Homework 1 Introduction to Microeconomics PDF

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INTRODUCTION TO MICROECONOMICS

Homework 1 EXERCISE 3.2 PRODUCTION FUNCTIONS 1. Draw a graph to show a production function that, unlike Alexei’s, becomes steeper as the input increases.

2. Can you think of an example of a production process that might have this shape? Why would the slope get steeper? There are, I believe, certain production processes that might adopt this shape. For instance, the process of learning a new ability, the so-called learning curve, often has a sigmoid shape, meaning that a slow beginning is followed by a steep acceleration in the learning process, although later on a plateau section is reached. Besides, a steep exponential-like production function may be clearly observed in some biological processes involving procreation, such as cell division. If we were to leave a small colony of bacteria on a fertile soil and in appropriate environmental conditions, it wouldn’t take more than a few hours for the bacteria to originate a massive population, that would continue to grow as long as food and proper conditions were supplied. However, it should be pointed out that, even though in a theoretical model the population would grow forever, in real life there are certain limiting factors that restrict perpetual growth, such as predators and physical space, which would sooner or later flatten the curve and stabilize the production function.

3. What can you say about the marginal and average products in this case?

The marginal product is defined as the additional amount of output that is produced if a particular input was increased by one unit, while holding all other inputs constant, whereas the average product is defined as the total output divided by the total input at a certain point. Graphically, the marginal product is the tangent line to the function at a certain point, that is, the derivative of the function at that point. On the other hand, the average product is the line that connects the given pint with the origin of the function. Having said that, in these exponential-like models, the marginal product line will be, at all times, steeper than the average product line. In other words, the rate at which the output is produced will always increase with time or whatever input we have.

EXERCISE 3.3: WHY INDIFFERENCE CURVES DO NOT CROSSS In the diagram below, IC1 is an indifference curve joining all the combinations that give the same level of utility as A. Combination B is not on IC1.

1. Does combination B give higher or lower utility than combination A? How do you know? Combinations A and B clearly belong to two distinct indifference curves and, therefore, they represent different utility levels. More precisely, if we were to draw the indifference curve containing point B, it would clearly be further away from the origin. Hence, we can affirm that combination B gives a higher utility level than combination A, for the first one is located on a higher-utility indifference curve than the second one.

2. Draw a sketch of the diagram, and add another indifference curve, IC2, that goes through B and crosses IC1. Label the point at which they cross as C.

3. Combinations B and C are both on IC2. What does that imply about their levels of utility? If two combinations are connected by a single indifference curve, then the levels of utility they provide must be equal. In fact, this is the definition itself of an indifference curve, which is an imaginary line formed by all combinations that are indifferent to an individual, meaning that they all provide the same level of utility. 4. Combinations C and A are both on IC1. What does that imply about their levels of utility? Identically as before, if combinations A and C are both on IC1, then they provide the exact same level of utility. 5. According to your answers to (3) and (4), how do the levels of utility at combinations A and B compare? According to the answers, combinations A and B have the same level of utility. If B = C and A = C, then necessarily A = B. 6. Now compare your answers to (1) and (5), and explain how you know that indifference curves can never cross. The answers given in (5) go against the initial answer in (1). That is because the sketch in (2) is clearly an impossible one, thus leading inevitably to wrong answers. Indifference curves can never cross, because that would be a paradox itself. If two distinct IC by definition represent different utility levels, then they cannot cross, for the crossing point would belong to both IC and would not represent a single defined level of utility.

Ex. Assume an individual has a utility function over consumption y and leisure l with the form: 𝑈(𝑦 𝑦, 𝑙) = 𝑦 2𝑙3. This person works for a given wage per hour w and uses all her income to pay for her consumption. The maximum hours a day she can work is 24. (a) What is the marginal rate of substitution between consumption and leisure for this individual? In order to properly answer this question, we must first define the marginal rate of substitution (MRS) as the trade-off a person is willing to make between two scarce goods. It is also, at any point, the slope of the indifference curves. Having said that, we know that the IC will be defined by the utility function and its slope at any point will be given by the derivative of the utility function. Taking into account all of that, the MRS formula is as follows: |MRSxy| = Δy/Δx = (du/dx)/(du/dy) = dy/dx = MUx/MUy In this particular case, the variables of the utility function are: consumption (y) and leisure (l). |MRSly| = Δy/Δl = (du/dl)/(du/dy) = dy/dl = MUl/MUy Now we apply the formula and derive with respect to y and l, which gives us the MRS. |MRSyl| = 3y2l2/2yl3 = 3y/2l (b) If the price of consumption is 2 for every unit of consumption and the wage is 2 for every hour worked, what is the choice of hours of work (and leisure) and of consumption in a given day? As we already know, in this type of problems, the wage will, by definition, be equal to the negative slope of the feasible frontier and also to the MRT. However, we must take into account that, even though the wage is 2, every unit of consumption is worth 2 and therefore the hourly wage will only be enough to pay for one unit of consumption. On the other hand, we also know that, in order to make a choice that maximizes utility, the condition MRT = MRS needs to be satisfied. In this case, the MRT will be equal to -1, that is, in order to have 1 unit of consumption, we must sacrifice 1 hour of leisure. Now, if MRS = -2l/3y and MRT = -1, provided that MRT = MRS, -2l/3y = -1. We get our first equation that will help us solve this problem: 2l = 3y. The second equation will be obtained from the consumption function. We know that c = w*(24 – l), 24 – l being the hours of work. Next, we substitute the wage and the consumption in the equation: 2y = 2(24 – l). We write 2y because the price for every unit of consumption is 2. We can simplify and write y = 24 – l. We now have our second equation. We can substitute y in the first equation and find a unique value of hours of leisure that maximizes utility. 2l = 3(24 – l) = 72 – 3l; 5l = 72; l = 72/5 = 14.4 hours of leisure. y = 24 - l = 24 – 14.4 = 9.6 units of consumption. Therefore, the combination that maximizes utility is at 14.4 hours of leisure and 9.6 units of consumption. (c) What would the optimal choices be if the price of consumption remained constant, but wages increased to 4? In this case, the MRT is affected and we need to calculate the new slope of the feasible frontier. If wages increase to 4, then we will be able to buy 2 units of consumption per hour instead of 1. In order

to have 1 hour of leisure, you now need to sacrifice 2 units of consumption, that is, the opportunity cost of 1 hour of leisure has doubled. The new MRT will be equal to -2. Therefore, the first equation will now be MRT = MRS, -2l/3y = -2; l = 3y. We will also need to transform the second expression. Wages have increased to 4, and we will substitute this new value in the consumption function. 2y = 4(24 – l). Now, when we simplify, we will get a slightly different equation: y = 48 – 2l. We substitute in the first equation as before. l = 3(48 – 2l) = 144 – 6l; 7l = 144; l = 20.57 hours of leisure. y = 48 – 2l = 48 – 2*20.57 = 6.86 units of consumption. We see that having doubled the wage, the individual would prefer to work less hours and reduce his consumption. (d) What would happen in questions b. and c. if there was a maximum legal work limit of 8 hours a day? If a maximum legal work limit of 8 hours were to be introduced, then the answer in question (b) would have to be adjusted. The optimal combination of free time and leisure that the individual has chosen is 14.4 hours of leisure and 9.6 units of consumption. However, this combination is no longer possible. At most, the individual can work for 8 hours, and therefore the minimum leisure time they can choose is 16 hours a day. Now we can substitute l = 16 in the MRS equation and we get 32 = 3y; y = 10.7 units of consumption. The utility level of this new combination would, of course, not be the same. However, it’s the closest we can get to our original combination while having the legal work limit constraint. In question (c), note that no readjustments would need to be made, as the optimal choice of 20.57 hours of leisure does not conflict with the maximum legal work limit.

(e) Draw a graph to show the optimal choices in parts b. and c. above and decompose the final effect of the wage raise into a substitution and an income effect.

As we can observe, in this case, the income and substitution effects work in opposite directions. On the one hand, the income effect by itself moves the optimal combination from (b) to (c)’. If there were additional unearned income, then the individual, according to their indifference curves, would choose to work less hours and subsequently consume less goods. However, the substitution effect moves the optimal combination from (c)’ to (c). The individual chooses to work more hours because the opportunity cost of leisure time has doubled. In short, the worker substitutes leisure time for consumption units. In this particular case, the income effect outweighs the substitution effect and, therefore, the individual takes more free time at the expense of some consumption units.

Ex. Read the article in The Economist 22 April 2014 Nice work if you can get out. Use the concepts of income and substitution effects to explain why the rich work more hours that the poor. Maximum of 200 words (Please, include the word count at the end of your answer). Throughout the most part of human history, leisure time has been perceived as an indicator of wealth and high status. Nonetheless, this long-established trend seems to have reverted over the past century. It seems that rich people are these days working longer hours than they ever did. The income and substitution effects may help us delve into the topic and understand how this change has occurred. First and foremost, over the past few decades, inequality has dramatically increased. The wealthy are wealthier than they have ever been, whereas low-income families have had their wages drastically reduced. In a few words, middle classes are disappearing. On the one hand, as wages rise, the income effect encourages people to work less hours and have more leisure time. However, when rich people earn more and more money, the substitution effect encourages them to work longer hours, as the opportunity cost of leisure time becomes enormous. If they take time off, they give up money. Moreover, the market may be more competitive now than it has ever been, thus leading to a “winner-takes-all” nature of economy that further amplifies the substitution effect and prompts educated and wealthy people to work hard so that they can later benefit from huge gains. 206 words...