## 经济代写|宏观经济学代写Macroeconomics代考|ECON6002

2022年10月7日

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## 经济代写|宏观经济学代写Macroeconomics代考|Macroeconomics and finance

We’ve come a long way in understanding consumption. Now it is time to see if what we have learnt can be used to help us understand what asset prices should be in equilibrium.

To understand this relationship, we can use Lucas’s (1978) metaphor: imagine a tree that provides the economy with a unique exogenous income source. What is this tree worth? Optimal consumption theory can be used to think about this question, except that we turn the analysis upside down. Typically, we would have the price of an asset and have the consumer choose how much to hold of it. But in the economy the amount held and the returns of those assets are given because they are what the economy produces. So here we will use the FOCs to derive what price makes those exogenous holdings optimal. By looking at the FOCs at a given equilibrium point as an asset pricing equation allows us to go from actual consumption levels to asset pricing. Let’s see an example.
Start with the first order condition for an asset that pays a random return $r_{t+1}^j$ :
$$u^{\prime}\left(c_t\right)=\frac{1}{1+\rho} E_t\left[\left(1+r_{t+1}^i\right) u^{\prime}\left(c_{t+1}\right)\right] \quad \forall i .$$
Remember that
$$\operatorname{cov}(x, y)=E(x y)-E(x) E(y),$$
so, applying this equation to (12.51), we have that
$$u^{\prime}\left(c_t\right)=\frac{1}{1+\rho}\left{E_t\left(1+r_{t+1}^i\right) E_t\left(u^{\prime}\left(c_{t+1}\right)\right)+\operatorname{cov}\left(1+r_{t+1}^i, u^{\prime}\left(c_{t+1}\right)\right)\right} .$$
This is a remarkable equation. It says that you really don’t care about the variance of the return of the asset, but about the covariance of this asset with marginal utility. The variance may be very large, but, if it is not correlated with marginal utility, the consumer will only care about expected values. The more positive the correlation between marginal utility and return means a higher right-hand side, and, therefore, a higher value (more utility). Notice that a positive correlation between marginal utility and return means that the return is high when your future consumption is low. Returns, in short, are better if they are negatively correlated with your income; and if they are, volatility is welcomed!
As simple as it is, this equation has a lot to say, for example, as to whether you should own your house, or whether you should own stocks of the company you work for. Take the example of your house. The return on the house are capital gains and the rental value of your house. Imagine the economy booms. Most likely, prices of property and the corresponding rental value goes up. In these cases your marginal utility is going down (since the boom means your income is going up), so the correlation between returns and marginal utility is negative. This means that you should expect housing to deliver a very high return (because it’s hedging properties are not that good). Well, that’s right on the dot. Remember our mention to Kaplan et al. (2014) in Chapter 8, who show that housing has an amazingly high return.

## 经济代写|宏观经济学代写Macroeconomics代考|What next?

Perfect or not, the idea of consumption smoothing has become pervasive in modern macroeconomics. Many of you may have been taught with an undergraduate textbook using a consumption function $C=a+b Y$, with a so-called marginal propensity to consume from income equal to $b$. Modern macroeconomics, both in the version with and without uncertainty, basically states that this equation does not make much sense. Consumption is not a function of current income, but of intertemporal wealth. The distinction is important because it affects how we think of the response of consumption to shocks or taxes. A permanent tax increase will imply a one to one reduction in consumption with no effect on aggregate spending, while transitory taxes have a more muted effect on consumption. These intertemporal differences are indistinguishable in the traditional setup but essential when thinking about policy.

The theory of consumption has a great tradition. The permanent income hypothesis was initially stated by Milton Friedman who thought understanding consumption was essential to modern macroeconomics. Most of his thinking on this issue is summarised in his 1957 book A Theory of the Consumption Function (Friedman (1957), though this text, today, would be only of historical interest. The life cycle hypothesis was presented by Modigliani and Brumberg (1954), again, a historical reference.

Perhaps a better starting point for those interested in consumption and savings is Angus Deaton’s (1992) Understanding Consumption.

For those interested in exploring value function estimations you can start easy be reviewing Chiang’s (1992) Elements of Dynamic Optimization, before diving into Ljungqvist and Sargent (2018) Recursive Macroeconomic Theory. Miranda and Fackler (2004) Applied Computational Economics and Finance is another useful reference. Eventually, you may want to check out Sargent and Stachurski (2014) Quantitative Economics, which is graciously available online at url:http://lectures. quantecon.org/.

There are also several computer programs available for solving dynamic programming models. The CompEcon toolbox (a MATLAB toolbox accompanying Miranda and Fackler (2004) textbook), and the quant-econ website by Sargent and Stachurski with Python and Julia scripts.

If interested in macrofinance, the obvious reference is Cochrane’s Asset Pricing (2009) of which there have been several editions. Sargent and Ljungqvist provide two nice chapters on asset pricing theory and asset pricing empirics that would be a wonderful next step to the issues discussed in this chapter. If you want a historical reference, the original Mehra and Prescott (1985) article is still worth reading.

# 宏观经济学代考

## 经济代写|宏观经济学代写宏观经济代考|宏观经济与金融

$$u^{\prime}\left(c_t\right)=\frac{1}{1+\rho} E_t\left[\left(1+r_{t+1}^i\right) u^{\prime}\left(c_{t+1}\right)\right] \quad \forall i .$$

$$\operatorname{cov}(x, y)=E(x y)-E(x) E(y),$$

$$u^{\prime}\left(c_t\right)=\frac{1}{1+\rho}\left{E_t\left(1+r_{t+1}^i\right) E_t\left(u^{\prime}\left(c_{t+1}\right)\right)+\operatorname{cov}\left(1+r_{t+1}^i, u^{\prime}\left(c_{t+1}\right)\right)\right} .$$

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## MATLAB代写

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