 # 宏观经济学macroeconomics代写|网课代修 Question 1: Short questions
(a) Suppose that, in a closed economy, the central bank adjusts the money supply in such a manner that the interest rate, $R$, is constant. Derive the slope of the AD curve. Show what happens to the LM curve as a result of the central bank’s policy.
(b) What do we mean with the notion that capital and labour are cooperative production factors? Can you give an example where this holds true?
(c) Assume that firm investment depends only weakly on the interest rate. Does that make the IS curve very steep or relatively flat? (In the usual diagram with the interest rate on the vertical axis and real output on the horizontal axis.)
(d) What are the two most important differences between the views of the classical and Keynesian economists?
(e) Consider the usual diagram with the real wage on the vertical axis and employment on the horizontal axis. Perfectly competitive firms use capital and labour to produce output. Why must competitive labour demand functions be downward sloping? Why must capital and labour be cooperative factors of production?

Question 1: Short questions
(a) The effective LM curve is horizontal at the interest rate $R_0$. The AD curve is vertical, i.e. does not depend on the price level. See Figure A1.1.
(b) If capital and labour are cooperative production factors, then more capital increases the marginal productivity of labour and more labour increases the marginal productivity of capital. A simple example (see page 4 in the textbook): The use of robot mixers in the kitchen enhances the productivity of the cooks.
(c) The IS-curve describes equilibrium on the goods-market $Y=C(Y)+I(R)+G$. Total differentiation gives the slope of the IS-curve (with $R$ the variable on the vertical axis and keeping $G$ constant):
$$d Y=C_Y d Y+I_R d R \quad \Rightarrow \quad\left[1-C_Y\right] d Y=I_R d R \quad \Rightarrow \quad \frac{d R}{d Y}=\frac{1-C_Y}{I_R}$$
If investment depends only weakly on the interest rate, then $I_R$ is very small, so $d R / d Y$ is large and the IS-curve is almost vertical (very steep).
(d) Their answers to two questions are different: (a) can the government affect the economy, and (b) should the government stabilize the economy. The classical economists say (a) maybe, and (b) no. The Keynesians say (a) yes, and (b) yes.
(e) Competitive labour demand functions take the form $W / P=F_N(N, K)$. They slope down in the $(W / P, N)$ space because there are diminishing returns to the labour input, $F_{N N}(N, K)<0$.

With constant returns to scale and only two factors of production, it must be the case that $F_{N K}>0$. This result is proved formally in Intermezzo 4.1.

Question 2: The Cobb-Douglas production function
Consider the Cobb-Douglas production function:
$$Y=F(N, \bar{K})=N^{\varepsilon} \bar{K}^{1-\varepsilon}, \quad 0<\varepsilon<1,$$
where $Y$ is output, $N$ is employment and $\bar{K}$ is the capital stock. The capital stock is fixed in the short run.
(a) Show that under perfect competition in the goods market the parameter $\varepsilon$ corresponds to the national income share of wages.
(b) Derive the short-run labour demand and goods supply schedules, both in levels and in terms of relative changes.
(c) What are the real wage elasticities of labour demand and the supply of goods?

Question 2: The Cobb-Douglas production function
(a) Under perfect competition the representative firm hires labour up to the point where the value of the marginal product of labour equals the nominal wage:
$$P F_N=W$$
From the Cobb-Douglas equation (Q1.1) we get:
$$F_N=\varepsilon N^{\varepsilon-1} \bar{K}^{1-\varepsilon}=\varepsilon \frac{Y}{N} .$$
By combing (A1.1) and (A1.2) we obtain:
$$\left[F_N=\right] \frac{W}{P}=\varepsilon \frac{Y}{N} \Rightarrow \varepsilon=\frac{W N}{P Y} .$$
Hence, $\varepsilon$ represents the national income share of wages.

(b) From equation (A1.1) and (A1.2) we get:
$$\varepsilon\left(\frac{N}{\bar{K}}\right)^{\varepsilon-1}=\frac{W}{P} \Rightarrow N=\bar{K}\left(\frac{W}{\varepsilon P}\right)^{\frac{1}{\varepsilon-1}} .$$
By substituting (A1.4) into the Cobb-Douglas production function (Q1.1) we obtain the short-run supply of output:
$$Y=N^{\varepsilon} \bar{K}^{1-\varepsilon}=\left[\bar{K}\left(\frac{W}{\varepsilon P}\right)^{\frac{1}{\varepsilon-1}}\right]^{\varepsilon} \bar{K}^{1-\varepsilon} \Rightarrow Y=\bar{K}\left(\frac{W}{\varepsilon P}\right)^{\frac{\varepsilon}{\varepsilon-1}}$$
In terms of relative changes we derive in a straightforward fashion from (A1.4) and (A1.5):
\begin{aligned} & \frac{d N}{N}=\frac{d \bar{K}}{\bar{K}}-\frac{1}{1-\varepsilon}\left[\frac{d W}{W}-\frac{d P}{P}\right], \ & \frac{d Y}{Y}=\frac{d \bar{K}}{\bar{K}}-\frac{\varepsilon}{1-\varepsilon}\left[\frac{d W}{W}-\frac{d P}{P}\right] . \end{aligned}
(c) The wage elasticities of labour demand and output supply are, respectively, $1 /(1-\varepsilon)$ and $\varepsilon /(1-\varepsilon)$ in absolute value.

Consider an economy with a representative profit maximizing producer with a CobbDouglas production function:
$$Y=N^\alpha \bar{K}^{1-\alpha}, \quad 0<\alpha<1,$$
where $Y$ is output, $N$ labour, $\bar{K}$ the (fixed) capital stock, and $\alpha$ a share parameter. This producer maximizes short-run profits $\Pi=P Y-W N$, with $W$ denoting the nominal wage rate and $P$ the price level.
(a) Derive the explicit expressions for the labour demand curve and the real wage elasticity of labour demand $\left(\varepsilon_D\right)$. Does the partial derivative have the correct sign (i.e., does labour demand decrease if the real wage rate goes up)?

The same economy has a representative consumer who maximizes utility $U$ that depends positively on consumption $C$ and negatively on labour supply $N$.
$$U=\mathrm{C}-\gamma \frac{N^{1+\sigma}}{1+\sigma}, \quad \gamma, \sigma>0 .$$
The consumer pays no taxes and cannot save or borrow, so she faces the expected budget constraint:
$$P^e \mathrm{C}=W N$$
(b) Derive the labour supply curve and the expected real wage elasticity of labour supply $\left(\varepsilon_S\right)$. Which effect dominates, the income effect or the substitution effect?
(c) Derive the aggregate supply curve.
Suppose that the demand side of this economy is described (in the neighbourhood of the equilibrium) by the aggregate demand curve:
$$\Upsilon=\xi+\theta \frac{\bar{M}}{P}$$
where $M / P$ is the real money supply.
(d) What is the interpretation and sign of $\theta$ ? Explain.
(e) Derive graphically the short run effect $\left(P^e\right.$ given) and the long run effect of an unexpected monetary shock assuming that the adaptive expectations hypothesis holds:
$$P_{t+1}^e=P_t^e+\lambda\left[P_t-P_t^e\right] .$$
Is the model stable? What happens if consumers are blessed with perfect foresight?

(a) The labour demand curve is obtained by setting marginal labour productivity equal to the real wage rate:
\begin{aligned} & F_N(N, \bar{K})=W / P \quad \Rightarrow \quad \alpha[\bar{K} / N]^{1-a}=W / P \Rightarrow \ & N=\bar{K}\left[\frac{W}{\alpha P}\right]^{\frac{1}{a-1}} \end{aligned}
The real wage elasticity of labour demand is:
$$\varepsilon_D=-\frac{W / P}{N} \frac{\partial N}{\partial(W / P)}$$
Differentiate the labour demand equation (A1.6) with respect to $W / P$ and we have:
$$\frac{\partial N}{\partial(W / P)}=-\frac{1}{1-\alpha} \frac{N}{W / P} \quad \Rightarrow \quad \varepsilon_D=\frac{1}{1-\alpha} .$$
Because $0<\alpha<1$, the real wage elasticity is always negative (real wages up, labour demand down, as expected).
(b) Substitute $C=W N / P^e$ into the utility function (Q1.3) and differentiate with respect to $N$, set the first order condition to 0 and we have the labour supply equation:
$$\frac{\partial U}{\partial N}=0: \gamma N^\sigma=W / P^e \Rightarrow W / P=\gamma \frac{P^e}{P} N^\sigma$$

The real wage elasticity of labour supply is:
$$\varepsilon_S=\frac{W / P}{N} \frac{\partial N}{\partial(W / P)}$$
Differentiate the labour supply equation (A1.7) with respect to $W / P$ and we have:
$$\frac{\partial N}{\partial(W / P)}=\frac{1}{\sigma} \frac{N}{W / P} \quad \Rightarrow \quad \varepsilon_S=\frac{1}{\sigma}>0$$
The substitution effect dominates the income effect.
(c) Substitute $W / P$ of the labour supply equation (A1.7) into the labour demand equation (A1.6):
$$N=\bar{K}\left[\frac{\gamma}{\alpha} \frac{P^e}{P} N^\sigma\right]^{\frac{1}{\alpha-1}} \Rightarrow N=\left[\frac{\alpha}{\gamma} \frac{P}{P^e} \bar{K}^{1-\alpha}\right]^{\frac{1}{1-\alpha+\sigma}}$$
This is the relation between $N$ and $P$. As you can see, if $P$ goes up, $N$ goes up. Substitute the equilibrium level of labour into the production function, do all the math correctly and we have the aggregate supply curve:
$$Y=B\left(\frac{P}{P^e}\right)^\delta$$
with:
$$\delta \equiv \frac{\alpha}{1-\alpha+\sigma}, \beta \equiv \alpha\left[\frac{1-\alpha}{1-\alpha+\sigma}+\frac{1-\alpha}{\alpha}\right], B \equiv\left[\frac{\alpha}{\gamma}\right]^\delta \bar{K}^\beta$$
(d) $\theta$ is the change in aggregate demand if real money balances increase 1 unit. From the IS-LM model we know that $\theta$ must be positive (LM curve shifts to the right).
(e) See Figure A1.2. The AD curve shifts to the right, households are surprised at impact and have not yet adjusted their expected price levels, the AS curve stays where it was. $P$ and $Y$ both increase from $P_0$ to $P_1$ resp. $Y_0$ to $Y_1$. In the following periods households adjust their expectations based on the previous forecasting error, and the AS-curve slowly shifts to the left until there is a new equilibrium at $\mathrm{E}{\infty}$. The model is stable. If people are blessed with perfect foresight, they always supply the correct amount of labour, so the AS-curve is vertical and prices immediately jump to the new equilibrium $\mathrm{E}{\infty}$.

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