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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).$$The first term says "if I sell one extra unit then I will receive an extra p in revenue". The larger is this effect, the higher is the MR. The second term says "in order to sell one extra unit, I will have to ... 5 Marginal utility (of x) in your case is U_x(x,y)=2xy^2. You use the sign of the derivative of MU, namely U_{xx}, to tell whether MU is increasing, constant, or decreasing. Specifically, you have increasing MU if U_{xx}>0, constant MU if U_{xx}=0, and decreasing MU if U_{xx}<0. In your case, assuming that y>0, you'd have U_{xx}&... 4 I'll offer a less algebraic alternative to Alecos's answer. In short, yes and no. The "no" part Normally the MC and AC curves would look like the following, with MC intersecting AC from below AC's minimum point. Suppose price P_0 were below this point. Then the firm would sell at a quantity below Q_1. But what does this imply for the firm's ... 4 You are missing the average cost curve in the same diagram. Basic algebra gives us the following. Let's find the minimum of the AC = C/Q. We have$$\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 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 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 ... 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 According to the orthodox economist, in the real world, firms do not consciously try to calculate MC or MR. Nor do they consciously try to produce/sell at the point where MC=MR. Instead, the theory you have learnt is simply theory. This theory argues that: If firms are maximizing profits, then they must be producing/selling at the point where MC=MR. ... 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$$where c is a constant (e.g. c=0). This is clearly different ... 3 The following is from Thomas Piketty and Gabriel Zucman (2015, From Handbook of Income Distribution, Volume 2, Chapter 15, Part 15.5.3 which is hard to link to directly but get it here): Take a CES production function$$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 ... 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. ... 2 Let C(y) denote the cost function and y the quantity. The average cost function is AC=C/y. This is increasing if its derivative is positive. Let C' denote the derivative of C. The marginal cost function is MC=C'. We have: \frac{\partial AC}{\partial y}= \frac{C'y - C}{y^2}>0 by using the quotient rule. As y^2>0, the condition becomes: ... 2 It is possible for firms to have constant marginal costs in monopolistic competition in theory. Nevertheless, they must also have fixed costs. The fixed costs prevent firms from entering in sufficient numbers such that you would have perfect competition. As to whether constant marginal costs are realistic in you scenario, that depends on what you believe ... 2 Consider that the following production function Y=F(K,L) excibits constant returns to scale ie F(x K, x L)=x F(K,L), for any arbitrary scalar x. This means in plain English that eg for x=2, doubling the inputs will produce double the output. The intuition behind constant returns to scale is that inputs and outputs are directly proportionate. In ... 2 This is true for quasilinear utility function. Suppose$$U = d + V(x), $$where d is the numeraire. The budget constraint is given by p_x x + d = I. The utility is then$$U = I - p_x x + V(x).$$Maximizing this, we have the following FOC: p_x = V'(x). That is, for a given x, my willingness to pay is V'(x), the marginal utility. 2 This seems like a homework question, so I'll just give hints. By definition, a quasilinear utility has the form u(x, y) = x + v(y) where y is a vector of all other goods and v(\cdot) is strictly concave. In this case, x is called the numeraire. From utility maximization, what's the first order condition that relates MU_x, MU_y, p_x, and p_y?... 2 Marshallian Theory is notoriously about computing consumer surplus (and then welfare changes). But it would be nonsensical to perform such calculations if at the level of one individual, the unit of valuation of surplus, i.e. the marginal utility of money (or the utility of one extra euro), were changing before and after changes in, say, prices. Also, what ... 2 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 ... 2 From U(W+11)−U(W)\le U(W)−U(W−10) we get that \frac{U(W+11)-U(W)}{11}\le\frac{10}{11}\frac{U(W)-U(W-10)}{10}, which is what the sentence before the bolded italic part says. Now by concavity of U(.) we know that MU(W-10)\ge\frac{U(W)-U(W-10)}{10}, and also that MU(W+11)\le\frac{U(W+11)-U(W)}{11}. Therefore MU(W+11)\le\frac{U(W+11)-U(W)}{11}\le\... 2 What your school economics textbook must say is that : "In a Perfectly Competitive Market (PC), firms maximize profits when MC= P ". This equality applies only to firms in a perfectly competitive market. This is because, in a Perfectly Competitive Market there is an infinite number of buyers and sellers and the prices are set by the overall market forces,... 1 Max Then you have: Max Which gives you: So L = 100, and you have a loss of 15000 1 The marginal cost is the cost added of producing an additional unit of product. In this case the company is paying a factory in China \3 per unit to produce the product. Assuming you have no other variable costs (i.e. costs that vary with level of output) then the cost of producing an additional unit is indeed 3. 1 The marginal tax rate is supposed to reflect incentives for the individual. In order to derive a meaningful metric, the calculation should hence try to reflect the actual decision an individual is facing. An increase in \1 pay is probably not something people are considering. If, however, someone earning \10 per hour considers to increase his/her labor ... 1 This is quite simple to answer given you know a bit of multivariable differential calculus. You're looking for partial derivatives of the utility function. So, given$$U(x,y) = x(y+1)$$we have$$\frac{\partial U}{\partial x} = y+1$$and$$\frac{\partial U}{\partial y} = x. These are the goods' marginal utilities.