## Homework Assignment 1

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC - Fall 2017

Due on Wed Oct 11 2017

Homework assignments will be turned-in on paper (Letter size), at the beginning of the corresponding lecture. Late assignments will not be accepted.

Only four randomly-chosen questions will be graded.

It is graded out of 100 points and 5 points will be subtracted if you do not staple your homework.

### Question 1:

You are waiting to board at an airport for a trip to Korea. You decided to buy cigarettes and Swiss chocolates for your friends at a shop in the airport. You have ${$}$30 only. A price for one pack of cigarettes is${$}$2 and a price for one bar of Swiss chocolates is ${$}$3. You also learned from a traveller's book that you can only bring 10 packs of cigarettes into the country at the maximum and are not allowed to buy more than that$($"cigarettes rationing"). 1. Write down the budget set and budget line in equations for the case without regulation on cigarettes amount. 2. Draw a budget line for each of two cases in a same graph: without the regulation on cigarettes amount and with the regulation in place. Put Swiss chocolates on the$y$axis and cigarettes on the$x$axis. Label$x$and$y$intercepts. 3. Suppose the price of a bar of Swiss chocolates increases to${$}$4 due to an increase in cocoa powder. Draw a new budget line with the cigarettes rationing under the new price in the same graph.
4. Can you afford an bundle $($cigarettes = 7 packs, chocolates = 5 bars$)$ under the old price of chocolates $($i.e. $p_2 = 3$ $)$? How about under the new price of chocolates $($i.e. $p_2 = 4$ $)$?

A consumer has an income of ${$}$1,000 to spend on food and medicine. The price of one unit of food is${$}$5 and the price of one unit of medicine is ${$}$10. For each question, write down the mathematical expression of the budget constraint and draw it carefully. Put food on the x-axis and medicine on the y-axis. Label the incercepts, slopes, and kinks. 1. Suppose the consumer receives coupons for 50 units of food and those can only be used to buy food. 2. Suppose the consumer receives coupons for 50 units of food and can sell the coupons at half of the market price of the food. 3. Instead of the coupon, the consumer receives a 20% discount for additional units of food that exceed 100 units. That is, the consumer pays${$}$5 for each of the first 100 units and ${$}$4 for each additional unit. ### Question 3: This question allows you to evaluate how to think about the welfare of consumers. Assume a consumer's welfare is driven by what he/she consumes. 1. Suppose there are only two types of goods to consume: food and leisure. An average Californian citizen has a daily income of${$}$100. The price of one meal of food is ${$}$20 and the price$($or value$)$of one unit of leisure is${$}$10. An average citizen in Mississippi has a daily income of ${$}$50. The price of one unit of food is${$}$10 and the price $($or value$)$ of one unit of leisure is ${$}$10. Who is more well-off in terms of the bundles of goods they can consume?$($hint: draw budget sets.$)$2. Suppose you earn${$}$100 per day. One day your boss increases your income by 50% solely due to your high performance at work. Are you more well-off, worse-off, the same, or cannot answer compared to before the change?

3. Suppose the boss had lied to you. The reason why she increased your salary by 50% is because all other prices also increased by 50%. Are you more well-off, worse-off, the same, or cannot answer compared to before the change?

4. Suppose you are a big fan of Japanese food and Chinese food and those are the only two goods you consume. Suppose your income does not change and the price of Japanese food decreases. However, the price of Chinese food increases. Are you more well-off, worse-off, the same, or cannot answer compared to before the changes in prices?

### Question 4:

Answer the following questions regarding preferences:

4.1. Suppose you have two distinct bundles $X$ and $Y$, and, for you, $X$ is strictly better than $Y$. Explain briefly using a graph and words why the two indifference curves associated to two bundles $X$ and $Y$ -- $I(X)$ and $I(Y)$ cannot cross each other.

4.2. Explain using a graph and words why if the assumption of monotonic preferences $($aka “more-is-better”$)$ implies that indifference curves are not thick and they must be downward sloped.

4.3. Provide three examples of goods that are typically regarded as perfect substitutes. Explain briefly.

4.4. Provide three examples of goods that are typically regarded as perfect complements. Explain briefly.

### Question 5:

For each of the following functions, i) pick three utility levels and draw the precise indifference curves that are associated with the levels of your choice, ii) label the utility level of the lines -- you cannot just draw random lines and assign arbitrary utility levels, and iii) give the name of preferences they represent (hint: see figures in textbook chapter 3).

1. $u(x_1,x_2) = x_1 + x_2$
2. $u(x_1,x_2) = min(x_1, x_2)$
3. $u(x_1,x_2) = x_1$
4. $u(x_1,x_2) = x_1 - x_2$
5. $u(x_1,x_2) = \sqrt{x_1 x_2}$
6. $u(x_1,x_2) = -${$(x_1 - 3)^2 + (x_2 - 2)^2$}

### Question 6:

Draw an indifference curve and its weakly preferred set for two goods that exhibit following preferences:

1. monotonic and convex preferences

2. monotonic and concave preferences

3. non-monotonic preferences

### Question 7:

Consider a utility function $u(x,y) = x^{a} y^{b}$, where $0 < a < 1$ and $0 < b < 1$. Also assume that $x,y > 0$.

7.1. Derive the marginal utility of $x$ and the marginal utility of $y$ and state whether or not the assumption that more is better is satisfied for both goods.

7.2. Does the marginal utility of $x$ diminish, remain constant, or increase as the consumer buys more $x$? What does it mean in words?

7.3. What is $MRS_{x,y}$?

7.4. Suppose $a,b = \frac{1}{2}$. What is $MRS$ at $(x,y) = (1,4)$ and what does it mean in words? What is $MRS$ at $(x,y) = (4,1)$ and what does it mean in words?

7.5. Continue assuming $a,b = \frac{1}{2}$. Draw two indifference curves of the utility function. To get full credit, draw one with the level of utility 2 and the other with the level of utility 3. For each indifference curve, label at least one point on the curve.

7.6. Draw tangent lines at $(x,y) = (1,4)$ and $(x,y) = (4,1)$ in the same graph and label the slopes you solved in $MRS(1,4)$ and $MRS(4,1)$.

### Question 8:

8.1. Consider a transformation of the utility function in Question 7 using $u' = ln (u)$. In other words, the new utility function $u' = ln (u) = ln(x^{a}y^{b}) = a \times ln(x) + b \times ln(y)$. What is $MRS_{x,y}$ of this new utility function? Is it the same as or different from $MRS_{x,y}$ you found in Q7.3? Explain.

8.2. Will the MRS be still the same for each of the following transformation? Explain without directly solving for MRS. a). $u' = u^2$ b). $u' = 1/u^2$ c). $u' = 1987 \times u - 507$ d). $u' = e^u$

8.3. Explain why taking a monotonic transformation of a utility function does not change the marginal rate of substitution (MRS).