经济代写|宏观经济学代写Macroeconomics代考|EC4505

经济代写|宏观经济学代写Macroeconomics代考|Bringing in the firm

We now need to solve for the path of investment and output. But we have done that already, when we solved the problem of the firm. We know that the firm’s behaviour can be summarised by the initial condition $k_0>0$, plus the pair of differential equations
$$k_t=\chi\left(q_t-1\right) k_t .$$
and
$$\dot{q}_t=r q_t-A f^{\prime}\left(k_t\right)-\frac{1}{2} \chi\left(q_t-1\right)^2 .$$
Recall that (13.37) and (13.38) are a system of two differential equations in two unknowns, which can be solved independently of the other equations in the system.
Recall also that profits are defined as
$$\pi_t=A f\left(k_t\right)-\psi\left(\dot{k}_t, k_t\right)-\dot{k}_t .$$

• Once we have a solution (13.37) and (13.38), we know what $k_t, \dot{k}_t, q_t$ and $\dot{q}_t$ are.
• With that information in hand, we can use (13.39) to figure out what the whole path for profits $\pi_t$ will be.
• Knowing that, we can use equation (13.36) to solve for consumption levels.
• Knowing output levels $A f\left(k_t\right)$, investment rates $\dot{k}_t$, and consumption levels $c_0$, we can use this information plus the budget constraint (13.31) to figure out what the current account $\dot{b}_t$ is and, therefore, the path of bond-holdings $b_t$.

经济代写|宏观经济学代写Macroeconomics代考|The surprising effects of productivity shocks

Suppose again that, starting at this steady state, the productivity parameter $A$ falls to a lower permanent level $A^{\prime}<A$. To make life simpler, suppose that initially bond-holdings are zero $\left(b_0=0\right)$.

Focus first on the evolution of the capital stock $k_t$ and its price $q_t$, shown in Figure 13.3. The drop in $A$ shifts the $\dot{q}=0$ schedule to the left. The other locus is unaffected. The steady state of the economy also moves left. The new steady state capital stock is $k^{* \prime}<k^*$.

At time zero, $q$ falls on impact all the way to the saddle path that goes through the new steady state. Over time the system moves along the new saddle path until it reaches the new steady state. Along this trajectory the rate of investment is negative, eventually hitting $\dot{k}_t=0$ as the economy lands on the new steady state.

Notice that net output, defined as $y_t-\dot{k}_t-\psi\left(\dot{k}_t, k_t\right)$, may either go up or down initially, depending on whether the fall in gross output $y_t$ is larger or smaller than the change in costs associated with the decline in investment. Over the long run, however, the effect is unambiguous: Net output is lower, since gross output falls and investment tends to zero. Figure $13.4$ shows net output falling initially, and then declining further to its new steady state level.
The level of consumption is constant and given by
$$c_0^{\prime}=r \int_0^{\infty}\left(A^{\prime} f\left(k_t\right)-\dot{k}_t-\psi\left(\dot{k}_t, k_t\right)\right) e^{-r t} d t<c_0 \quad t \geq 0 .$$
Graphically (in terms of Figure 13.4), consumption is determined by the condition that the thatched areas above and below it be equal in present value.

What about the current account? Since initial bonds are zero, having consumption below net output (see Figure 13.4) must imply that the economy is initially running a current account surplus, saving in anticipation of the lower output in the future. In the new steady state, the net foreign asset position of the economy is that of a creditor: $b^{* \prime}>0$ and the current account goes back to zero.

宏观经济学代考

经济代写|宏观经济学代写宏观经济学代考|引入公司

$$k_t=\chi\left(q_t-1\right) k_t .$$

$$\dot{q}_t=r q_t-A f^{\prime}\left(k_t\right)-\frac{1}{2} \chi\left(q_t-1\right)^2 .$$

$$\pi_t=A f\left(k_t\right)-\psi\left(\dot{k}_t, k_t\right)-\dot{k}_t .$$

经济代写|宏观经济学代写宏观经济学代考|生产率冲击的惊人影响

$$c_0^{\prime}=r \int_0^{\infty}\left(A^{\prime} f\left(k_t\right)-\dot{k}_t-\psi\left(\dot{k}_t, k_t\right)\right) e^{-r t} d t<c_0 \quad t \geq 0 .$$

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