For more information on these topics, see Allen, Doherty, Weigelt, and Mansfield Chapter 5: Production Theory.
First, here are some practice questions to review what we covered in Classwork 10.
The law of diminishing marginal returns states that as additional units of an input (like labor) are added to a fixed input (such as capital or land), the additional output produced by each new unit of input will eventually decrease. For example: consider the production process for apples. If you only have a few workers, the workers will pick only the apples that are easiest to reach. If you add another worker at that point, their marginal contribution will be high because there are plenty of apples to pick that are close to the ground.
But if you have tons of workers, they’ll try to pick every apple off every tree, no matter how hard it is to reach. You can imagine if you add another worker at that point, their marginal contribution would be much lower: they’d spend all day looking and only be able to find a couple of apples to pick. This is the law of diminishing marginal returns. This principle is fundamental in production theory because it illustrates that there are limits to how much can be produced by simply increasing one type of input.
An isoquant is a curve that represents all the efficient combinations of inputs that can produce a specific level of output. In consumer theory, the equivalent concept is an indifference curve, which shows all the combinations of goods that provide the consumer with the same level of utility.
For example, imagine you own an apple orchard. You can either plant more trees or hire more apple pickers, as these inputs can partially substitute for each other in the production process. To achieve the same output, you could plant a lot of trees and hire fewer workers, plant fewer trees and hire more workers, or be somewhere in the middle (a preview: the profit maximizing choice of inputs will depend on the cost of labor versus the cost of trees).
If production is given by \(Q(X_1, X_2) = X_1 X_2\), a couple of isoquants would look like this (blue line: output level 1, purple: output level 2, red: output level 3):
library(tidyverse)
ggplot() +
stat_function(fun = function(x) 1 / x, color = "blue") +
stat_function(fun = function(x) 2 / x, color = "blueviolet") +
stat_function(fun = function(x) 3 / x, color = "violetred1") +
ylim(0, 5) +
xlim(0, 5) +
labs(title = "Isoquants", x = "X1", y = "X2")
stat_function with ggplot to plot the isoquant where Q = 100, 200, and 300. Let \(X_1\) be on the x-axis and \(X_2\) be on the y-axis.library(tidyverse)
ggplot() +
stat_function(___) +
stat_function(___) +
stat_function(___) +
xlim(0, 500) +
ylim(0, 500) +
labs(title = expression("Isoquants for " * Q == 0.8 * X[1]^{0.3} * X[2]^{0.8}),
x = "X1", y = "X2")The marginal rate of technical substitution (MRTS) is -1 times the slope of the isoquant. Recall that a slope is just “rise over run”, so the MRTS tells you how much of \(X_2\) (“rise”) you could substitute for one unit of \(X_1\) (“run”), to hold your output \(Q\) at a constant level. Calculate the MRTS for \(Q = 0.8 {X_1}^{0.3} {X_2}^{0.8}\): the formula is \(\text{MRTS} = \frac{\partial Q / \partial X_1}{\partial Q / \partial X_2}\). You might start by taking natural logs of both sides: \(\ln Q = \ln 0.8 + 0.3 \ln X_1 + 0.8 \ln X_2\). Evaluate and interpret the MRTS at \((X_1, X_2) = (1, 1)\). When you are using equal amounts of inputs 1 and 2 in the production process, you’d be able to hold production constant by either getting ___ units of \(X_2\), or getting ___ units of \(X_1\).
Continuing from the previous question, if you’re using equal amounts of inputs 1 and 2 in your production process and if they cost the same amount, what would be the lower cost way to expand production: start using more of input 1, or start using more of input 2? Why?
Show that if \(Q\) has this form: \(Q = L^a K^b c\) and you increase \(L\) and \(K\) by 1% each, the output elasticity is the sum of \(a\) and \(b\). That is, if \(a + b = 1\), a 1% increase in \(L\) and \(K\) lead to a 1% increase in \(Q\). If \(a + b \gt 1\) (\(\lt\)), a 1% increase in \(L\) and \(K\) lead to a greater (less) than 1% increase in \(Q\). Hints:
Until now, we haven’t considered costs. If input 1 costs \(P_1\) and input 2 costs \(P_2\), then your costs \(M\) given your choice of input quantities \(X_1\) and \(X_2\) is equal to \(P_1 X_1 + P_2 X_2\). Take your ggplot from part a) and add the isocost line of input combinations you can obtain with \(P_1 = 1\), \(P_2 = 2\), and \(M = 450\). By eyeballing, what is the highest isoquant you can reach given this isocost curve? How much of \(X_1\) and \(X_2\) are used in production?
Mathematically, the point of tangency is where the slope of the isoquant curve (you know the \(\text{MRTS} = \frac{\partial Q / \partial X_1}{\partial Q / \partial X_2}\)) is equal to the slope of the isocost curve (-\(\frac{p_1}{p_2}\)). Solve exactly for the producer’s choice from part g: when the producer spends $450 on inputs, what’s the maximum quantity they can produce and how much of each input do they purchase? Show your work.
The tangency condition says that the MRTS is equal to the price ratio: \(\frac{\partial Q / \partial X_1}{\partial Q / \partial X_2} = \frac{p_1}{p_2}\). Knowing that the partial derivative of Q with respect to an input is that input’s marginal product, we can simplify: \(\frac{MP_1}{MP_2} = \frac{p_1}{p_2}\). Rearranging, we have that: \(\frac{MP_1}{p_1} = \frac{MP_2}{p_2}\). That is, producers will choose an input bundle where the marginal product per dollar spent for each input is identical. If there are more than two inputs, we have:
\[\frac{MP_1}{p_1} = \frac{MP_2}{p_2} = \frac{MP_3}{p_3} = \ ... = \frac{MP_n}{p_n}\] If a producer uses 2 inputs: A and B, and they find that A has a marginal product of $80 and the B has a marginal product of $250, would they move closer to profit maximization by employing more of A relative to B, or more of B relative to A? Why?