Question

The country of Podunk produces only wheat and cloth, using as inputs land and labor. Both are produced by constant retums-to-scale production functions. Wheat is the relatively landintensive commodity. a. Explain, in words or with diagrams, how the price of wheat relative to cloth (p) deter mines the land-labor ratio in each of the two industries. b. Suppose that \(p\) is given by external forces (this would be the case if Podunk were a "small" country trading freely with a "large" world). Show, using the Edgeworth box, that if the supply of labor increases in Podunk, the output of cloth will rise and the output of wheat will fall.

Short answer

Answer: When the supply of labor increases in Podunk, the output of cloth will rise and the output of wheat will fall.

Step-by-step solution

Understand the constant returns-to-scale production functions

Constant returns-to-scale production functions imply that doubling the inputs (land and labor) would result in doubling the output. Since wheat production is land-intensive, we can assume that the production function of wheat is \(Q_w = F_w(L_w, T_w)\), whereas the production function of cloth is \(Q_c = F_c(L_c, T_c)\), where \(L_w, L_c\) represent labor and \(T_w, T_c\) represent land employed in wheat and cloth industries respectively.

Analyze the effect of relative price on land-labor ratio

The relative price of wheat (\(p\)) is the price of wheat divided by the price of cloth. A higher relative price of wheat would increase the profitability of producing wheat and would lead to a higher land-labor ratio \((\frac{T_w}{L_w})\) in the wheat industry, and lower land-labor ratio \((\frac{T_c}{L_c})\) in the cloth industry. On the other hand, a lower relative price of wheat would reduce the profitability of producing wheat, leading to lower land-labor ratio in the wheat industry and higher land-labor ratio in the cloth industry.

Introduce the Edgeworth box and its assumptions

The Edgeworth box is a graphical tool used to analyze the allocation of factors of production (land and labor) between two industries (wheat and cloth). The width of the box represents the total land available \((T_w + T_c)\), and the height of the box represents the total labor available \((L_w + L_c)\). The origin of the box representing the wheat industry is at the lower-left corner, while the origin of the box for the cloth industry is at the upper-right corner.

Analyze the impact of an increase in labor supply

Suppose that the relative price of wheat (\(p\)) is given by external forces, and the labor supply in Podunk increases. This increase in labor would change the shape of the Edgeworth box, making it taller while keeping the width the same. This new shape of the box would give a new allocation in land and labor across industries, which satisfies the condition of equal returns to factors in both industries. We could observe that with the increase in labor supply (height of the box) and the equalization of returns to factors, the new equilibrium would involve the cloth industry employing more labor and land (thus increasing its output), while the wheat industry would employ less labor and land (thus decreasing its output). This is because cloth production is labor-intensive, and the additional labor would result in a higher output of cloth, while the wheat industry employs less labor and land due to the given relative price. In conclusion, when the supply of labor increases in Podunk, the output of cloth will rise and the output of wheat will fall, given the external relative price of wheat and the constant returns-to-scale production functions.

Key concepts

Constant Returns-to-Scale Production Functions

In the context of an economy like Podunk's, constant returns-to-scale production functions are an important concept to grasp. This principle indicates that if you double all inputs used in production, such as labor and land, the output will also double. This relationship holds true for any proportional increase in inputs.

For students, it's essential to understand that this concept is not about changing the combination of inputs, but rather scaling all inputs equally. In practice, if Podunk's wheat farms were to double both the amount of land and labor they use, then the wheat production would also double, assuming a constant returns-to-scale production function.

It’s important to note that constant returns to scale represent an ideal scenario. Real-world phenomena such as managerial challenges or limits on efficiency often mean that actual production may experience decreasing or increasing returns to scale as the scale of production grows.

Land-Labor Ratio

The land-labor ratio is a measure that compares the amount of land to the amount of labor used in the production of a good. For example, wheat, being a land-intensive commodity, would have a higher land-labor ratio compared to cloth, indicating that wheat production requires relatively more land per unit of labor.

Podunk's decision on how much land and labor to allocate to wheat versus cloth production is directly influenced by this ratio. A high land-labor ratio signifies that more land is being used with respect to labor, which is optimal for wheat production. On the other hand, cloth production, which is less land-intensive, would have a lower land-labor ratio favoring more labor use. Students should recognize that changes in the price of wheat relative to cloth will shift the optimal land-labor ratio in each industry, impacting how resources are distributed between the two sectors.

Relative Price Effect

The relative price effect refers to how the price of one good compared to another influences economic decisions. In Podunk's case, if external forces set the relative price of wheat higher, then farmers are incentivized to produce more wheat because it yields more revenue per unit sold. This decision will alter the land and labor allocation leading to a higher land-labor ratio for wheat.

Conversely, if the relative price of wheat decreases, it becomes less profitable, and farmers might shift resources towards producing more cloth, thus changing the land-labor ratio again. For students, visualizing this concept might involve plotting the prices and quantities on a graph to observe how relative changes affect production choices.

To dive deeper, it would mean understanding that the price change can lead to a reallocation of resources within the Edgeworth box model, ultimately affecting the outputs for both wheat and cloth in the economy. The emphasis here is on the responsiveness of production decisions to changes in relative prices within the framework of constant returns-to-scale production functions.