a. Consider the production function Q = 0.8X10.3X20.8. Use stat_function with ggplot to plot
the isoquant where Q = 100, 200, and 300. Let X₁ be on the x-axis and X2 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")
80/9X1
b. 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 X2 ("rise") you could
substitute for one unit of X₁ ("run"), to hold your output Q at a constant level. Calculate the
MRTS for Q = 0.8X10.3 X 20.8: the formula is MRTS = 20/ax, You might start by taking
natural logs of both sides: In Q = ln 0.8 +0.3 ln X1 + 0.8 ln X2. Evaluate and interpret the
MRTS at (X1, X2) = (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 X2,
or getting ____ units of X1.
c. 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?
