Chapter 3, The Budget Constraint and the Consumer'sOptimal Choice Video Solutions, A Short Course in Intermediate Microeconomics with Calculus

Problem 1

The consumer's original budget equation is $p_1 x_1+p_2 x_2=M$, where $p_1$ and $p_2$ are the original prices and $M$ is the original income level.
(a) If $p_1$ doubles and $p_2$ falls by half, what is the consumer's new budget equation? How has the slope of the budget line changed?
(b) If $p_1$ doubles and $M$ triples, what is the equation for the new budget line? How has the slope of the budget line changed?

Akash M

Numerade Educator

01:34

Problem 2

The consumer's utility function is $u\left(x_1, x_2\right)=x_1 x_2^2$.
(a) Graph his budget constraint for $p_1=3, p_2=2$ and $M=900$, and write down the equation for his budget line.
(b) Using the $M R S=M U_1 / M U_2=p_1 / p_2$ tangency condition, find his optimal con-

Nick Johnson

Numerade Educator

03:10

Problem 3

George enjoys apples ( $a$ ) and bananas $(b)$. If he spends his entire allowance, he can afford 10 apples and 30 bananas. Alternatively, he can afford 15 apples and 15 bananas. The price of an apple is $$\$ 3$$.
(a) Calculate George's allowance, and the price of bananas.
(b) Assume his utility function is $u(a, b)=a+b$. How many apples and bananas will he consume?

Maryam Shahid

Numerade Educator

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Problem 4

There are two goods in the world, pumpkins $\left(x_1\right)$, and apple cider $\left(x_2\right)$. Pumpkins are $$\$ 2$$ each. Cider is $$\$7$$ per gallon for the first two gallons. After the second gallon, the price of cider drops to $$\$ 4$$ per gallon.
(a) Peter's income is $$\$54$$. Draw his budget line. Solve for the intercepts on the $x_1$ and $x_2$ axes, and the kink in the budget line. Show these in your graph.
(b) Peter's utility function is $u\left(x_1, x_2\right)=x_1+3 x_2$. Sketch some indifference curves in your graph. Find Peter's optimal consumption bundle $\left(x_1^*, x_2^*\right)$.
(c) Paul's income is $$\$22$$. Draw his budget line in a new graph. Solve for the intercepts on the $x_1$ and $x_2$ axes, and the kink in the budget line. Show these in your graph for Paul.
(d) Paul's utility function is $u\left(x_1, x_2\right)=\min \left(3 x_1, 2 x_2\right)$. Sketch some indifference curves in your graph for Paul. Find Paul's optimal consumption bundle $\left(x_1^*, x_2^*\right)$.

Oluwadamilola Ameobi

Numerade Educator

01:33

Problem 5

Olivia gets an allowance of $$\$ 50$$ this week, but it will have to last her for two weeks, as Mom pays her every other week. Let $c_1$ be her consumption this week (measured in units of stuff), and let $c_2$ be her consumption next week (measured in the same units). The price of one unit of stuff this week is $$\$ 1$$. Next week the price will be higher, because of inflation. Assume the inflation rate $\pi$ is 1 percent per week, or 0.01 per week when expressed as a decimal. (About notation: When we use the symbol $\pi$ in this book we do not mean 3.14. In this exercise and in Chapter 5, $\pi$ means inflation. From Chapter 8 onward, $\pi$ means profit.) Therefore, the price of one unit of stuff next week will be $$\$ 1(1+\pi)=\$ 1.01$$. Olivia can borrow or save at the local bank; whether she is borrowing or saving, the interest rate $i$ is 1 percent per week, or 0.01 when expressed as a decimal. Olivia's utility function is $u\left(c_1, c_2\right)=\ln \left(c_1\right)+\ln \left(c_2\right)$.
(a) Write down Olivia's budget constraint, first in the abstract (with $M$, $\pi$, and $i$ ), and then with the given values incorporated.
(b) Find her optimal consumption bundle $\left(c_1^*, c_2^*\right)$.
(c) Assume the inflation rate rises to 10 percent, and the interest rate drops to zero. Find her new optimal consumption bundle.

Jennifer Stoner

Numerade Educator

Problem 6

Sylvester's preferences for consumption this period $\left(c_1\right)$ and consumption next period $\left(c_2\right)$ are given by the utility function $u\left(c_1, c_2\right)=c_1^2 c_2$. Suppose the price of consumption this period is $$\$ 1$$ per unit. Assume the interest rate $i$ is 10 percent; and the inflation rate $\pi$ is 5 percent. Sylvester has income of $$\$ 100$$ this period and $$\$ 100$$ next period.
(a) Write down the equation for his budget line, and show it in a graph. What are the intercepts of the budget line on the $c_1$ and $c_2$ axes? Label them in your graph. What is the slope of the budget line? Explain it briefly. Where is the zero savings point? Explain it briefly.
(b) Find Sylvester's optimal consumption bundle $\left(c_1^*, c_2^*\right)$. We'll call this $S$ for short. Is Sylvester a saver or a borrower? Show $S$ on your graph, and include the indifference curve passing through it.
(c) Suppose the interest rate $i$ falls to 5 percent. Show the new budget line in your graph. Find Sylvester's new optimal consumption bundle $S^{\prime}$.
(d) Is Sylvester better off or worse off at the new point $S^{\prime}$ ?

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