28

One possible answer is that Mankiw's argument takes consumer demand for airline tickets as fixed and given. I would speculate that cheap last-minute tickets are a substitute good for regularly priced tickets, and that if enough people came to prefer them, demand for regularly priced tickets would fall and the airlines would lose revenue. In other words, ...


20

Do airlines actually do anything like the above? Yes, in fact now you will see on many airports specialized companies/windows that will offer last-minute flights very cheap. For example, in the past, I used to frequently fly to Vienna Schwechat airport, and there used to be a window where you could get cheap tourist flight but the catch was you could not ...


13

Pricing of last minute tickets for airlines is a tricky problem. Yes, discounting fares may attract customers who would not have flown otherwise. But buying a flight is a bit more complex than buying some gadget on discount on Amazon. Few customers can make such last-minute decisions. They usually need that flight, but also a flight back, and the schedules ...


7

One problem with trying to price goods or services at their marginal cost is that customers may hold off on making immediate purchases of goods or services they expect to be available cheaper at a later time. If there aren't enough people who would have any interest in a particular flight to fill it even if tickets were only \$1 each, but the number of ...


6

As noted in the comments, it is not true that homothetic preferences must have constant marginal rates of substitution. To see this, recall that preferences given by the utility function $$ u(x,y) = x^\alpha y^{1-\alpha} $$ are homothetic. (More generally, Cobb-Douglas preferences are homothetic.) However, the marginal rate of substitution is $$ \text{...


6

It is perfectly consistent for the marginal revenue to increase in $q$, even if the demand curve decreases. Marginal revenue is $$p(q)+ q p'(q).$$ The first term says "if I sell one extra unit then I will receive an extra $p$ in revenue". The larger is this effect, the higher is the MR. The second term says "in order to sell one extra unit, I will have to ...


5

Marginal utility (of $x$) in your case is $U_x(x,y)=2xy^2$. You use the sign of the derivative of MU, namely $U_{xx}$, to tell whether MU is increasing, constant, or decreasing. Specifically, you have increasing MU if $U_{xx}>0$, constant MU if $U_{xx}=0$, and decreasing MU if $U_{xx}<0$. In your case, assuming that $y>0$, you'd have $U_{xx}&...


5

Your intuition is correct. First, you're right that "marginal cost only depends on variable cost", since \begin{equation} MC(q)=\frac{\mathrm dTC(q)}{\mathrm dq}=\frac{\mathrm d(FC+VC(q))}{\mathrm dq}=\frac{\mathrm dVC(q)}{\mathrm dq}. \end{equation} Next, if marginal cost is some constant $k$, then variable cost must be $VC(q)=kq$, because we can ...


5

I think the other answers covered well why airlines don't usually do quite what Mankiw described. But everything I have read about airline pricing says they are acutely aware of marginal costs. Even if the discounted seats are sold well before the flight, it can be seen as charging less than otherwise desired to avoid ending up with an empty seat. The ...


4

I'll offer a less algebraic alternative to Alecos's answer. In short, yes and no. The "no" part Normally the MC and AC curves would look like the following, with MC intersecting AC from below AC's minimum point. Suppose price $P_0$ were below this point. Then the firm would sell at a quantity below $Q_1$. But what does this imply for the firm's ...


4

You are missing the average cost curve in the same diagram. Basic algebra gives us the following. Let's find the minimum of the $AC = C/Q$. We have $$\frac {\partial AC}{\partial Q} = \frac {MC\cdot Q - C}{Q^2}$$ For this to be equal to zero, we must have $MC \cdot Q = C \implies MC = AC$. So when $AC$ is at its minimum, it equals $MC$. But we also ...


4

The reason why marginal benefit is measured in cans of soda is that this economy only has two goods: pizza and soda. So instead of using money we may as well use soda. Alternatively, in the absence of money this economy is an exchange economy, and the only way to pay for pizza in that case is with soda. When you move beyond two goods willingness to pay is ...


4

Utility functions as ordinarily used are not a measure of well-being comparable among people, but a representation of preferences. Moreover, preferences could principally be elicited from choice experiments. A utility function assigns real numbers to alternatives so that one alternative is preferred to the other if and only if it is assigned a higher number. ...


3

Marginal utility tells you how the utility changes as you alter x. That is the first derivative, which here is a function of x. This means it is increasing. The rate of that increase is constant as long as y is fixed (second derivative).


3

The marginal cost is 3. Marginal costs do not depend on the fixed cost, and when your variable costs are constant, then the marginal cost and the variable cost are the same. Note that your total cost is $C=FC+3q$ and the marginal cost is always the derivative of your total cost, in this case, $3$. As for the fixed costs, 4000 is definitely part of it, but ...


3

In general, you are right to be mystified: specifying a point (consumption bundle) isn't enough to compute MRS and indifference curves. However, in this problem, I would suggest you take the first sentence seriously as a description of her preferences. She likes fries. (She doesn't care about what box they come in!) Let $v(f)$ represent her utility as ...


3

The marginal profit you calculate is correct. We can rearrange the solution of the problem you are given. This is equivalent to $$ \frac{dP}{dq} = 192 -176q + 48q^2 -4q^3$$ This derivative has as primitive function the following profit function: $$P(q) = c + 192q -88q^2 + 16q^3 - q^4$$ where $c$ is a constant (e.g. $c=0$). This is clearly different ...


3

Having worked on these issues myself for quite some time, I am not aware of any study on this for New Zealand. For a back-of-the-envelope calculation of the revenue-neutral flat tax rate, you'd need the following: The tax base $Y$, i.e. the amount of income subject to income tax. Most probably, the New Zealand Treasury has published statistics on this. Note ...


3

The following is from Thomas Piketty and Gabriel Zucman (2015, From Handbook of Income Distribution, Volume 2, Chapter 15, Part 15.5.3 which is hard to link to directly but get it here): Take a CES production function $$Y=F(K,L)=(a⋅K^{\frac{\sigma-1}{\sigma}}+(1−a)⋅L^{\frac{\sigma-1}{\sigma}})^{\frac{\sigma}{1-\sigma}}$$ Me: $\sigma$ is the elasticity of ...


3

Seems like the only function $f$ that fits your description $$ \forall i: \frac{\partial f(\mathbf{x})}{\partial x_i} = c_i $$ is $$ f(\mathbf{x}) = A + \sum x_i c_i. $$ (Frequently $f(\mathbf{0}) = 0$ is assumed. The assumption is referred to as "no free lunch".) Then you can apply the definition of returns to scale.


3

Your marginal revenue is not calculated correctly. Marginal revenue $(MR)$ is the derivate of total revenue which is equal price times quantity $TR=PQ$. In your case $TR$ should be: $$TR=(k+aQ)Q \implies MR = \frac{dTR}{dQ} = k+2aQ$$ If the demand is given as: $P = 120−2Q$ then: $$TR= (120-2Q)Q \implies MR = \frac{dTR}{dQ} = 120-4Q $$ Also made a graph for ...


3

The Cobb Douglas production function with constants returns to scale $$y = \prod_i x_i^{\alpha_i} = A \prod_i \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i} ,$$ where $A:= \prod_i \alpha_i^{\alpha_i}$ annoying constant. Cost minimization with perfect competition $$\min_x \ \ p^\top x\ \lvert \ y = \prod \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i},$$ implies ...


3

You can think of it as the marginal benefit per person. The first piece of pizza you eat is amazing, it has a very high marginal benefit. But say you eat five whole pizzas in one sitting and then are immediately offered another slice. That next slice doesn’t seem so appealing, does it? You’re already full and starting to get sick of it. That slice has a low ...


3

Still, not 100% clear whether I get the question right, but in models with constant returns to scale and decreasing marginal productivity (question to others, are these conditions even necessary?), if firms are price takers in product and labor markets (no pricing power in which case you could look into monopsony models, see for instance the textbook "...


3

I figured it out: The first-order condition of the cost minimization problem for, say, material inputs $m_{it}$ gives: $ \lambda \frac{\partial F}{\partial M} = P_M $ Where F is the production function, $P_M$ the material input prices. Multiply by $\frac{M}{F}$ and rearrange, $ \lambda = \frac{P_M M}{\beta_M F} $, where $\beta_M$ is the output elasticity ...


2

The Marginal Rate of Substitution is not just the "ratio of the partial derivatives": it represents the slope of an indifference curve. In order to obtain it, you must guarantee that you remain on the same indifference curve. How do we do that? One way is by taking the total differential of the utility function and requiring this total differential to be ...


2

Converting my comments to an answer: I think your answers to 1 and 2 are correct. For 3.1, your intuition that "as well off as" should be interpreted "indifferent" is correct, and indifferent is about comparing utility levels, not MRS. You answer to 3.2 looks correct.


2

The Prime Minister of Australia seems to earn around AUD $500,000 according to Wikipedia This would give him a marginal tax rate of 45% plus supplements The average for full-time employees seems to be close to AUD $80,000 which would make typical marginal rates either 32.5% or 37% So the Prime Minister of Australia pays a higher marginal tax ate


2

We can break this into three parts: (1) price elasticity, (2) substitutes, and (3) marginal utility. Price elasticity measures price sensitivity (how much a change in price affects quantity consumed). In the example in the book, the product is elastic, which means that a decrease in price increases consumption proportionally more than the decrease in price. ...


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