CLASS 15: MONETARY POLICY, PRODUCTION, AND EMPLOYMENT

• So far we assumed that output was like `fruits from the trees', i.e. was given and not affected by monetary policy.
• For many of the issues that we studied (inflation, exchange rate determination) that assumption was perhaps fine, but it left us unable to address a crucial question: What are the effects of monetary policy on output? on employment?
• This question is crucial for policy making: from Federal Reserve officials quoting the Phillips curve when deciding interest rate policy, to European policy makers invoking a monetary relaxation to stimulate the economy, much of the policy talk seems to assume that monetary policy has some effect on output - one way or the other.

• Does it? in the models that will be analyzed here we will see that the answer depends on the degree of price/wage rigidity.
• We will first study the production function, labor demand, and labor supply.
• Then we will study the equilibrium with flexible prices, and finally we will discuss the equilibrium with price rigidity.

The Production Function, Labor Demand and Labor Supply

• To keep the problem simple, we will assume that the only input in output production is labor, and we will therefore neglect capital accumulation (it is not hard to generalize the problem to the situation where the supply of capital is fixed, as in the case we studied in the first class, but to keep notation simple we will not do it).
• The production function (per capita) is as follows:
  
  y_t=f(h_t),
  
  
  with f'>0 and f''<0,
  where y_t is per capita output, and h_t is the amount of hours used in production by the representative agent.
• There is perfect competition in the labor market, and in the market for the consumption good. The representative firm (there are many of these firms, and they are all the same) takes as given both the nominal wage per hour, w_t, and the price of the consumption good, p_t.
• The firm chooses the optimal amount of labor (labor demand) so to maximize profits, which are defined as:
  
  $$\pi_t=p_tf(h_t)-w_t h_t$$
  
  

• The problem of the firm is then:
  
  $$max_{h_t} \pi_t=p_tf(h_t)-w_t h_t$$
  
  
  and the associated first order condition is, not surprisingly:
  
  $$f'(h_t)=\frac{w_t}{p_t}$$
  
  
  The firm is going to hire labor until the real wage is equal to the marginal productivity of labor: this condition determines Labor Demand.
• What about Labor Supply? If the agent had no dis-utility from working, she would work all the time. However she likes to enjoy leisure, so that her utility function is defined as:
  
  $$\sum_{t=0}^\infty \beta^t u(c_t,m_t,l_t)$$
  
  

• If we call $\bar{l}$ the amount of hours the agent has available per period, the amount of leisure she can enjoy is inversely related to the hours she spends working:
  
  $$h_t=\bar{l}-l_t$$
  
  

• Let us study the budget constraint of the agent. This is:
  
  $$M_{t+1} \leq w_t (\bar{l}-l_{t}) +M_t +\pi_t +T_t- p_t c_{t}$$
  
  
  where $w_t (\bar{l}-l_{t})$ represents the paycheck, and all the other terms all well known except for $\pi_t$.
• What is $\pi_t$ doing in the agents budget constraint? The underlying assumption is that the agent owns the firm, and so receives the profit from the firm.
• Note that, if we substitute the definitions of h_t and $\pi_t$ into the budget constraint, we obtain:
  
  $$M_{t+1} \leq p_tf(h_t) +M_t +T_t- p_t c_{t}$$
  
  
  given that
  
  $$w_t h_{t} +\pi_t =p_tf(h_t) .$$
  
  

• The agent, however, takes $\pi_t$ as given in its maximization process, that is, it does not behave as it it could affect the profits of the firm. Why? Let us say that each of the many identical representative agents owns a different firm from the one she works for. Then, as a firm owner, she goes after maximizing profits and solves the firm's problem above. As a worker, she decides how much time to allocate between labor and leisure, taking the profits of her firm as given. Alternatively, think of each firm as run by some board of trustees, and of each agent as stockholder.
  In any case, in equilibrium, since all firms and all agents are identical and therefore behave identically, it will have to be the case that:
  
  $$w_t h_{t} +\pi_t =p_tf(h_t) .$$
  
  

• The problem of the agent is very much the same problem as in the standard Sidrauski- Brock model:
  
  $$max_{\{M_{t+1},c_t,l_t\}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t},l_t)$$
  
  
  subject to
  
  $$M_{t+1} \leq w_t (\bar{l}-l_{t}) +M_t +\pi_t +T_t- p_t c_{t} .$$
  
  

• The budget constraint can be rewritten as:
  
  $$c_t\leq \frac{w_t}{p_t}(\bar{l}-l_{t})+\frac{\pi_t}{p_t} +\tau_t-\frac{M_{t+1}-M_t}{p_t}$$
  
  
  and substituted into the objective function, so that the agent's problem becomes:
  
  $$max_{\{M_{t+1},c_t,l_t\}_{t=0}^{\infty}}$$
  
  
  
  
  $$\sum_{t=0}^\infty \beta^t u(\frac{w_t}{p_t}(\bar{l}-l_{t})+\f... ...i_t}{p_t} +\tau_t-\frac{M_{t+1}-M_t}{p_t} ,\frac{M_t}{p_t},l_t)$$
  
  

• The FOC with respect to M_t+1 is the same old one:
  
  $$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t},l_t)=$$
  
  
  
  
  $$=\beta \frac{1}{p_{t+1}}[u_m(c_{t+1},\frac{M_{t+1}}{p_{t+1}},l_t)+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}},l_t)]$$
  
  

• The novelty is the FOC with respect to l_t, which is:
  
  $$u_l(c_t,m_t,l_t)=\frac{w_t}{p_t}u_c(c_t,m_t,l_t)$$
  
  

• This condition can as usual be interpreted as MRT=MRS. The marginal rate of transformation of spending one more hour working rather than watching TV is the real wage, and the marginal rate of substitution is given by the ratio of the utility of leisure and consumption:
  
  $$\frac{u_l(c_t,m_t,l_t)}{u_c(c_t,m_t,l_t)}=\frac{w_t}{p_t}.$$
  
  
  This condition can be thought as determining Labor Supply

The Equilibrium with Constant Money Supply

• Let us now consider the equilibrium with constant money supply:
  
  $$M^s_t=M^s_0 \mbox{, all t}$$
  
  

• Note that by repeated substitution we obtain the usual formula:
  
  $$\frac{1}{p_t} u_c(c_t,m_t,l_t)= \sum_{i=1}^{T} \beta^i \frac{1}{p_{t+i}}u_m(c_{t+i},m_{t+i},l_{t+i})+$$
  
  
  
  
  $$+\frac{1}{p_{t+T}}u_c(c_{t+T},m_{t+T},l_{t+T})$$
  
  
  and by imposing the transversality condition
  
  $$lim_{T \rightarrow \infty} \frac{1}{p_{t+T}}u_c(c_{t+T},m_{t+T},l_{t+T})=0$$
  
  
  we obtain:
  
  $$\frac{1}{p_t} = \frac{1}{u_c(c_t,m_t,l_t)} \sum_{i=1}^{\infty} \beta^i \frac{1}{p_{t+i}} u_m(c_{t+i},m_{t+i},l_{t+i}).$$
  
  
  We know that this condition yields to the standard LM equation, also called Equilibrium in the Money/Asset Market.
• A second equilibrium condition is obtained by substituting the relationship determining labor demand, $f'(h_t)=\frac{w_t}{p_t}$, into the condition for labor supply:
  
  $$f'(\bar{l}-l_{t})=\frac{u_l(c_t,m_t,l_t)}{u_c(c_t,m_t,l_t)}.$$
  
  
  We could call this condition Equilibrium in the Labor Market.
• The third and last equilibrium condition can be obtained from the budget constraint:
  
  $$M_{t+1} \leq w_t (\bar{l}-l_{t}) +M_t +\pi_t +T_t- p_t c_{t}$$
  
  
  after recognizing that
  
  $$w_t (\bar{l}-l_{t}) +\pi_t =p_tf(h_t)$$
  
  
  and that, from the government budget constraint:
  
  T_t=M_t+1-Mt=0.
  
  

• This equilibrium condition becomes:
  
  $$c_t=f(\bar{l}-l_{t})$$
  
  
  and can be interpreted as Equilibrium in the Goods Market: demand for the consumption good (c_t) equal to supply (f(h_t)).
• Let us rewrite all three equilibrium condition:
  
  $$\mbox{(1) } \frac{1}{p_t} = \frac{1}{u_c(c_t,m_t,l_t)} \sum_{... ...{\infty} \beta^i \frac{1}{p_{t+i}} u_m(c_{t+i},m_{t+i},l_{t+i})$$
  
  
  
  
  $$\mbox{(2) } f'(\bar{l}-l_{t})=\frac{u_l(c_t,m_t,l_t)}{u_c(c_t,m_t,l_t)}$$
  
  
  
  
  $$\mbox{(3) } c_t=f(\bar{l}-l_{t}).$$
  
  

• Let us guess that we have an equilibrium in which l_t is constant over time:
  
  l_t=l^*
  
  

• If this is the case, also c_t is constant over time, and equal to:
  
  $$c_t=c^*=f(\bar{l}-l^*)$$
  
  

• But if both l_t and c_t are constant, we know that (1) becomes:
  
  $$\frac{1}{p_t} = \frac{1}{u_c(c^*,m_t,l^*)} \sum_{i=1}^{\infty} \beta^i \frac{1}{p_{t+i}} u_m(c^*,m_{t+i},l^*).$$
  
  
  We can multiply both sides by M^s and obtain:
  
  $$m_t = \frac{1}{u_c(c^*,m_t,l^*)} \sum_{i=1}^{\infty} \beta^i m_{t+i} u_m(c^*,m_{t+i},l^*).$$
  
  

• But we know that this relationship has an equilibrium for constant m, and the relationship that determines the level of m is:
  
  $$u_c(c^*,m^*,l^*)-\frac{\beta}{1-\beta}u_m(c^*,m^*,l^*)=0.$$
  
  

• In particular, we realize that the equilibrium level of real money balances is:
  
  m_t=m^*(0)
  
  

• Finally, our initial guess that l_t is constant is indeed verified. From
  
  $$f'(\bar{l}-l_{t})=\frac{u_l(c^*,m^*(0),l_t)}{u_c(c^*,m^*(0),l_t)}$$
  
  
  we see that, if m and c are constant, l_t has to be constant over time as well.
• So the three condition jointly determining m^*(0), c^*, and l^* are:
  
  $$u_c(c^*,m^*(0),l^*)-\frac{\beta}{1-\beta}u_m(c^*,m^*(0),l^*)=0$$
  
  
  
  
  $$f'(\bar{l}-l^*)=\frac{u_l(c^*,m^*(0),l^*)}{u_c(c^*,m^*(0),l^*)}$$
  
  
  
  
  $$c^*=f(\bar{l}-l^*)$$
  
  

• We know that this implies that inflation is zero:
  
  $$p_t=\frac{M^s_0}{m^*(0)}$$
  
  
  and that the real wage is: $\frac{w_t}{p_t}=f'(\bar{l}-l^*).$
• Last, note that the level of money supply does not affect the equilibrium in real variable (employment, real money balances, production, consumption): Money is Neutral.
• Indeed, the level of real money balances does not enter in any of the three relationship determining the equilibrium for m^*(0), c^*, and l^*, namely:
  
  $$u_c(c^*,m^*(0),l^*)-\frac{\beta}{1-\beta}u_m(c^*,m^*(0),l^*)=0$$
  
  
  
  
  $$f'(\bar{l}-l^*)=\frac{u_l(c^*,m^*(0),l^*)}{u_c(c^*,m^*(0),l^*)}$$
  
  
  
  
  $$c^*=f(\bar{l}-l^*)$$
  
  

• It only enters the relationship determining the price level:
  
  $$p_t=\frac{M^s_0}{m^*(0)}.$$
  
  

• However, note that this is true only if money supply is constant over time: that is, we are not saying that changes in the level of money supply have no effect. We are saying that if you take two economies with different -but constant over time- nominal money supplies, the equilibrium for all `real' variables in both economies is the same.