the price index. When the elasticity of substitution is high, the goods are closer substitutes, the power is the smaller negative number, the price index is larger, , the demand is lower.
Substituting elasticities of substitution for the preference parameters the expression becomes
Several interesting properties of the above expression. Use the demand to construct the indirect utility. Note that here we use the formulation where the summation sign is raised to power .
Notice how the whole expression is raised to the power
Further simplify to get
Utility can be thought of as the real wage where the price level captures changes in real wages. Noting that if we define utility as the CES quantity index
the total income (or total expenditure) at home can be expressed as the
Assume that all products are equally priced the utility can be expressed as a function of the number of products
Note the role of the elasticity of substitution . It also reflects the degree of love for variety. Larger moves the products closer to perfect substitutes and reduces by moving it closer to 1. That is any number of varieties would give the same utility as any one of them, perfect substitutes.
Pricing
Krugman follows Dixit, Stiglitz’s (1977) assumption of monopolistic competition and ignores strategic interaction between the firms. Each firm effectively disregards its effect on the combined price index. Firm’s profit is given by
Note that constant markup also implies that the variable profits of the firm
Are constant share of the total revenue
or
This result is useful because it implies that for a given fixed cost the firm that can collect higher revenue is also the firm that will have higher profit.
Free entry
Firms enter until their profits are equal to zero
Substituting for the price gives us
So each firm produces
Larger fixed cost leads to higher output per firm, and so does smaller variable cost. Higher elasticity of demand means that markups are smaller and to cover a given fixed cost firm will have to produce higher quantity.
Number of varieties.
After determining the output per firm we can determine the amount of labor that each firm consumes:
or
This give us the number of varieties that a country can produce in equilibrium
and
Higher fixed cost implies fewer firms. Higher love for variety (smaller ) also means more firms.
World trade in the frictionless world.
In the absence of transportation cost each country will produce the same number of varieties. This means that each consumer will have access to more varieties. The wage in terms of any product remains the same because the price is still determined by the monopoly pricing but the real wage increases because it is a function of the number of varieties.
Why larger country has higher wage in equilibrium.
It is useful to define relative total demand (in quantity terms) for a representative imported versus a representative local variety. Note that for units of the foreign variety to be consumed by the domestic consumers the foreign producers need to produce to account for the part that melts in the way:
Using the expressions for relative demands, for the home country the expression above is
Similarly for the foreign country the expression is
Notice that good that uses relatively cheaper labor commands a higher total expenditure than the good that uses relatively more expensive labor which translates into:
1)
From the budget constraint at home we get
The demand for the foreign variety in the home country can be expressed in value terms relative to the foreign wage as
This is the value of home country’s imports of each foreign variety.
Following a similar procedure we can derive the foreign demand for the domestic varieties in value terms, which will give us the value of home country’s exports. Start with the foreign budget constraint
So the demand in value terms is
The trade balance of the home country in terms of foreign wage is therefore:
Using the expression for the number of varieties produced in each country the expression can be further simplified to
Then to
Expressing labor endowment in relative terms and dividing both sides by L we get
2)
Rewrite
The first term is a scaled version of the export value and the second term is a scaled version of the import value. We know that the relative share increases and decreases with an increase in relative wage. The first term is unambiguously decreasing in , the sign of the second term depends on the effect of on . Using the definition of we get , which is decreasing in because . The second term is therefore also decreasing in . This gives us that
.
So, if we start with equal wages, and sign the trade balance condition, we will know which way the wage needs to adjust to restore equilibrium. Start with equality of wages and show by contradiction that it is not an equilibrium. In case we know that and the expression for the trade balance becomes
So the trade balance condition for the case of equal wages, , becomes:
The case of is familiar. It is the case of no transportation cost. When there is no transportation cost any amount of the final good that is shipped arrives completely, , so that . When transportation costs are present , trade is not balanced if the countries differ in size. For convenience, think that Home is the larger country: . Then trade balance is positive at equal wages. To realize that the relative wages adjust upwards to the equilibrium, note that the balance of trade is decreasing in relative wage. Both terms in the expression for the trade balance (the value of exports and the value of imports) are decreasing in relative wage. To see this rewrite as
Applying inequalities yields:
So the relative wage would have to adjust upwards from one to bring the trade balance back to zero. Hence, in equilibrium the larger country has a larger wage.
The home market effect
A useful way to rewrite equation (25) from the paper in terms of shares.
note that
To check intuition, note that when endowments are equal the expression becomes:
.
12