Midterm
Solutions

1) True. This is the definition of Ricardian Equivalence, which holds in a closed economy as well.
2) True. CA=Y-G-C. If Ricardian Equivalence holds, the time pattern of taxes does not affect consumption, and therefore it does not affect the Current Account (we have not discussed it in class yet, but the time pattern of taxes does not affect investment either, so the statement is true even if we are not in the endowment economy case)
3) False. Income effect may go in the opposite direction.
4) True. We know that with perfect capital mobility the level of investment depends only on the productivity of capital, and not on how much domestic agents decide to save.
5) True. One of the assumptions of Ricardian Equivalence is that individuals can borrow freely. If they cannot, then the government, by lowering taxes today and raising them tomorrow, is effectively borrowing on behalf of agents.
6) True: $CA=Y1-\bar{Y}-(G1-\bar{G})-I$. If a country is growing its permanent income is higher than its current income, $\bar{Y}>Y1$, so CA<0.
7) False if you assume that Ricardian Equivalence holds, because of the answer to question (2). But it may be true if you think it does not hold: consumption will decrease due to the increase in taxes, and the CA will improve.
8) False. Because of the difference in the relative price of tradables and non-tradables across countries, the same per capita income is US$ may buy different consumption bundles.
9) True. In a closed economy C1=Y1, C2=Y2, so the relation MRS=MRT becomes $\frac{U_1(Y1,Y2)}{U_2(Y1,Y2)}=(1+r)$. When Y1 is high, consumption is high and U₁ is low, so (1+r) is low (assuming that consumption today and consumption tomorrow are substitutes, so U₂ decreases when C1 is high, or is not affected at all, as in the case of logarithmic utility). Viceversa when Y1 is low.
10) True. KA=-CA. If KA=0, then it must be that CA=0, which means exports=imports (neglecting net factor payments).
Analytical question.
a) The firm's problem is:

max_N 2N-wN

and the FOC is:

2-w=0.

Since the marginal productivity of labor is constant, the labor demand curve is flat: the firm is willing to hire any amount of labor at the wage w=2.
b) The agent's problem is:

$$max_{\{C,L\}} \gamma C+(1-\gamma)\ln(L)$$

s.t.

C=wN

$$0 \leq N \leq 1$$

We will take care of the second constraint by checking whether it is satisfied ``ex-post''. Substituting the first constraint into the objective function we obtain:

$$max_{\{L\}} \gamma w(1-L)+(1-\gamma)\ln(L)$$

and the FOC gives:

$$-\gamma w+(1-\gamma)\frac1L=0$$

or (with $\gamma=\frac12$)

$$w=\frac1L.$$

The labor supply curve is therefore:

$$w=\frac1{1-N}$$

For w<1, the FOC would imply negative N: the agent would like to do negative work. But this violates the non-negativity constraint. So for w<1 we have N=0. For $w\geq 1$ the labor supply curve is increasing, and has an asymptote for N=1. See figure 1.
c) The equilibrium is w=2 and (from the labor supply curve) $N=\frac12$.
d) When utility over consumption and leisure is defined as follows:

$$U(C,L)=\left\{ \begin{array}{cc} -\infty \mbox{ for } C<\bar{... ...\ln(L) \mbox{ for } C\geq \bar{C} \end{array} \right. \mbox{. }$$

It means that the agent ``dies'' (has infinite negative utility) if she does not provide at least consumption $\bar{C}$ - say because she has to maintain her family. The agent's problem is:

$$max_{\{C,L\}} \left\{ \begin{array}{cc} -\infty \mbox{ for } ... ...(1-\gamma)\ln(L) \mbox{ for } C\geq \bar{C} \end{array} \right.$$

s.t.

C=wN

$$0 \leq N \leq 1$$

The first order condition for the problem is the same as before:

$$w=\frac1{1-N}$$

Of course, this first order condition holds only if the implied N is enough that $C=wN \geq \bar{C}$. This other condition implies:

$$N \geq \frac{\frac12}{w}.$$

So for $w<\frac12$ the labor supply curve is not defined: even if the agent works all day (N=1) she cannot reach the ``survival'' amount of consumption. For $\frac12<w \leq \frac32$ the curve $N \geq \frac{\frac12}{w}$ is binding. So the agent works all she can to meet her consumption needs: for any given wage $N = \frac{\frac12}{w}$. At $w=\frac32$ the two curves ( $N = \frac{\frac12}{w}$ and $w=\frac1{1-N}$ intersect): this means when wages reach that level the agent does not need to work all she can to survive, and can start considering the trade-off between leisure and consumption. For $w \geq \frac32$ the labor supply curve is the same one studied above. Since the equilibrium wage is $w=2>\frac32$, the equilibrium labor supply is as before, $N=\frac12$. See figure 2.