7

Does Modern Monetary Theory (MMT) provide a useful insight into how to manage the economy? That depends on your definition of MMT, because it is not generally agreed on what it even is. You will find some arguing it is just a macro/monetary theory (such as the Wikipedia page) but then I seen MMT proponents on this site arguing it is a whole new paradigm ...


6

The $\gamma$ on the RHS comes from applying chain rule when differentiating the second term with respect to $a_i$. Regarding elasticity, note that with a differentiable function $f$, the ratio $f'(x)/f(x)$ can be interpreted as the percentage change in the value of $f$ around $x$. So isolating $\gamma$ from the FOC, you'll get an expression for elasticity, ...


6

It just comes from the derivative of profit function. I assume that $a_i$ is the choice variable here so the derivative of $\pi$ wrt $a_i$ is (step by step): $$\frac{\partial \pi}{ \partial a_i} = \frac{\partial \pi}{ \partial a_i} [ \ln R(a_i) ] + \frac{\partial \pi}{ \partial a_i} [ \ln N_i(\gamma a_i, \gamma a_{-i}) ] \\ = \frac{1}{ R(a_i)} R'(a_i) ...


5

$$y'\frac{x}{y}= -\frac{f_1'(x,y)}{f_2'(x,y)}\frac{x}{y} = - \frac{y^2\exp(x+1/y)}{2y\exp(x+1/y)+y^2\exp(x+1/y)(-y^{-2})}\,\frac{x}{y}=\\=\frac{y^2}{1-2y}\,\frac{x}{y}=\frac{xy}{1-2y}.$$


5

When people talk about 'elasticity of demand' without further qualification they normally mean 'price elasticity of demand': $$\text{E}_p=\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}$$ However, note your calculations are not entirely correct as in this case the elasticity should be: $$\text{E}_p=-\frac{1}{\gamma}\frac{P_t}{(X_t-P_t)/ \gamma} = - \frac{...


4

From the shares equation, we obtain: $$ 1 + \frac{rK}{wL} = \frac{1}{s_L} \to tk = \frac{1 - s_L}{s_L} $$ where I defined $t = \frac{r}{w}$ and $k = \frac{K}{L}$. then taking logs gives: $$ \ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L) $$ Then take the derivative of this expression with respect to $\ln(t)$: $$ \begin{align*} -\sigma &= -1 - \dfrac{\dfrac{\...


4

Income elasticity of demand Let $q(y)$ be the Engel curve for a good, i.e. it gives the demanded quantity for a given level of income $y$ (keeping prices fixed). The income elasticity of demand is then given by: $$ \varepsilon^y_q = \frac{\partial q}{\partial y} \frac{y}{q} $$ It measures the percentage point change in demand $q(y)$, due to a 1$\%$ increase ...


4

Yes. I have to include at least 30 characters in an answer, so let me repeat: Yes.


3

yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income. In some cases, however, you can make stronger statments. For example, homothetic preferences always lead to normal demands as $$ x_1(p_1, p_2,m) = x_1(p_1, ...


3

This will be a weird answer: As we know that $Q*P=const.$ for Cobb-Douglas preferences. $$ QP=const. \implies 0=d(PQ)=Q\ dP+P\ dQ \implies \frac{dQ}{Q}=-\frac{dP}{P} $$ thus we can conclude that $\frac{dQ/Q}{dP/P}$ is always $-1$.


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 ...


3

Homothetic goods is not a widely used term (as far as I know). It is not true that when preferences are homothetic $$ \frac{\Delta x}{\Delta m} = 1 $$ always holds. Instead $$ \frac{\Delta x}{\Delta m} = \text{constant} \cdot \frac{1}{p_x} $$ (where the constant is the share of income spent on $x$) or $$ \frac{\Delta x}{\Delta m} \frac{m}{x} = 1 $$ hold in ...


3

A monopolist maximizes profit. For me, it is usually easier to do this in the quantity space. So you rearange the demand and maximize $$\max_{Q_p,Q_r} \quad P_p(Q_p)Q_p + P_r(Q_r)Q_r - TC(Q_p+Q_r)$$ $$\max_{Q_p,Q_r} \quad (Q_p-10)Q_p + (28-2Q_r)Q_r - 5-2(Q_r+Q_p) -\frac{(Q_r+Q_p)^2}{8}$$ The FOC gives you two equations with two unknowns, $Q_r$ and $Q_p$, ...


3

According to advocates of MMT, the primary risk once the economy reaches full employment is inflation, which can be addressed by gathering taxes to reduce the spending capacity of the private sector. This statement is in accord with MMT, and it can be traced back to the concept of Functional Finance. One could do a search for Abba Lerner’s articles on ...


3

You can find demand functions like this in textbook and exam problems. Just a random example from internet: $$Q = \ln 4 - 0.5 \ln P$$ which is just a special case of $Q=−1/a \ln (P/b)$ where a is 1 and b is also 1 and there is some additional constant $\ln 4$. We just had an exam where I also saw one of the other examples. If I remember right it was $Q = \...


2

Price elasticity of supply is: $\frac{dQ_s}{dp}\frac{p}{Q_s}$. Because supply slopes upward $\frac{dQ_s}{dp}>0$. $Q_s$ and $p$ are also positive so $\frac{dQ_s}{dp}\frac{p}{Q_s}>0$.


2

For starters, elasticity is the percentage change in quantity (demanded or supplied, depending on what we're looking at) due to a one percent change in price. Or more vaguely: elasticity measures the sensitivity to price of behavior. From this definition it is clear that stating that a 'price is elastic' is nonsense. Trivially, the elasticity of price with ...


2

The standard elasticity still makes sense. Elasticity is not necessarily constant. Also, I am not sure what sort of formula you learned at your college but elasticity is rigorously defined (and also taught at college level for econ majors) as $EL = \frac{df(x)}{dx}\frac{x}{f(x)}$ for some function $f(x)$ (see Essential Mathematics for Economic Analysis by ...


2

Premise of your question is simply false. You state (emphasis mine): Price elasticity of demand $\mathrm{e_{D,P} = \dfrac{dD}{dP}. \dfrac{P}{Q^*}}$ clearly depends on the levels of price and quantity. Incorrect. Consider trivial counter example. A perfectly reasonable demand can be given by: $$ Q = A p^{-\epsilon} $$ where $Q$ is quantity, $p$ price and $...


2

Since you have discrete datapoints and want to calculate PED, you will have to make assumptions, like "the data represents the general behavior of the consumption function". (You can also get more data or throw your hands up in despair.) If you know that price has no effect on the quantity demanded, you can apply the PED formula, and get a very ...


1

Demand is a function In economics, "demand" is an unfortunate shorthand for "demand function". Less frequently and even more unfortunately it is also used for quantity demanded at the current price. Many textbooks make the effort to use "demand schedule" instead of "demand". Functions and operators The elasticity of ...


1

What is the proper way to read and/or express “price elasticity of demand”? Price elasticity of demand is for simple demand function defined as following: $$EL_D = \frac{\partial Q(p)}{\partial p} \frac{p}{Q(p)}$$ so the price elasticity of demand is a product of a slope of a demand function evaluated at some specific price and a ratio of that price to the ...


1

You are right that this is a bit of an abuse of a terminology textbooks do but it is actually not without any sense at all. For example consider two linear demand function functions: $$ Q = 10 - 2p \tag{1}$$ $$ Q = 10 -1/2p \tag{2}$$ In a intro textbook such as in the Mankiws principles or any other undergraduate textbook the demand given by 1 would be ...


1

Elasticity of demand is normally considered in relation to market demand for a good, that is, the sum of individual customer demands. Different customers will probably respond in different ways to a price increase. If the increase is 5%, especially on a low value item, very likely many customers will not notice, some will notice but not change their buying ...


1

Price elasticity of demand of $-0.5$ means that if price increases by $1\%$ demand decreases by $0.5\%$ (and vice versa in case of decrease). Consequently, if the equilibrium quantity with floor is $100$ and the price of lettuce is $25\%$ that means that eliminating the price flow - which will offset the $25\%$ will increase the quantity demanded by: $$0.5\...


1

For elasticity calculation why don't you try this: \begin{align} \frac{x_2^*}{x_1^*}&=\frac{\beta w_1}{\alpha w_2} \\ \ln\bigg(\frac{x_2^*}{x_1^*}\bigg) &= c - \ln\bigg(\frac{w_2}{w_1}\bigg) \tag{$c=\ln(\beta/\alpha)$} \end{align} It's easy to see from above that elasticity is $-1$ For second part, cost of production: \begin{align} C&=w_1x_1^* + ...


1

The first order conditions equate marginal revenue per factor to the price of that factor: \begin{align} p\cdot\alpha\frac{y}{x_1} &= w_1\\ p\cdot\beta\frac{y}{x_2} &= w_2, \end{align} Where I used the property of power function $(x^n)'_n = n \frac{x^n}{x}$. Divide the second FOC by the first to get the relation between the relative prices and the ...


1

this was answer to the original question before user completely edited it to something else You are asking several different questions: Do barriers to entry increase the (collective) market power of incumbent suppliers? Not generally. For example, in Bertrand price competition firms will have no market power even if there are barriers to entry. Even ...


1

Suppose $c_t$ is some composite function of interest rate $r$, e.g. $c_t(G(r_t))$. First the intertemporal elasticity of substitution is actually (IES) given by $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial r}$ (or also $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial \ln( u'(c_{t+1})/u'(c_t))}$). You can take the derivative above using chain rule for composite ...


1

Assuming that $p_i \neq p_j$ you just apply the formula; $$\epsilon_i(p_i,I,P) =-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)} \\ = -\left( \frac{-\sigma p_i^{-\sigma-1} I}{P^{1-\sigma}} \right)\frac{p_i}{\frac{p_i^{-\sigma} I}{P^{1-\sigma}}} =\sigma$$. The way how you gave the problem neither $P$ or $I$ contain $p_i$ so you treat them as ...


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