ECON 8040 Final exam Lastrapes Fall 2007 Answer all eight questions on this exam. 1. Write out a static model of the macroeconomy that is capable of predicting that money is non-neutral. Your model should have three endogenous variables: the price level (P ), real output (y), and the interest rate (R). Describe each of the equations in the model, prove that it implies that money is non-neutral, and explain why money is non-neutral. 2. Suppose that the macro economy can be described by the following linear dynamic model, where all variables are in logs and are defined as in class, and all parameters are positive: y t = ỹ + a 1 (p t p t ) y t = d 0 + d 1 (m t p t ). Here, p t represents agents expectations of the price level, conditional on information at time t 1. a. Assume that p t = p t 1. How will a permanent increase in the stock of money affect real output in the short-run and the long-run? Explain. b. Assume that p t = E t 1 p t, the mathematical expectation of the price level, conditional on all information up to time t 1. How will a permanent increase in the stock of money affect real output in the short-run and the long-run? Explain. 3. Assume that all markets clear, agents respond to relative prices, and expectations are formed rationally in the economy. Markets, however, are separated, so that information about aggregate nominal shocks and relative price shocks is incomplete when decisions are made, even when local prices are observed. How might an aggregate shock to the money supply affect the price level and real aggregate production in such an economy, and why? As local prices become more informative about aggregate nominal shocks, what will happen to the response of output and prices to money supply shocks? 4. Consider a household s problem of optimal intertemporal consumption, where interest rates and income are known and exogenous. Suppose that the government sector imposes a proportional tax on the interest income earned by the household (i.e. the net return from owning capital); the tax rate is ρ. Write the household s budget constraint that includes the effects of this tax, and fully describe the effects on the optimal consumption path of an increase in ρ. 5. Write a short essay on the Ricardian Equivalence theory. In particular, state the basic predictions of this theory and the conditions under which the predictions will hold. 1
6. Suppose long-run growth in an economy is characterized by Solow s neoclassical model, where the labor force grows at rate n and technology grows at rate g. Assume that a quick wave of immigration causes the economy s labor force to jump at a point in time (while the steady state population growth remains constant at n). Explain the effects of this immigration wave on both the level and growth rates of capital per effective worker and output per effective worker, in the short run and the long run. Use a graph if it aids your explanation. 7. A central planner chooses consumption (c) and capital (k) paths to maximize t=0 ( ) c β t 1 θ 1 θ subject to k α t = k t+1 (1 δ)k t + c t, where the initial level of capital is given, capital can never be negative and β and α take on values between 0 and 1. a. Derive the Euler equation for this optimization problem. b. Use a phase diagram to describe the dynamics of the economy if the capital stock begins at a level higher than its steady-state value. 8. In the AK model of optimal (endogenous) growth, we derived the following steady-state relationship: r = i + θg, where r is the market interest rate, i is the rate of time preference, 1 θ is the elasticity of intertemporal substitution, and g is the growth rate of consumption. a. Explain where this equation comes from and what it means in the context of the model. b. What happens to steady-state growth in this model if both i and the (constant) marginal product of capital fall. Explain. 2
Guide to answers 1. Here is the simplest possible model that works: y = y(p), y (p) > 0 y = D(r), D (r) < 0 M/p = m(r, y), m r < 0, m y > 0 where y is output, p is the price level and r is the interest rate. M is exogenous money supply. The first equation is aggregate supply, reflecting the effect of the price level on production. The positive derivative of the supply function presumes some sort of rigidity that yields the upward sloping supply. The second equation (the IS curve) shows that interest rates have a negative effect on aggregate demand, presumably through consumption and investment. The third equation is money market equilibrium: real money supply equals real money demand. Money is non-neutral if changes in its quantity have real effects, on output in this model. It is easy to show, given the constraints on the slopes, that y M > 0. As money increases, say, the price level tends to rise, providing incentives for increased production through the aggregate supply curve (e.g. nominal wages might be exogenous so real wage falls as price rises). As output rises, the interest rate will fall to ensure that the demand for output rises to accommodate the increased supply. Money works through a liquidity effect. 2. Phillips curve dynamics. The model assumes that households are imperfectly informed about the price level, so expected price shows up in the supply curve. a. This is the case of a very simple version of adaptive expectations. It is easy to show that the equilibrium price level is a first-order difference equation. Solving this difference equation and substituting into the demand curve yields the reduced form for equilibrium output, which shows that output depends on current and lagged values of the nominal money stock. The dynamic multipliers show that output rises in the short run as money rises, but gradually falls to its initial level in the long-run. In the long-run p = p. b. This is the case of Rational Expectations. In this case, the nature of the change in money matters. In particular, if the change in money is anticipated, then there will be no effect on output in the short-run or the long-run. If, however, the change is unanticipated, output will be positively related to money in the short-run. Since there are no other dynamics in the model, this real effect is short-lived: output will return to its original value after only one period, since the change is unanticipated for only one period. 3. The conditions described are consistent with Lucas s theory of the business cycle. Suppose that the monetary authority increases the money supply uniformly throughout the economy, but this action is not known by individual producers when their production decisions are made. Because money has risen everywhere, each producer will observe the price of his or her good rising. However, each producer believes, given the history of prior shocks, that, at least to some extent, the rising price reflects an increase in his or her relative price. Therefore, the 3
representative producer has an incentive to increase output in response to this (perceived) relative price increase. Because all producers are similarly myopic, the aggregate shock to the money supply results in an increase in aggregate output (and a smaller increase in the price level than would have been if the shock had been perceived. Lucas s model assumes that producers know the probability distribution of shocks, and use this knowledge to rationally predict the price level from past forecasts and currently observed prices. As the variance of aggregate shocks increases, local prices will better reflect the source of shocks to the market; producers will place more weight on local price in predicting the price level. Thus, since producers are less fooled when an aggregate shock comes along, they will respond by increasing output less in the face of a positive aggregate shock. The aggregate supply curve becomes steeper. 4. If there is a tax on interest income, the households budget constraint becomes A t+1 = (A t + y t c t )[1 + (1 ρ)r] where A are financial assets, y is labor income, c is consumption, r is the interest rate on assets, and ρ is the tax rate. If we define the after tax interest rate as r = (1 ρ)r, then the problem is identical to the one discussed in class, except we replace the interest rate with the corresponding after tax rate. Thus, an increase in ρ ceteris paribus in effect lowers the return to saving, causing the budget line to rotate counterclockwise around the no borrowingno lending point. The effects of this change on consumption paths depends on whether the household is originally a net lender or borrower. If a lender, then future consumption will fall unambiguously, while current consumption will rise or fall depending on the relative strength of the income and substitution effects. If a borrower, then current consumption unambiguously rises (the rise in the tax rate lowers the effective interest rate paid on loans) because both the income and substitution effects enforce each other, while the effect of future consumption is ambiguous. 5. Ricardian equivalence is based on the assumption that both households and government must satisfy their respective budget constraints in the sense ultimately having to pay back debt with future income or future tax revenues. Given other conditions, this assumption implies that lump sum taxes and government debt are equivalent in the following sense: suppose that taxes rise (and the government debt falls), holding the present value of government spending constant (an important assumption). Households will respond to this increase by reducing saving dollar for dollar to offset the reduction in debt, knowing that future taxes will fall. Households are thus able to maintain their original consumption path. In other words, fiscal policy that changes the timing of taxes and thus debt, has no effect on household decisions. Other assumptions that must be satisfied: infinite lifetime or bequest motive, perfect capital markets, lump sum taxes, no overlapping generations. 6. The immigration wave causes the capital per effective worker k = K/AL to fall immediately (it jumps discretely to the left). Because nothing else changes (recall the assumption that labor continues to grow at rate n), the new level of capital per effective worker is smaller in the 4
short-run than its steady-state value. Therefore, the rate of capital accumulation exceeds the rate at which it is deteriorating (due to depreciation, labor growth and technology growth). The capital stock will thus rise in the short-run. Eventually, given diminishing returns to capital, capital per effective worker will approach its original steady-state value. Output per effective worker will follow the same pattern as capital per effective worker. But note that in the new steady-state, even though k and y are the same as before the immigration wave, both the overall capital stock and output will have grown proportionally to the rise in the labor force. 7. Optimal growth. Λ = β t c1 θ t 1 θ + λ t [kt α k t+1 + (1 δ)k t c t ] Take derivatives with respect to c t and k t+1 and eliminate λ to get β( c t ) θ = [αkt+1 α 1 + (1 δ)] 1 c t+1 If the capital stock is too big relative to its steady-state, then consumption must jump discretely to the saddlepath, since this path is the only one that satisfies the Euler equation above and the transversality condition. Because this point on the saddlepath must lie above the constantcapital locus and to the right of the constant-consumption locus, the capital stock will begin to fall over time, and consumption will gradually fall as well. Ultimately, the system will approach its steady-state where both the capital stock and consumption are lower than the initial values. 8. AK model. a. This equation is simply the Euler equation from the representative agents dynamic optimization problem, in which the agent maximizes lifetime utility subject to his or her capital accumulation constraint and the production technology (think of this problem as one of a central planner). The distinguishing feature of the model is that the marginal product of capital does not diminish; there are constant returns to capital. g then becomes the endogenous growth rate of consumption that satisfies the tangency condition implied by the Euler equation. That is, consumption must grow at rate g so that the marginal benefit of increasing current consumption just equals the discounted marginal cost of decreasing future consumption, given technology and preferences. b. A decrease in i tends to raise steady-state growth (more patience allows capital to accumulate faster), while a decline in the marginal product of capital decreases growth (this reduces the return to saving and thus accumulating capital). Thus, the effect on growth will depend on the relative magnitudes of the change in each parameter. 5