Question

Assume that a country is endowed with equal amounts of capital (K) and land (T). The two goods in this economy are x-ray machines (X) and yucca (Y). Prices are Px = Py = $8 and the country's total labor force is L = Lx + Ly. Labor is fully mobile between the two sectors. K is specific to the x-ray sector and T is specific to the yucca sector. Marginal products are given as:

MPₗˣ = 2K/Lₓ and MPₗʸ = T/Ly. MPₖˣ -Lₓ/K and MPₜ ʸ - Ly/4T.
a) Draw a graph to illustrate the allocation of labor in equilibrium. Your graph should have two vertical axes, one representing the price of labor in the yucca sector and the other representing the price of labor in the x-ray machine sector
b) How is labor distributed between the x-ray and yucca sectors in equilibrium?
c) Now assume, the prices of both X-rays and yucca suddenly decrease by 50%. What is the new labor distribution?
d) Has the output of x-rays and yucca changed?
e) Under the same price change, what happens to:
i) The real wage of workers?
ii) The real return to capital?
iii) The real return to landowners?

Answer 1

In equilibrium, labor is evenly distributed between sectors to equalize wages, based on the marginal product and market price. A 50% decrease in the price of the goods (x-rays and yucca) decreases the demand for labor, changing output and labor distribution. The real wage, return to capital, and return to landowners are also negatively impacted.

Step-by-step explanation:

Understanding the Impact of Price Changes on Labor Distribution and Output

The allocation of labor in equilibrium is guided by the principle that labor moves between sectors to equalize the wage across sectors. As labor is mobile, it will be distributed in such a way that the value of the marginal product of labor (VMP) is equal across sectors. The VMP is dependent on the market price of the product and the marginal product of labor.

In equilibrium, labor will be allocated so that the wage in the x-ray sector equals the wage in the yucca sector: $8 x MP₁ₓₓₓ - Wₓ = $8 x MP₁ʸ = Wʸ. If both the prices of x-rays and yucca decrease by 50%, the VMP will fall, and the demand for labor in both sectors will decrease, leading to a new equilibrium of labor distribution. Additionally, the output of x-rays and yucca will change, as the decrease in price affects the profitability of producing each good, which in turn affects the allocation of labor.

Regarding the effects of the price change:

• The real wage of workers will decline as the VMP decreases with lower prices.
• The real return to capital and landowners will also decrease, reflecting the lower prices of products specific to their respective sectors.