Homework Assignment 1

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC

Instructions:

You will scan your homework into a single PDF file to be upload.
Only legible assignments will be graded.
Late assignments will not be accepted.

Only two randomly-chosen questions will be graded.

Question 1:

(Question statements are summarized)

You decided to buy cigarettes and Swiss chocolates. You have ${$}$400. A price for one pack of cigarettes is ${$}$20 and a price for one bar of Swiss chocolates is ${$}$40. You can only bring up to 10 packs of cigarettes (cigarettes rationing).

  1. Write down the budget set and budget line without the regulation that limits the quantity of cigarettes amount.

  2. In a same graph: draw budget lines without the regulation and with the regulation. 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 ${$}$50. 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. $p_2 = 40$ $)$? How about under the new price of chocolates $($i.e. $p_2 = 50$ $)$?

Solutions.

  1. Budget set: $$Cigarettes \times p_1 + Chocolates \times p_2 \leq 400$$ Budget line:$$Cigarettes \times p_1 + Chocolates \times p_2 = 400$$

  2. See the graph below.

  3. See the graph below.
  4. Let us see whether the necessary spending of each bundle is affordable or not: $7 \times 20 + 5 \times 40 = 340 < 400 \implies$ affordable
    $7 \times 20 + 5 \times 50 = 390 < 400 \implies$ affordable

Question 2:

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 ${$}$400. The price of one meal of food is ${$}$40 and the price $($or value$)$ of one unit of leisure is ${$}$10. An average citizen in Mississippi has a daily income of ${$}$100. The price of one unit of food is ${$}$10 and the price $($or value$)$ of one unit of leisure is ${$}$10. Who is better-off in terms of the bundles of goods they can consume? $($hint: draw budget sets.$)$

  2. Suppose you earn ${$}$200 per day. One day your boss increases your income by 50% solely due to your high performance at work. Are you better-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 better-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 better-off, worse-off, the same, or cannot answer compared to before the changes in prices?

Solutions.

  1. The budget line for a representative person living in Californian is $$ meal \times 40 + leisure \times 10 = 400\\ meal \times 1 + leisure \times 0.25 = 10 $$ and the budget line for a representative person living in Mississippi is $$ meal \times 10 + leisure \times 10 = 100\\ meal \times 1 + leisure \times 1 = 10\\ $$ Since the budget set of a person in Mississippi is inside of the budget set of a person living in California, the person is California is more well-off than the person in Mississippi in terms of the amount of food and leisure that can be consumed.
  2. Well-off because you can afford more of every good. The budget set is wider.
  3. The same because you can afford just as much as before. The budget set did not change.
  4. We cannot answer without more information. It depends on which food you like more than the other which can be represented by an indifference curve.

Question 3:

Answer the following questions regarding preferences:

  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.

  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.

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

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

Solutions.

  1. From textbook, "To prove this, let us choose three bundles of goods, X, Y , and Z, such that X lies only on one indifference curve, Y lies only on the other indifference curve, and Z lies at the intersection of the indifference curves. By assumption the indifference curves represent distinct levels of preference, so one of the bundles, say X, is strictly preferred to the other bundle, Y . We know that X $\sim$ Z and Z $\sim$ Y , and the axiom of transitivity therefore implies that X $\sim$ Y . But this contradicts the assumption that X $\succ$ Y . This contradiction establishes the result—indifference curves representing distinct levels of preference cannot cross.''

  2. Thick indifference curve: Suppose the monotonicity holds even with the thick indifference curve. In the graph, the bundle X has more of good 1 and good 2 than than bundle Y, but the thick indifference curve implies that the bundle X is equally preferred to Y. This contradicts with the monotonicity assumption. Downward sloping: Consider the indifference curve that is not downward sloping. By the definition of indifference curve, B is equally preferred to A. However, this again violates the monotonicity assumption because B has more of good 1 and more of good 2.

  1. Example1: a consumer choosing between a blue pen and black pen who doesn't care about color. Example2: a consumer choosing between a coke and a pepsi who cannot tell the difference. Example3: a consumer choosing between an organic milk with 1% fat produced by farm A and an organic milk with 1% fat produced by farm B.

  2. Example1: a consumer who drinks a cup of coffee only with 2 teaspoons sugar. Example2: the number of left shoes and right shoes for a consumer who can only use pairs of shoes. Example3: the number of nuts and bolts.

Question 4:

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. $u(x_1,x_2) = x_1 + 2 x_2$
  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. $u(x_1,x_2) = -${$(x_1 - 3)^2 + (x_2 - 2)^2$}

Solutions.

Choose level of U, e.g. 5, and then determine combinations of $(x_1, x_2)$ that can produce value given provided functions.

See the graphs below for precise level curves.

1: perfect substitutes preferences $($textbook Fig 3.3$)$

$ u(x_1,x_2) = x_1 + 2 x_2 \\ \bar{u} = x_1 + 2 x_2 \\ x_2 = \frac{1}{2} \bar{u} - \frac{1}{2} x_1 \\ \text{Pick levels of } \bar{u}: 4, 7, 10 $

2: perfect complements preferences $($textbook Fig 3.4$)$

$ u(x_1,x_2) = min(x_1, x_2) \\ \bar{u} = min(x_1, x_2) \\ \text{Pick levels of } \bar{u}: 1, 2, 3 $

3: preferences with commodity 2 being neutral $($textbook Fig 3.6$)$

$ u(x_1,x_2) = x_1 \\ \bar{u} = x_1 \\ \text{Pick levels of } \bar{u}: 1, 2, 3 $

4: preferences with commodity 2 being bad $($textbook Fig 3.5$)$

$ u(x_1,x_2) = x_1 - x_2 \\ \bar{u} = x_1 - x_2 \\ x_2 = x_1 - \bar{u} \\ \text{Pick levels of } \bar{u}: -3, 0, 3 $

5: Cobb-douglas preferences $($textbook Fig 4.5$)$

$ u(x_1,x_2) = \sqrt{x_1 x_2} \\ \bar{u} = \sqrt{x_1 x_2} \\ \sqrt{x_2} = \frac{\bar{u}}{\sqrt{x_1}} \\ x_2 = \frac{\bar{u}^2}{x_1} \\ \text{Pick levels of }: \bar{u}: 1, 2, 3 \\ $

6: preferences with $(3,2)$ as a satiation point $($textbook Fig 3.7$)$

$u(x_1,x_2) = - $ { $(x_1 - 3)^2 + (x_2 - 2)^2 $}
$\bar{u} = - ${$(x_1 - 3)^2 + (x_2 - 2)^2$ }
Pick levels of $\bar{u}: -1, -4, -9 $ Why do we pick negative numbers? Because the right-hand side of the utility function is necessarily negative so utility values can only be negative in this case.

For example, if we choose $ \bar{u} = -1$ this implies $1^2 = (x_1 - 3)^2 + (x_2 - 2)^2$ We cannot easily solve for x_2 this time, but we can remember that $ c^2 = (x - a)^2 + (x - b)^2 $ is the formula of a circle around $ (a,b) $, and that is all we need.

Question 5:

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

Solutions:

1: Monotonic & Convex Preferences

2: Monotonic & Concave Preferences

3: Non-monotonic and Convex

Question 6:

Consider a utility function $ u(x,y) = \frac{1}{2} x^{a} y^{b} $, where $ 0 < a < 1 $ and $ 0 < b < 1 $. Also assume that $ x,y > 0 $.

  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.

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

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

  4. Suppose $a,b = \frac{1}{4}$. What is $MRS$ at $(x,y) = (3,12)$ and what does it mean in words? What is $MRS$ at $(x,y) = (12,3)$ and what does it mean in words?

  5. Suppose $a,b = \frac{1}{3}$. 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.

  6. Draw tangent lines at $ (x,y) = (4,16) $ and $ (x,y) = (16,4) $ in the same graph and label the slopes you solved in $ MRS(4,16) $ and $ MRS(16,4) $.

Solutions.

6.1. $MU_x = \frac{1}{2}ax^{a-1}y^b > 0$ and $MU_y = \frac{1}{2}bx^{a}y^{b-1} > 0$, because $x,y,a,b > 0$. Now, since an increase in $x$ or $y$ increases utility, ''more is better'' assumption is satisfied.

6.2. To see if something is increasing in x we need to take its derivative wrt x. In this case, we want to see whether MUx is increasing in x so we take the derivative of $MU_x$:

$$ \frac{\partial MU_x}{\partial x} = \frac{1}{2}a(a-1)x^{a-2}y^b < 0 $$

This is negative because $(a-1) < 0 $ (since $0 < a < 1$). Therefore, marginal utility of $x$ is decreasing in $x$. It means additional units of $x$ increase utility, but at a decreasing rate.

Notice that the derivative of MUx is simply the second derivative of $U(x_1,x_2)$

6.3. $MRS(x,y) = \frac{d y}{d x} = - \frac{MU_x}{MU_y} = -\frac{\frac{1}{2}ax^{a-1}y^b}{\frac{1}{2}bx^{a}y^{b-1}} = -\frac{a}{b}\frac{y}{x}$

6.4. $MRS(3,12) = -4$ and $MRS(12,3) = -0.25$. The MRS measures the amount of good 2 that one is willing to give up for a marginal amount of extra consumption of good 1. When the consumer is consuming a bundle $(3,12)$, he is willing to pay four units of good 2 for a little more of good 1. When the consumer is consuming a bundle $(12,3)$, he is willing to pay 0.25 units of good 2 for a little more of good 1. He is now willing to pay much less of good 2 because he now has more of good 1.

6.5. See the graph below.

6.6. See the graph below.

Question 7:

7.1. Consider a transformation of the utility function in Question 6 using $ u' = ln (u) $. In other words, the new utility function $ u' = ln (u) = ln(\frac{1}{2}x^{a}y^{b}) = ln(\frac{1}{2}) + 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 Q6.3? Explain.

7.2. Will the MRS be still the same for each of the following transformation? Explain without directly solving for MRS.
a). $u' = u^2$

b). $u' = 1/u^3$

c). $u' = 1987 \times u - 507$

d). $u' = e^u$

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

Solutions.

7.1. $MRS(x,y) = -\frac{MU_x}{MU_y} = - \frac{\frac{a}{x}}{\frac{b}{y}} = -\frac{a}{b}\frac{y}{x}.$ This is equal to the MRS in Q6.3.

7.2.

a). $v = u^2$ If u, as in question 6, can only take positive values u>0, then $v = u^2$ will yield the same indifference curves and $v$ will represent the same preferences. This is because, for positive values of u, $v = u^2$ is an increasing transformation of u.

b). $u' = 1/u^3$ If u, as in question 6, can only take positive values u>0, then $v = 1/u^3$ will yield the same indifference curves BUT $v$ will NOT represent the same preferences. This is because, for positive values of u, $v = 1/u^3$ is NOT an increasing transformation of u.

c). $v = 1987 \times u - 507$ yields the same indifference curves as $u$. This is because $v$ is an increasing transformation of u (that is: $∂(1987u - 507)/∂u > 0$).

d). $v = e^u$ yields the same indifference curves as $u$. This is because $v$ is an increasing transformation of u (that is: $∂(e^u)/∂u > 0$).

7.3. Suppose $v$ is simply a monotonic transformation of $u$. A monotonic transformation preserves the ranking of bundles, hence $v$ represents the same preferences as $u$. Since MRS fully describe preferences, then the MRS of u and those of v should be the same and the proof is shown below.

$ v=f(u) \Rightarrow \\ \text{MRS of v} = -\frac{MU_x}{MU_y} = - \frac{\partial v / \partial x}{\partial v / \partial y} = - \frac{f'(u) \partial u / \partial x}{f'(u) \partial u / \partial y} = - \frac{\partial u / \partial x}{\partial u / \partial y} = \text{MRS of u} $