Question

Consider two countries, A and B. Each country has 120 total labour
hours to produce two goods X and Y , and each country has preferences
given by the utility function
U = min(x,y)
No other inputs are required to produce these goods. Their production
technologies are the following. In country A, production of one unit of
X requires 2 labour hours and that of Y requires 1 labour hour. In
country B, production of one unit of X requires 1 labour hour and that
of Y requires 2 labour hours.
(a) Find the equilibrium allocation of labour hours between the goods
in each country under a no trade situation.
(b) Now suppose that each country has specialized in the activity in
which it has a comparative advantage and then they trade at a
`one for one' price. Find the optimal consumption of two goods
by each country.
(c) Draw the production possibility frontier of each country and that
of the joint production possibility frontier explaining their shapes
and slopes.

Answer 1

a) in AUTARKY

country A. , Labour endowment is 120 hours. This can be seen analogous to income . We have 120 labour hours to allocate between production of good x and y. For production of a unit of x , 2 labour hour is required and for y , 1 labour hour is required. This can be understood as analogous to prices.

Number of x produced * 2 + number of y produced 120

2x + y 120

And U = min( x,y )

Utility can oy be maximised when x= y

To maximize utility whole of 120 labour hours is used.

2x + y = 120 or 2x + x = 120 or x = 40 , y = x = 40

2*40 = 80 hours of labour is allocated to x and 40 to y

Country B

Here 1 unit of x requires 1 unit of labour, 1 unit of y requires 2 units of labour .

Labour hour endowment is same as in country A 120 hours

Similarly , we construct constraint x + 2 y 120.

Utility function is same as for country A therefore to maximize utility we chooses , x= y .

Every labour hour should be used in order to maximize budget constraint. x + 2y = 120 or x + 2x = 120 or x = 40 and y = x = 40 units.

Labour hours allocated in x ' s production = 40 hours

And that allocated to y's production = 2*40 = 80 hours.

b) price of x in country A \price of y in country A .= labour hour required to prduce a unit of x / labour hour required to produce a unit of y = {Px/Py}_A = 2

Similarly price of x in country B / price of x in country A ={ Px /Py}_B= 1/2

Clearly , {Px/Py}_A > {Px/Py}_B

Country A has comparative advantage in Y and Country B has comparative advantage in X . Therefore country A will allocate all its labour hours in production of y . A unit of y can be produced using 1 labour hour , therefore using 120 labour hours , 120 units of y can be produced

Similarly , country B will produce 120 units of x .

Given price to be one to one, Total income of Country A and B will be 120 .

Demand function will be same in country A and B as they have same utility function given by x = y = M/(Px+ Py ) = 120/(1+1) = 60

Both country A and B consumes (x,y) = (60,60)

c) slope of production possibility frontier shows the opportunity cost . Since production of a unit of X requires twice labour hours than production of a unit of Y, opportunity cost of producing a unit of X is 2 units of Y. Slope = -2 . Intercept at Y axis (0,120) shows the total units of Y that can be produced when all labour hours are employed in production of Y . Similarly intercept at X axis ( 60, 0) shows total amount of X that can be produced by employing total labour hours endowment in producing X. Fig 1 shows production possibility frontier of country A

Fig 1

Similarly slope of Production possibility frontier of country B represents the opportunity cost of producing a unit of X i.e.,1/2 . Slope = -1/2. Intercept at Y axis (0,60) shows total units of Y that can be produced when all labour hours are employed in production of Y . Similarly intercept at X axis (120,0) shows total amount of X that can be produced by employing total labour hours endowment in producing X. Fig 2 shows production possibility frontier of country B.

fig 2

Joint production possibility frontier is the production possibility frontier after trade show. In figure 3.

Fig 3