Say’s Law argues that the ability to purchase something depends on the ability to produce and the wealth. In order to have something to buy, the buyer has to get something to produce and sell. Thus, the source of demand is production, not the money itself. Therefore, production drives economic growth.
Say drew four main conclusions.
More producers would boost the economy.
If members of the society do not produce would drag the society.
Business entities with trading are benefitial when they near each other.
Encouraging consumption is harmful. Production adn accumulation of goods constitutes prosperity, but consuming without producing eat away the economy.
The implication is that government should support and control production rather than consumption.
Global Cooperation (Prof. Stiglitz is upset about the global cooperation): unable to supply vaccine, cannot egt the low cost of production
Country’s Government conduct government purchases (about 25% U.S. GDP), But developing countries and emerging markets do not have ability of finance the economy recovery by “G”.
SDR of CB: about to move money from CB to Treasury
Make funds recyclable
Problems of debts. We nned deep or restructure and cooperate between private and public sectors to solve the problems of extra debts.
Climate Risks
Weather uncertainty. Extreme weather.
Transformation to neclear energy.
Maket not works well. Asset price changes as well.
Systematic Consequence.
Wrong valuation of financial sectors
Banks need to ensure not over-exposure to value risks
Gov needs to response by policy that not allow risks to be tranfered to private sectors. i.e. ensure all mortgages are green mortgage.
Financial Secors commisions to allocate.
Companies also need to allocate and disclose financial risks.
China – Debt Finance + Real Estate
The economy relies highly on real estate, so highly exposed to debts problems. Therefore, alternative modes can be chosen.
more equity finance than debt finance. However, equity finance needs more information, and then better regulation. Equity based finance is better to absorb risks.
Focus on more small business lending. Use small business loan to stimulate the economy.
More public investment. Private sectores are polluted (driven by tendency of profitability). Thus, public sectors need to work and help movement from rural to city and further make city better. (common prosperity).
A transformation to service dominant mode.
Public investment is the engine of economy.
A country with rapid tranformation would be the winner.
朱民 （清华大学国家金融研究院院长，前IMF副总裁，央行副行长，中行副行长）
2022通胀之剑+央行的挑战
p.s. 参考Dalio经济weather取决于1. 经济增长; 2. 通胀。
Facts:
通胀呈现国际国与国之间不同，国内产业间不同的趋势。
全球：大宗商品价格上升 PPI上升，能源价格上升. PPI+CPI剪刀差导致企业利润被严重压缩。
问题：通胀，暂时性 or 持久性？— Ideas：持久性
Demand side: 需求持续上升，由于Fiscal & Monetary Policy的传递最终传向demand side
Supply side: cannot be stable coz covid continues + fluctuations of econ
大宗商品价格上升 （主要 油+能源+稀有金属），价格上升的来源为：1. supply structure changes；2. Targets of carbon neturality
Labour Market： 劳动力参与率低 low skill workers 导致unemployment。但是demand高，i.e. 卡车司机+港口集装箱问题。 最终导致low skill workers wage 上升 => 导致 all wages 上升。
Summary: structural inflation becomes persistent because both supply and demand sides changes. Fed are less affected to the inflation.
Future
U.S. debts 提升 interest rate下降 => achieve a balance。 但是当Fed要提高 interest rate收缩经济时，U.S. unable to pay back debts because of higher cost of interest payment. 最终导致 interest 难以提升的问题。
Mechanism Design: start with the outcomes, then work backward to find what mechanism can create that outcomes.
E.G. Seperate a cake to two kids. Aims to make those kids think they get same size. Ideal way to do so is halve the cake. However, kids may not consider sizes are same. We the mechanism design does is 1. as one kid to cut, and then 2. ask the other kid to choose a piece.
Two Realistic Problems can be solved by mechansim design.
1. Supply Chain Disruption: Economy is complex that there are goods and inputs, and producers get multi-sources.
Problems: producer does not seek multi-sources, coz they would prefer protect itself & its downstream by using single source supply chain. => supply side market is inefficient.
Solve: Government conduct i.e. 1. government subsidies to achieve multi-sources; 2. government encourage producers through other ways.
By government is the mechanism
Climate changes: Human emission gas into atmosphere. A firm that uses coal to generate electicity has no incentives to switch to clean energy, coz 1. high cost of tranformation; 2. CO2 has less effects on that certain firm.
Solution: conduct carbon tax. 1. reduce carbon uses; 2. high cost of coal encourage firms to switch to clean power.
Carbon Tax is the mechanism
Chinese Government did good:
carbon trade system (,which is similar to carbon tax). Also, what banks can do higher rate for high carbon emission firms (my simple idea).
ban Cryptocurrency (加密货币). Cryptocurrency would reduce the effects of monetary policy, coz people would instead use Cryptocurrency.
陆磊 （国家外汇管理局副局长）
PPI 上涨来自 supply shocks，上游利润上升，CPI上涨来自demand shocks，下游利润下降（利润被上游收割）。
高质量发展： 工具+动力
工具：
需求侧：宏观调控需要continuous & persistent & transparent
供给侧：增强competitivity，竞争性由technology带来。
结构：优化分配（见李扬）。
Green Finance。
动力：
改革创新 – 灵活有效的fiscal policy
发展源泉激励
Focus on also the liquidity risk (, which exceeds the credit risks already)
Jeffrey Sachs
In the long-term future, there is a fundamental changes of the realtionship between China and U.S.. Need strengthen the cooperation.
At the end of North Atlantic dominant, Asia takes doninant instead. Geographic driven by economic divergence,
Environmental Crisis – Fragile ecosystem
Demographic changes. Massive urbanisation.
Common prospertiy – demand for social inclusion.
Smart machines and digital socity
Wealth and well-being. How to shift the focus from wealth to well-being.
All countries, governments need to consider the global environment regulation goal.
Smart chiens and technology declines labour demands => reallocation of wealth (conflicts exist that the wealthy people rejects to pay higher tax)
The Arrow-Pratt coefficient of absolute risk aversion
Definition (Arrow-Pratt coefficient of absolute risk aversion). Given a twice differentiable Bernoullio utility function \(u(\cdot)\),
$$ A_u(x):=-\frac{u”(x)}{u'(x)} $$
Risk-aversion is related to concavity of \(u(\cdot)\); a “more concave” function has a smaller (more negative) second derivative hence a larger \(u”(x)\).
Normalisation by \(u'(x)\) takes care of the fact that \(au(\cdot)+b\) represents the same preferences as \(u(\cdot)\).
In probability premium
Consider a risk-averse consumer:
1. prefers \(x\) for certain to a 50-50 gamble between \(x+\epsilon\) and \(x-\epsilon\).
2. If we want to convince the agent to take the gamble, it could not be 50-50 – we need to make the \(x+\epsilon\) payout more likely.
3. Consider the gamble G such that the agent is indifferent between G and receiving x for certain, where
$$G= \begin{cases} x+\epsilon, & \text{with probability $\frac{1}{2}+\pi$}.\\ x-\epsilon, & \text{with probability $\frac{1}{2}-\pi$ } \end{cases}$$
4. It turns out that \(A_u(x)\) is proportional to \(\pi/\epsilon\) as \(\epsilon \rightarrow 0\); i.e., \(A_u(x)\) tells us the “premium” measured in probability that the decision-maker demands per unit of spread \(\epsilon\).
ARA.
Decreasing Absolute Risk Aversion. The Bernoulli function \(u\cdot)\) has decreasing absolute risk aversion iff \(A_u(\cdot)\) is a decreasing function of \(x\). Increasing Absolute Risk Aversion… Constant Absolute Risk Aversion – Bernoulli utility function has constant absolute risk aversion iff \(A_u(\cdot)\) is a constant function of \(x\).
Relative Risk Aversion
Definition (coefficient of relative risk aversion). Given a twice differentiable Bernoulli utility function \(u(\cdot)\),
$$ R_u(x):=-x\frac{u”(x)}{u'(x)}=xA_u(x) $$
There could be decreasing/increasing/constant relative risk aversion as above.
Implication: DARA means that if I take a 10 gamble when poor, I will take a10 gamble when risk. DRRA means that if I gamble 10% of my wealth when poor, I will gamble 10% when rich.
Ramsey (1928), followed much later by Cass (1965) and Koopmans (1965), formulated the canonical model of optimal growth for an economy with exogenous ‘labour augmenting technological progress. The R.C.K model (or called. Ramsey (Neo-classical model) can be considered as an extension of the Solow model but without an assumption of a constant exogenous saving rate.
Assumptions
Firms
Identical Firms.
Markets, factors markets and outputs markets, are competitive.
Profits distributed to households.
Production fucntion with labour augmented techonological progress, \(Y=F(K,AL)\). (Three properties of the production: 1. CRTS; 2. Diminishing Outputs, second derivative<0; 3. Inada Condition.)
\(A\) is same as in Solow model, \(\frac{\dot{A}}{A}=g\). Techonology grows at an exogenous rate “g”.
Households
Identical households.
Number of households grows at “n”.
Households supply labour, supply capital (borrowed by firms).
The initial capital holdings is \(\frac{K(0)}{H}\)., where \(K(0)\) is the initial capital, and \(H\) is the initial number of households.
Assume no depreciation of capital.
Households maximise their lifetime utility.
The utility fucntion is constant-relative-risk-aversion (CRRA).
The lifetime utility for a certain household is represented by,
We denote \(w(t)=W(t)/A(t)\) as the efficient wage rate, then we get,
$$ w(t)=f(k(t))-k(t)f'(k(t)) $$
Another key assumption of this model is,
$$ \dot{k}(t)=f(k(t))-c(t)-(n+g)k(t) $$
, which represents the actual investment (outputs minus consumptions), \(f(k(t))-c(t)\); and break-even investment, \((n+g)k(t)\). The implication is that population growth and technology progress would dilute the capital per efficient work.
The difference with the Solow model is that we do not assume constant saving rate “s” in \(sf(k)\) now, instead we assume the investment as \(f(k)-c\).
Households
The budget constraint of households is that: the PV of lifetime consumption cannot exceed the initial wealth and the lifetime labour incomes.
, where \(R(t):=\int_{\tau=0}^ t r(\tau) d\tau\) to represent the discount rate overtime. When \(r\) is a constant, \(R(t)=r\cdot t\) and \(A(t)=A(0)e^{rt}=A(0)e^{R(t)}\).
We plug the \( \begin{cases} C(t)=A(t)c(t)\\L(t)=L(0)e^{nt}\\A(t)=A(0)e^{gt} \end{cases} \) into households’ lifetime utility function (objective function), and then get the,
, where \(B:=A(0)^{1-\theta} \frac{L(0)}{H}\) and \(\beta:=\rho-n-(1-\theta)g\) (we need \(\beta>0\) to make the utility function convergence), and the utility function is, \(u(c_t)=\frac{c_t^{1-\theta}}{1-\theta}\).
The budget constraint is the households’ lifetime budget constraints divided by \(A(0)\) and \(L(0)\),
Therefore, we find the time-path of consumption depends on \(f'(k)\). We define \(k*\) is the solution when \(f'(k)=\rho+\theta g\). So, at \(k^*\), the numerator of RHS equals zero.
As \(f(k)\) is an increasing function but with diminishing returns, so \(f”(k)<0\) and that means \(f'(k)\) is a decreasing function in k. Thus,
at \(k<k^*\), \(f(k)>\rho+\theta g\) and \( \frac{\dot{c_t}}{c_t} >0\);
at \(k>k^*\), \(f(k)<\rho+\theta g\) and \( \frac{\dot{c_t}}{c_t} <0\).
Figure 1
The dynamics of k
We recall the assumption,
$$ \dot{k}(t)=f(k(t))-c(t)-(n+g)k(t) $$
At \(\dot{k}=0\), consumption, \(c(t)=f(k(t))-(n+g)k(t)\), equals outputs minus break-even investment.
We now consider the Solow model without the depreciation term. Recall the difference between the RCK model and the Solow model is that we do not assume a constant saving rate over time, but other things keep similar. Thus, the term \(c(t)\) is “equivalent” to \(sy\) in Solow model.
Figure 2
In the Solow model, changes in saving rate would change the magnitude of \(sy\) curve. The Golden Rule saving rate is “s” that maximises consumption (the difference between Y and the interaction between \(sy\) and \(k(g+n)\)). The shape of the production function determines the property of the Golden Rule saving rate.
An equilibrium level of consumption is determined in mainly two steps. 1. the interaction between saving \(sy\) and \((n+g)k\) determines the \(k^*\). 2. plug \(k^*\) back to \(sy\) and find the difference between outputs and savings to get consumption.
We here focus on the second step, the equilibrium level of \(k^*\) determines consumption and thus consumption is a function of \(k^*\). At a lower saving rate (see Figure 2), \(k^*\) is too small, so there is less consumption. At a higher saving rate, \(k^*\) is too large, so there is also less consumption. Therefore, we can find that consumption is in a quadratic form w.r.t. \(k^*\).
Figure 3. \(c(t)=f(k(t))-(n+g)k(t),\ \dot{k}=0\)
$$ (s\downarrow) \Leftrightarrow k \downarrow \to c \downarrow$$
$$ (s\uparrow) \Leftrightarrow k \uparrow \to c \downarrow$$
Or, we can consider consumption as the difference between \(y\) and \((n+g)k\). The wedge like area gets large and then shrinks.
Overall, the above facts make the \(\dot{k}=0\) curve.
Recall \( \dot{k}(t)=f(k(t))-c(t)-(n+g)k(t) \).
Above the curve where \(c\) is large, then \(\dot{k}<0\) so \(k\) decreases. Below the curve where \(c\) is small, then \(\dot{k}>0\) so \(k\) increases. Implication is that if less consumption, then more saving, \(\dot{k}\) increases.
Phase Diagram.
Figure 4
Combining Figure 1 and Figure 3, we get the above Phase Diagram. The equilibrium is shown in the figure above.
P.S. We can prove that the equilibrium is less than Golden Rule level \(k^*_{GoldenRule}\) (which is the maximum point in the quadratic shaped curve). The proof is the following,
The value of k at \(\dot{c}=0\) is \(f'(k)-\rho-\theta g=0\), and the Golden Rule level is \(c=f(k)-(n+g)k\) (as we illustrated before), and take f.o.c. w.r.t. k to solve the Golden rule k. \(\frac{\partial c}{\partial k}=0 \to f'(k)=n+g\). Therefore, we get,
$$ \begin{cases} f_1:=f'(k_{equilibrium})=\rho+\theta g \quad\text{equilibrium in phase diagram}\\ f_2:=f'(k_{GoldenRule})=n+g\quad\text{golden rule level}\end{cases}$$
$$ \rho+\theta g>n+g \quad \text{by our assumption of \beta convergence}$$
So, we get,
$$ f_1>f_2 $$
$$ k_{equilibrium}< k_{GoldenRule} $$
Thus, we find the equilibrium level capital per efficient workers, \(k_{equilibrium}\), must be less than the Golden Rule level \( k_{GoldenRule} \).
From the phase diagram, we can get the saddle path that can achieve equilibrium.
BGP
At the Balanced Growth Path, the economy is in equilibrium. So the time-paths satisfy \( \frac{\dot{c}}{c}=0, and \frac{\dot{k}}{k}=0 \). Therefore, we can get the BGP of others,
Assumption: government conducts government purchases, \(G(t)\). The government purchases do not affect the utility of private sectors, and future outputs. Government finances, G(t), by lump-sum taxes.
We can consider the crowding-out effect. Under full employment, government purchases take away part of the consumptions. In our case, government spending takes away some of the savings. Therefore, the dynamics of capital per efficient workers become (the minus government spending term shifts the curve downward by G(t)),
$$\dot{k}(t)=f(k(t))-c(t)-G(t)-(n+g)k(t)$$
In short, government purchases would make the economy achieve a new equilibrium where there is less consumption but the same capital (investment from the private sector) level. Also, the saddle path moves downward.
\(\rho\) Changes
A fall in \(\rho\) can be considered as the effect of monetary policy. A fall in \(\rho\) would result in a movement of \(\dot{c}=0\) curve to the right by the equality, \(\frac{\dot{c}}{c}= \frac{r(t)-\rho-\theta g}{\theta} =\frac{f'(k)-\rho-\theta g}{\theta}\).
P.S. \(\dot{c}=0 \Leftrightarrow f'(k)=\rho-\theta g \to k=f’^{-1}( \rho-\theta g )\)
As \(\rho\) changes, a new path generates. The economy is at \(E^1\), and then follows the new path moving to \(E^{new}\). We would finally end up with a new equilibrium with higher consumption.
We replace this non-linear equation with the linear approximation, so we take the first order Taylor approximation around the equilibrium \(k^*\) and \(c^*\).
We then replace \( \dot{c}=\frac{f'(k)-\rho-\theta g}{\theta}c \) ) and \( \dot{k}=f(k(t))-c(t)-(n+g)k(t) \) into \( \dot{k}_{approx} \) and \( \dot{c}_{approx} \). Also, we denote \(\tilde{c}=c-c^*\) and \(\tilde{k}=k-k^*\).
From the above two equations we can find growth rate of \(\dot{\tilde{c}}_{approx}\) and \( \dot{\tilde{k}}_{approx} \) depend only on the ratio, \(\frac{\tilde{k}}{\tilde{c}}\).
Later, we apply a very strong assumption that \(\tilde{c}\) and \(\tilde{k}\) changes at the same rate, and also the rate make LHS of two equations equal. By this assumption, we denote,
We can see \(\mu\) must be negative, otherwise the economy cannot converge (see the path BB). If \(\mu<0\), the economy would be in the path AA instead. The path is the saddle path of R.C.K. model.
Applying such as the Cobb-Douglas form production, we can plug second derivatives of the production into \(\mu\) and get the speed of adjustment.
The current mostly used Solow model always have a depreciation term, and thus the law of motion becomes, \(\dot{K}=I-\delta K\).
The mainstream model has different assumptions about the production function as well. For example, technological progress is generally added. 1. \(Y=AF(K,L)\) in which technology is exogenous, and it could be called Hicks-neutral; 2. \(Y=F(K,AL)\) that can represent the efficient workers, labour-augmented, or Harrow-neutral; 3. \(Y=F(AK,L)\) in which the technological progress is capital augmented.
Applying for example the labour-augmented technology and \( \frac{\dot{A}}{A}=g\) , we can simply solve the Solow model as the following,
, where \(y=\frac{Y}{AL}\) and \(\frac{K}{AL}\) represent the output/capital per efficient works. Therefore, if \(\dot{k}=0\), then \(sy=(\delta+g+n)k\).
The stable point of k is \(k^*\) in which \(sf(k)=(\delta+n+g)k\).
We always the Cobb-Douglas function to represent the production function, because it satisfies CRTS, increasing and diminishing assumptions, and the Inada conditions (\(\lim_{k\rightarrow0}f'(k)=\infty; \lim_{k\rightarrow \infty}f'(k)=0\), Inada, 1963 ).
In the following, we would all analyse the model using efficient works to do analysis.
Balance Growth Path
All the following is assuming the economy is at the steady state or stable point.
For \( \frac{\dot{K}}{K} \),
$$ k=\frac{K}{AL} $$
By taking logritham,
$$ ln(k)=ln(K)-ln(A)-ln(L) $$
By taking differentiation and set \(\dot{k}=0\) (based on our previous derivations of finding the steady state condition).
In summary, the BGP is a situation in which each variable of the model is growing at a constant rate. On the balanced growth path, the growth rate of output per worker is determined solely by the rate of growth of technology.
P.S. Technology Independent of Labour And Capital
Applying for example the Type 1 case and \( \frac{\dot{A}}{A}=g\) , we can simply solve the Solow model as the following,
We would not use capital per efficient worker here, because labour is not technology-augmented by assumption. Instead, we simply assume capital per capita, \(k=\frac{K}{L}\). We can easily get the relationship,
We expand output per se (demoninator) by Euler’s Theorem \(Y=AF’_1K+AF’_2L\) (A is now outside the production function), and then calculate the percentage changes of outputs,
We now consider first how does changes in the saving rate affect those factors.
The determinants of saving rate are, for example, uncertainty or decrease in expected income, and required pension rate.
See the following figures,
An increase in the saving rate would result in an increase in the investment curve. \(\dot{K}=I-\delta K\) tells that there would be a huge increase in \(\dot{K}\) initially, and by the shape of production function, the difference diminishes until achieving the new stable point \(k^*_{new}\).
As \(\dot{k}\) is a derivative of \(k\) w.r.t. \(t\), we can easily get the time path of \(k\) as the following,
Another important factor is the growth rate of output per capita,
Also \(ln(Y/L)\),
For this one, we can prove that the slope of \(ln(Y/L)\) is \(\dot{ln(Y/L)}=\frac{\partial}{\partial t}[ln(Y)-ln(L)]=(g+n)-n=g\), so it grows constantly at rate “g” before \(t_0\). Later growth rate of Y jumps makes the slope of \(ln(Y/L)\) increases, but \(ln(Y/L)=g\) when achieves a new steady state and \(ln(Y/L)\) keeps growing at “g” in the long run.
The Speed of Convergence
Way 1
We follow our Solow model with labour-augmented technology. The time path of changes of capital per efficient works is,
$$ \dot{k}=sy-(\delta+n+g)k$$
$$ \dot{k}=sy-(\delta+n+g)k$$
At the steady state, \(\dot{k}=0\), so \( sy-(\delta+n+g)k \). We then plug in the Cobb-Doglas production function and denote \(y=\frac{Y}{AL}=\frac{K^{\alpha}(AL)^{1-\alpha}}{AL}=k^{\alpha}\), we can find the \(k^*\),
To find the speed of convergence, we would focus on the time path of k around \(k^*\). Or approximate the time-path by taking first-order Taylor expansionaround \(k^*\) to approximate,
$$ G(k)\approx G(k^*)+G'(k^*)(k-k^*) $$
As \(G(k^*)=0\) by our proof of steady state condition, thus,
Therefore, we find the mathematic expression of the convergence speed, \( (1-\alpha)(\delta+g+n) \). It is the measure of how quickly k changes when k diviates from \(k^*\). Also, we find that the growth rate \( G(k)=\frac{\dot{k}}{k} \) depends on both the convergence speed and \( \frac{k-k^*}{k^*} \), which is how far k deviates from its steady state level.
Take also a Taylor expansion to \(ln(k)\) at \(k^*\), we would get,
So we get \(g_y=\alpha g_k\), and \(\beta= (1-\alpha)(\delta+g+n) \) is the speed of convergence. It measures how quickly \(y\) increases when \(y<y^*\). The growth rate of y depends on the speed of convergence, \(\beta\), and the log-difference between \(y\) and \(y^*\).
Way 2
We take first order Taylor approximation to \(f(k)=\dot{k}\) around \(k=k^*\).
By definition of steady state condition, the first term of RHS is zero. So,
$$ \dot{k}\approx -\lambda \cdot (k-k^*) $$
We denote \(-\frac{\partial \dot{k}}{\partial k}|_{k=k^*}\:=\lambda\) as the speed of convergence. As \(\dot{k}=sy-(\delta+g+n)k=sk^{\alpha}- (\delta+g+n)k\), so,
To see why we denote \(\lambda\)as the speed of convergence, solve the differential equation, \( \dot{k}\approx -\lambda \cdot (k-k^*) \), by restrict time from 0 to t.
Definition (Homogeneity of degree \(k\)). A utility function \(u:\mathbb{r}^n\rightarrow \mathbb{R}\) is homogeneous of degree \(k\) if and only if for all \(x \in \mathbb{R}^n\) and all \(\lambda>0\), \(u(\lambda x)=\lambda^ku(x)\).
Constant Return to Scale: CRTS production function is homogenous of degree 1. IRTS is homogenous of degree \(k>1\). DRTS is homogenous of degree \(k<1\).
The Marishallian demand is homogeneous of degree zero. \(x(\lambda p,\lambda w)=x(p,w)\). (Maximise \(u(x)\) s.t. \(px<w\). “No Money Illusion”.
Excess demand is also homogeneous degree of zero. Easy to prove by the Marshallian Demand.
$$CRTS:\quad F(aK,aL)=aF(K,L) \quad a>0$$
$$IRTS:\quad F(aK,aL)>aF(K,L) \quad a>1$$
$$DRTS:\quad F(aK,aL)<aF(K,L) \quad a>1$$
2. Euler’s Theorem
Theorem (Euler’s Theorem) Let \(f(x_1,…,x_n)\) be a function that is homogeneous of degree k. Then,
Proof: Differentiate \(f(tx_1,…,tx_n)=t^k f(x_1,…,x_n)\) w.r.t \(t\) and then set \(t=1\).
P.S. We use Euler’s Theorem in the proof of the Solow Model.
3. Envelop Theorem
Motivation:
Given \(y=ax^2+bx+c, a>0, b,c \in \mathbb{R}\), we need to know how does a change in the parameter \(a\) affect the maximum value of \(y\), \(y^*\)?
We first define \(y^*=\max_{x} y= \max_{x} ax^2+bx+c \). The solution is \(x^*=-\frac{b}{2a}\), and plug it back into \(y\), we get \(y^*=f(x^*)=\frac{4ac-b^2}{4a}\). Now, we take derivative w.r.t. \(a\). \(\frac{\partial y^*}{\partial a}=\frac{b^2}{4a^2}\). We would find that,
Think of the ET as an application of the chain rule and then F.O.C., our goal is to find how does parameter affect the already maximised function \(v(q)=f(x^*(q),q)\).
A formal expression
Theorem (Envelope Theorem). Consider a constrained optimisation problem \(v(\theta)=\max_x f(x,\theta)\) such that \(g_1(x,\theta)\geq0,…,g_K(x,\theta)\geq0\).
Comparative statics on the value function are given by: (\(v(\theta)=f(x,\theta)|_{x=x^*(\theta)}=f(x^*(\theta),\theta)\))
(for Lagrangian \(\mathcal{L}(x,\theta,\lambda)\equiv f(x,\theta)+\sum_{k}\lambda_k g_k(x,\theta)\)) for all \(\theta\) such that the set of binding constraints does not change in an open neighborhood.
Roughly, the derivative of the value function is the derivative of the Lagrangian w.r.t. parameters, \(\theta\), while argmax those unknows (\(x=x^*\)).
4. Hicksian and Marshallian demand + Shepherd’s Lemma
A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function \(f(x)\) about point \(x=a\) is given by,
, where those coefficients are free to change, and the magnitude of those coefficients would affect how the approcimated curve looks like.
To get a better approximation, we would adjust those coefficients. Thus, we consider using different orders of derivatives to simulate our target function.
We need the first order derivate of \(cos'(x)|_{x=0}=sin(x)|_{x=0}\) to be zero, so we set the first-order derivative of our polynomial function to equal to zero as well!
Let’s go one more step. As the second derivative of \(cos^{(2)}(x)=-1\), we need the second derivative (, which is also the second derivate of the second-order term of our constructed polynomial function) of our polynomial function to be also -1.
We adjust that to be negative one, so \(c_2=-\frac{1}{2}\).
Therefore, we get,
$$cos(x)|_{x=0}\approx P(x)=c_0+c_1 x +c_2 x^2 = 1-\frac{1}{2}x^2$$
Great! If we need a more accurate approximation, then we keep on going to more derivates and calculate the coefficient of the higher-order term. However, I would do that, so I just simply add a term \(O(x^3)\) to represent there are other terms that are less equal than \(x^3\). (There are accurate descriptions that I will update in later posts).
Based on many years of study of the Solow model, I learned different versions of it. Although all of them are talking about the same theory, there are some differences in the learning structure and functional forms of the model. Now, I revisit the Solow model mainly based on the original paper (Solow, 1956). Here is a summary of the Solow model.
Robert Solow was awarded the Nobel Prize in 1987 for his contributions to the theory of Economic Growth.
Start
All theory depends on assumptions which are not quite true. That is what makes it theory. The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive. 1 A “crucial” assumption is one on which the conclusions do depend sensitively, and it is important that crucial assumptions be reasonably realistic. When the results of a theory seem to flow specifically from a special crucial assumption, then if the assumption is dubious, the results are suspect.
The above paragraph is the beginning of his paper. That is the first time I read this paper. It really shocks me and brings me a different understanding about economic theory.
Assumption
Only one commodity, output as a whole. \(Y(t)\), by which output equals income.
For each agent, \(Y(t)=consumption+saving/investing\). That is equivalent to assuming no government and net export, \(Y=C+I+G+NX\).
Net investment equals saving. (1) \(\frac{\partial K}{\partial t}=\dot{K}=s\cdot Y(t)\).
Output is produced by two factors, labour and capital. (2) \(Y=F(K,L)\). Also, the production function is homogenous of degree one and CRTS.
Solve
Insert (1) into (2), we would get
(3)\(\quad \dot{K}=sF(K,L) \)
As \(s\) is exogenous, thus (3) has two unknowns, so it is unable to be solved. To solve (3), we need to know something about labour.
There are mainly two ways of finding labour. Way 1: \(MPL=\frac{W}{P}\), which is the marginal productivity of labour equals real wage rate. Then, we can solve the labour supply. Way 2: take a general form. Making labour supply \(=f(W/P\). In any case, there are three unknowns, \(W,P,\) and \(L\).
However, we here assume an exogenous population growth rate \(n\). Thus,
(4) \(\quad L(t)=L_0\cdot e^{nt} \)
(easy to show \(ln(\frac{L_t}{L_{t-1}})=n\) and \(\frac{\dot{L_t}}{L_t}=n\).)
This term can be considered as the labour supply. It says that an exponentially growing labour force is offered. Employment is completely inelastic.
Put (4) into (3), we would get,
(5) \(\dot{K}=sF(K,L_0e^{nt})\).
, which is the time-path of capital accumulation under full employment.
(5) is a differential equation, and \(K(t)\) is the only variable. The solution could tell us, (all the followings are under full employment)
the time path of capital.
\(MPL=w\), and \(MPK=r\). The time-path of real wage rate and capital rent rate.
By given saving rate, we would also know saving and consumptions.
We introduce a new variable\(k=\frac{K}{L}\), which is capital per capita.
Thus, \(K=kL=kL_0e^{nt}\). The, differentiate w.r.t. \(t\), we get,
Easy to see that if the production function is increasing and diminishing and shaped as the figure above, then the interaction \(k^*\) would be a stable value (easy to show by discussing before \(k^*\) and after. e.g. if before k* then \(\dot{k}>0\), there is the accumulation of capital per capita).
When\( \frac{\dot{k}}{k} =0\), then \( \frac{\dot{K}}{K} = \frac{\dot{L}}{L}=n\). The implication is that at \(k^*\) (two lines interact, and make \( \frac{\dot{k}}{k} =0\) ) the time-path of labour equal that of capital.
The introduction of Solow (1956) ends there (there are analysis of applying different functional forms of production in his paper, but I would not discuss here), the following are a summary of my previous learning.
P.S. there is no depreciation in the original Solow model.
In summary, the key assumptions are: 1, the law of motion of capital accumulation \(\dot{K}=I\); 2, the shape of the production function.
Solow followed this paper with another pioneering artical, “Technical Change and the Aggregate Production Function.” Before it was published, economists had believed that capital and labour were the main causes of economic growth. But Solow showed that half of economic growth cannot be accounted for by increases in capital and labour.This unaccounted-for portion of economic growth—now called the “Solow residual”—he attributed to technological innovation.
