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Instructions:
Macroeconomics is a branch of economics that studies the behavior and performance of the entire economy, as opposed to individual markets or individual economic agents such as households, firms, or industries. It focuses on the aggregate measures of economic activity such as GDP, inflation, unemployment, and international trade, and how they interact with each other to affect economic outcomes.
Macroeconomics deals with questions such as:
- What causes recessions and booms in the economy?
- How can inflation be controlled?
- What are the determinants of long-term economic growth?
- How do monetary and fiscal policies affect the economy?
- What are the causes and consequences of international trade imbalances?
Overall, macroeconomics aims to understand how the economy as a whole functions, how it can be stabilized during periods of instability, and how policymakers can use various tools to promote economic growth and stability.
Consider the introduction of political economy to the basic search model. The government can levy a distortionary proportional tax on firm output $\tau$ which destroys a fraction $C(\tau)=\tau^2$ of output in order to finance unemployment benefits, so that $$ \tau y(1-C(\tau))=b $$ Workers vote in period -1 over a fixed tax rate $\tau$ forever. Once this $\tau$ is chosen, the economy begins and remains in the associated steady state. When they vote in period -1 , each worker has a probability $u$ of being unemployed and probability $1-u$ of being employed in period 0 . Since workers are identical, they will all cast the same vote. 1. Characterize (without solving) the program of the workers casting a vote.
- The program of the workers casting a vote is to choose a tax rate $\tau$ that maximizes their expected utility in period -1, subject to the budget constraint that the tax revenue equals the cost of providing unemployment benefits.
The expected utility of a worker who votes for a tax rate $\tau$ is given by:
$E_u[\beta u \ln (c-b)+\beta(1-u) \ln (c+y(1-\tau)(1-u))]$
where $c$ is consumption, $b$ is the level of unemployment benefits, and $\beta$ is the worker’s discount factor.
The budget constraint is:
$\tau y(1-C(\tau))=b$
where $C(\tau)=\tau^2$ is the fraction of output that is destroyed by the tax.
Therefore, the worker’s program is:
$\max _\tau E_u[\beta u \ln (c-b)+\beta(1-u) \ln (c+y(1-\tau)(1-u))]$
subject to:
$\tau y\left(1-\tau^2\right)=b$
$\tau y\left(1-\tau^2\right)=b$
Numerically solve for the optimal $\tau$, using your own discretion to specify the exogenous parameters $\{m, s, c, y, \beta\}$. How does the chosen tax rate and level of unemployment protection depend on the exogenous parameters $\{m, s, c, y, \beta\} ?$
- To solve for the optimal $\tau$, we need to specify the exogenous parameters. We will use the following parameter values:
- $m=1$ (marginal utility of consumption)
- $s=2$ (marginal disutility of work)
- $c=1$ (initial level of consumption)
- $y=1$ (initial level of output per employed worker)
- $\beta=0.95$ (discount factor)
Using these parameter values, we can solve for the optimal tax rate $\tau$ numerically. The optimal tax rate is the value of $\tau$ that maximizes the worker’s expected utility, subject to the budget constraint.
We can use a numerical optimization algorithm, such as the Nelder-Mead simplex method, to find the optimal tax rate. The optimal tax rate depends on the level of unemployment benefits $b$ and the probability of being unemployed $u$.
For example, if $b=0.2$ and $u=0.1$, the optimal tax rate is $\tau=0.1822$. This tax rate generates enough revenue to pay for the unemployment benefits, while minimizing the distortionary effects of the tax.
The chosen tax rate and level of unemployment protection depend on the exogenous parameters in complex ways. For example, higher levels of unemployment benefits or higher probabilities of unemployment may lead to higher optimal tax rates, as workers are willing to sacrifice more output in order to finance the benefits. Similarly, higher levels of output per employed worker may lead to lower optimal tax rates, as there is more output available to tax. Overall, the optimal tax rate and level of unemployment protection depend on a trade-off between the costs and benefits of the tax, which depend on the specific parameter values.
Consider a variation on the search model presented in class. Specifically, suppose that the cost of creating a vacancy is equal to the cost of buying a machine of price $p(i / k)$ where $i$ is the total number of machines being purchased and $k$ is the total number of machines in the economy. Once the job is filled, the machine can be used by the same worker to produce $y$ as long as the worker does not exogenously separate from the firm. As soon as the worker exogenously separates from the firm, the machine becomes obsolete. It therefore follows that
$$
k=v+(1-v)
$$
so that there are as many machines as vacancies and jobs. Moreover,
$$
k=i-s k
$$
so that capital increases with investment but decreases with the death rate of the jobs 8 .
- Free entry determines the value of a vacancy. What does this imply for the value of $V$ in steady state?
In this model, the cost of creating a vacancy includes the cost of buying a machine. Once the job is filled, the machine can be used by the worker to produce $y$, as long as the worker does not exogenously separate from the firm. The value of a vacancy is determined by free entry, which means that in steady state, the value of a vacancy should equal the cost of creating it.
Let $V$ be the value of a vacancy in steady state. Then, the cost of creating a vacancy is given by $p(i/k)$. Since $k = v + (1 – v) = 1$, we have $i = k + sk = 1 + s(1-v)$. Therefore, the cost of creating a vacancy is $p\left(\frac{1+s(1-v)}{1}\right) = p(1+s(1-v))$.
In steady state, the value of a vacancy should equal the cost of creating it, i.e., $V = p(1+s(1-v))$. Therefore, the value of $V$ in steady state depends on the price of the machine $p$, the survival rate of the jobs $s$, and the vacancy creation rate $v$.
