Homework 4-
More Real and Nominal prices-
1) Last week we had you research careers and directed you to a data set that contained information about salaries, including a breakdown of salaries by state. However, just as the purchasing power of a dollar varies over time, so too does purchasing power varies across state. The table below contains median salary data on an occupation from 2013, along with some fictitious data on the cost of a basket of goods and services in each state. Use the data to compute a price index and a real salary for each state, using Wisconsin as the base location (this means that the Wisconsin base index will be 100 if we measure the price index on a 100 point scale). The same technique you use for data over time will work here as well! Does adjusting for cost of living matter?
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Median salary in 2013
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Cost of basket in 2013
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Price Index in 2013 with Wisconsin the base location
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Real Salary (measured in Wisconsin $)
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Hawaii
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$87,000
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1000
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Idaho
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$70,000
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500
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New York
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$105,000
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1000
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Wisconsin
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$90,000
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600
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2) The Bureau of Labor Statistics (BLS) is the US government agency that compiles CPI data. They provide an inflation calculator at https://www.bls.gov/data/inflation_calculator.htm that performs inflation adjustments automatically. Use this to perform an interesting comparison. For instance, you could ask someone much older than you how much they were paid during their first job and convert that to 2014 dollars.
Consumer Theory-
3) Nick earns $20,000 every year, and he spends his money on golf (g) and water-skiing (w). The price of playing golf and water-skiing are $100 and $200, respectively.
a. Given this information, what is the equation for Nick's budget line? Graph it with the number of times he plays golf on the x axis. Label this budget line BL1.
b. Suppose Nick's income increases to $30,000 a year. Draw his new budget line and label it BL2.
c. Suppose Nick's income is $30,000 per year and the price of water-skiing falls to $150. Draw his new budget line in your graph and label it BL3.
d. Go back to the starting situation (income of $20,000 per year, prices of $100 and $200 for golf and water-skiing respectively). To attract Nick to play golf more frequently, the manager offers him a new price package (the Mid-night VIP Package). For the first 100 units of golf-playing he purchases, the price is only $40 per game of golf, and for the extra units the price is $160 per game of golf. Draw a graph that illustrates Nick's new budget line given this information. (Hint: this budget line will definitely not be one linear segment: you will need to think about this carefully!) Label this budget line BL4. Will he be worse off under this new pricing arrangement than he was initially? Explain your answer.
4) Bob is organizing a large camping expedition and needs to acquire some flashlights. Each flashlight requires two batteries in order to work: without batteries a flashlight is useless and batteries are useless unless put inside a flashlight. Thus a flashlight and two batteries are perfect complements. Suppose a flashlight costs $15 and each battery costs $2.50 (so 2 batteries costs $5).
a. If Bob is trying to maximize the number of workingflashlights for the trip, what has to be true about the number of flashlights and the number of batteries he buys? Express this in words. Also, if F is the number of flashlights and B is the number of batteries, give an equation relating F and B if Bob is maximizing the number of flashlights with batteries in them. We call this an "optimality condition".
b. Bob has a budget of $300 to spend on flashlights. What has to be true about the amount of money Bob spends on flashlights and batteries compared to his budget? Express this in words and as an equation.
c. We will now find the optimal number of flashlights and batteries for Bob to buy. Since we are after two numbers (number of flashlights and number of batteries) we require two pieces of information. Use the optimality condition from (a) and the budget constraint from (b) to calculate Bob's optimal purchasing plan.
d. Now suppose that flashlights are on sale this week and the price is actually $10. What is Bob's new optimal consumption bundle?
e. Fill in the following table. Is Bob's decision making consistent with the law of demand?
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Price of flashlights
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Number of flashlights purchased
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$15
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$10
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f. Illustrate parts a-d on a diagram with the number of flashlights on the x axis. That is, show the indifference curves and budget lines that lead to Bob choosing the bundles in (c) and (d). Did we use the tangency condition given in lectures?
5) Jeff Albertson enjoys consuming comic books (C) and DVD's (D). His total utility depends on the product of the number of comic books and number of DVD's he owns. For instance, if he owns 4 comic books and 3 DVD's his total utility is 3 x 4 = 12. Mathematically we can write this as u(C,D) = CD. His marginal utility from an additional comic book depends on the number of DVD's he owns and is given by MUC¬ = D. For instance, if he owns 3 DVD's then an extra comic book improves his utility by 3, whereas if he has 5 DVD's an extra comic book improves his utility by 5. This captures the fact that Jeff likes reading comic books that involve characters he has seen in a DVD: if he recognizes someone in a comic book from a DVD then that comic book is extra awesome! Similarly his marginal utility from an extra DVD depends on the number of comic books he owns, so MUD = C. Suppose comic books cost $2.50 each and DVD's cost $10 each.
a. Given the above equation (u = CD) we know that 2 comic books and 5 DVD's gives Jeff a total utility of 10. Plot Jeff's indifference curve at this utility level, i.e. a curve representing all bundles that give him a utility of 10. In your graph measure comic books on the x axis. (Hint: when in doubt, plot some points! For instance, if Jeff has 10 comic books, how many DVD's does he need to give him a utility of 10? Try some numbers and see what happens!)
b. In the previous question the optimality rule Bob followed was very simple. In this case the indifference curves are smooth, so we need to use the tangency result given in lectures. What has to be true about the slope of Jeff's indifference curve and the slope of Jeff's budget line at the optimal bundle of comic books and DVD's? State this in words and express it as an equation involving C and D.
c. Suppose Jeff has a budget of $400 to spend on DVD's and comic books. Write down the equation of his budget constraint given this information.
d. As in part (c) of the previous question, use Jeff's optimality condition and his budget constraint to calculate his optimal consumption bundle of comic books and DVD's. How much utility does he obtain?
e. Suppose a natural disaster greatly reduces the supply of comic books, sharply driving up the price of comic books to $10 per comic book, while the price of DVD's remains unchanged. Find Jeff's new optimal consumption bundle and new utility. (Hint: you will have to adjust both Jeff's optimality condition and his budget constraint.)
f. Illustrate parts (b)-(e) on a well labelled diagram. As in lectures label the point found under the original prices A and the point under the new prices B, etc.
g. We now want to compute the income and substitution effects of the price change in part (e). To do this we must compute the point "C" that lies on an "imaginary" budget line. There are two features of point C: (i) it gives the same utility as the bundle chosen under the original prices, so it lies on the same indifference curve as point A; (ii) the imaginary budget line has the same slope as the second budget line, so the slope of the indifference curve at point C is equal to the slope of budget line 2. Importantly that means that the imaginary budget line does not pass through point A! Indeed, we do not know the income of the imaginary budget line, though we do know its slope. This information about slope gives you an optimality condition and the second equation you need is that the utility from the bundle at C is the same as the utility from the bundle at A. Use these two equations to find point C and illustrate on a diagram. Once you find the coordinates of bundle C, calculate the income of this imaginary budget line.
h. Use point C to calculate the income and substitution effects of the increase in comic book prices (in terms of the change in the quantity of comic books consumed).
i. Jeff is understandably annoyed by the increase in comic book prices. How much money would he need to be paid to return him to his old level of utility under the new prices? That is, by how much would his income have to change to be able to afford the bundle at point C?
Production and Costs-
6) (Hint: on this question you will likely want to use a calculator!) Alice owns and runs a factory that creates widgets using capital and labor. The table below summarizes the production and cost functions of Alice's factory, where q, K and L respectively represent the quantity of widgets, capital and labor, and costs are given in dollars. Before starting this question it would be a good idea to compile all the relevant formulae on a single piece of paper!
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K
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L
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q
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FC
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VC
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TC
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AFC
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AVC
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ATC
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MC
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MPL
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2
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0
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0
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$18
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2
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1
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$2/unit of output
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0.25 units of output/unit of labor
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2
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6
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2
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2
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9
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3
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2
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4
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2.5
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2
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22
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5
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29
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2
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44
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5
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2
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63
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9
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0.091
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2
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75
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10
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0.083
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a. Given the above information, what is the cost of one unit of capital to Alice?
b. How much labor is required to produce the first unit of output? What is the cost of a unit of labor? (Hint: you can figure this out using marginal cost and the marginal product of labor.)
c. Fill in the blank cells in the table. Check to make sure that your numbers are consistent with each other!
d. Does Alice's factory exhibit diminishing marginal productivity of labor? How do you know?
e. Is Alice's factory operating in the short run or the long run? How do you know?
f. If the price of a widget was $2.50, what quantity of widgets would Alice's factory produce (assuming that Alice's decisions do not affect the market price)? How large would her profit or loss be?
g. What price is the shutdown price for Alice's factory? (Hint: it is not $2!)
h. If the above costs were also true in the long run, what price would widgets have to be sold for in order for Alice to break even?
i. What quantity minimizes Alice's average total costs?
Perfect Competition-
7) Black Swan Clothing is a firm producing costumes for residents in Econ Town. The total cost of the firm is given by TC = 0.5q2+q+2 where q is the number of costumes produced by Black Swan Clothing. You are also told that the marginal cost curve (MC) for Black Swan Clothing is given by the equation MC = q + 1.Furthermore, assume Black Swan Clothing takes prices as given when choosing the quantity it will produce.
a. How many costumes will the firm produce when price, P, is given as P = $2? How many costumes will the firm produce is P = $1?
b. Draw a graph that represents Black Swan Clothing's marginal cost curve (MC), average variable cost (AVC) and average total cost(ATC) curves. Measure marginal cost, average variable cost, and average total cost on the vertical axis (all three of these cost curves are measured as "$ per unit"). Measure the quantity of costumes on the horizontal axis. Label your graph carefully and completely.
c. What is the firm's breakeven point? Mark it on your diagram from (b).
d. Suppose there are 10 firms in the market, including Black Swan Clothing, which all have the same cost curves. Assume that this market is perfectly competitive. What is the market supply curve given this information? (Hint: first find Black Swan Clothing's supply curve, then think about the market supply curve.)
e. Suppose demand is given by the equation Qd = 50-10P. Calculate the equilibrium price and quantity. Is the market in long run equilibrium at this price and quantity? Explain your answer.
f. Suppose demand increases to Qd = 90-10P. What is the new market equilibrium price and quantity in the short-run?
g. Given the situation in (f), how much profit in the short-run does Black Swan Clothing make now? What do you predict will happen to the number of firms in the long run? Explain your prediction.
h. If demand is as is given in (f), how many firms will be in the market in the long run, assuming all firms still have the same cost curves as Black Swan Clothing? (Hint: what does the price have to be if the market is in long run equilibrium?)