Consider the economy of the Queensland that produces juice and soup. Each indust

Question

Consider the economy of the Queensland that produces juice and soup. Each industry employs workers, but land is specific to the production of juice, and dishes a specific factor in the soup industry. The amounts of land (L) and dishes (D) are L=1 and D=16, while the population size of the Queensland is 10 workers. The production functions and the marginal products of labor are:

J=(Wj)^0.5(L)^0.5, MPWj=0.5(L)^0.5/(Wj)^0.5
S=(Ws)^0.5(D)^0.5, MPWs=0.5(D)^0.5/(Ws)^0.5

where J is the juice output in litres, S is the soup output in litres, and Wj and Ws are the numbers of workers employed in the juice and soup sectors, respectively.

If the price of 1 litre of juice is $2 and the price of 1 litre of soup is $8, what is the wage rate paid to workers in the Queensland? How many workers are in the juice sector? In the soup sector?

Thank you for your help!

Explanation / Answer

As we know in equilibrium,

MPWj/MPWs = Ps/Pj

=> (0.5(L)^0.5/(Wj)^0.5)/(0.5(D)^0.5/(Ws)^0.5) = 8/2

=> 0.5*1^0.5 /Wj^0.5 /(0.5(16)^0.5/(Ws)^0.5) =4

=> Ws/Wj)^0.5 = 1/16

=> Ws/Wj = 1/256 ---(1)

as Ws+Wj =10---(2)

=> solving (1) and (2) we get Ws = 1 ,Wj =9 ( here Ws and wj will come in fraction, but we approaximated it)

So no of workers in Soup sector = 1 , No of workers in juice sector =9

wage/price = Marginal product

so for juice sector,

given marginal product(MPW) = 0.5(L)^0.5/(Wj)^0.5 and price = $1 and L=1,D=16

=> wagej/2 =0.5*(1)^0.5 /(9)^0.5

=> Wagej= $0.333

so wage in juicew sector =$0.333

and for soup,

wage(s)/8 = 0.5*16^0.5/1^0.5

=>wage(s) = $16

=> Wage in Soup sector = $16

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