# EC 101B Final

### Introduction

These notes are for Economics 101B: Macroeconomics, based on Sanjay K Chugh’s Modern Macroeconomics, which is here. Notes on the midterm are here.

#### Chapter 26: Neoclassical Growth

While the Solow growth framework assumes a constant savings rate as an exogenous parameter, the neoclassical growth framework determines the savings rate endogenously through a consumption-savings optimality condition, maximizing lifetime utility $$\max \sum^\infty_{t=1} \beta^{t-1} u(c_t)$$ subject to $$c_t + k_{t+1} - (1-\delta)k_t = k_t^\alpha$$

Solving the Lagrangian yields the optimality condition $$\frac{u'(c_t)}{\beta u'(c_{t+1})} = 1 + \alpha k^{\alpha -1}_{t+1} - \delta$$ where the left hand side is the marginal cross-period rate of substitution and the right hand side is the gross real return on physical capital.

We reach equilibrium whenever aggregate savings equals aggregate investment, with steady state expression $$\frac{1}{\beta} = 1 + \alpha(k^*)^{\alpha -1}-\delta$$ yielding $$k^*_{\text{neo}} = [\frac{1}{\alpha} (\frac{1}{\beta} - (1-\delta))]^{1/(\alpha -1)}$$ which contains the discount factor. We can also rearrange to yield $$c^*_{\text{neo}} = [\frac{1}{\alpha} (\frac{1}{\beta} - (1-\delta))]^{\alpha/(\alpha -1)} + \delta [\frac{1}{\alpha} (\frac{1}{\beta} - (1-\delta))]^{1/(\alpha -1)}$$ and a long-run savings rate of $$s = \frac{\delta k^*}{y} = \delta (k^*)^{1-\alpha}$$ The Golden Rule (maximizing long run consumption) yields a higher savings rate than the neoclassical model, which maximizes lifetime utility.

In the long term, aggregate economic activity tends to settle down. Although business cycles induce ups and downs, the average is the long run. The consumption-savings optimality condition at the heart of macro is $$\frac{1}{\beta} = 1+r$$ relating the subjective discount factor to the real interest rate. The modern view is that impatience induced a positive real interest rate.

Side note: Examples of TFP includes technology, education, misallocation, management institutions, financial development, culture, political stability, regulations, and more!

#### Chapter 6: Dynamic Firms

A firm makes profit-maximizing decisions at the beginning across all period, using labor and capital goods to produce final goods. Capital goods include machines, factories, computers, trucks, and more; crucially, it is a accumulation (stock) quantity, so it takes time to build up. If we invest in capital at the beginning of period 1, then the capital will not be ready until period 2. Production is subject to a positive, diminishing marginal product with respect to each input.

The dynamic/lifetime/intertemporal profit function is $$\sum_t P_t f(k_t) - P_t(k_{t+1} - k_t) - P_t w_t n_t$$ representing the profit from production minus the costs of capital and labor.

The FOCs yield two insights. First, $$w_t = f_n(k_t, n_t)$$. Firms choose a quantity of labor such that the real wage is equal to the marginal product of labor. Second, $$f_k(k_t, n_t) = r$$. Firms choose an amount of future capital such that the real interest rate is equal to the marginal product of capital.

Natural experiments from post-WWII college education and the black death suggest that decreases in a factor increase marginal product and income.

Some relationships:

• Capital stock increases (e.g. from FDI) → capital/labor ratio increases → MLP increases → MPK decreases
• TFP increases (e.g. from innovation) → MPL increases & MPK increases
• Labor increases (e.g. from immigration) → capital/labor ratio decreases → MPK increases
• Experiments show that MPK may increase enough that wages don’t decrease

#### Chapter 7: Fiscal Policy

Fiscal gap is comprised of formal debt and debt-like liabilities (such as pensions, social security, etc.), minus government revenues. A fiscal surplus denotes positive government savings. A fiscal deficit denotes negative government savings.

Ricardian equivalence states that if the government spends along a fixed path, the timing of taxes does not affect national savings or the equilibrium real interest rate. Individuals anticipate an increase in future taxes, so they save the full tax cut, independent of their preferences.

The government can finance spending through taxes and debt. We have the budget constraints: $$g_1 + b_1 = (1+r) b_0 + t_1\\ g_2 + b_2 = (1+r) b_1 + t_2$$ which can combine and rearrange to yield the LBC $$g_1 + \frac{g_2}{1+r} = t_1 + \frac{t_2}{1+r} + (1+r)b_0$$ which is analogous to the consumer LBC. Combining the two yields the economy-wide resource frontier: $$c_1 + \frac{c_2}(1+r) = y_1 - g_1 + \frac{y_2 - g_2}{1+r} + (1+r)(a_0 + b_0)$$ and national savings $$s_{\text{nat}} = s_{\text{priv}} + s_{\text{govt}} = y - t -c + t - g = y +-c - g$$ which clarifies how private savings and government savings oppose one another. This serves as the basis for the Ricardian equivalence theorem.

In practice, tax rebates are often saved. Consumers are myopic, face borrowing constraints, and may ignore future generation. Ricardian equivalence is a theoretical benchmark.

#### Chapter 2: Labor Markets

Labor is relevant for household budget constraints and individual utility functions. For a household, we equate consumption plus investment to the sum of labor and capital income: $$GDP_t = C_t + A_T - A_{t-1} = iA_{t-1} + Y_t$$ Since assets are concentrated in a handful of households, the mean household receives more capital income than the median household. Asset pricing calculates the price of an asset based on the present value of future income streams. We can measure human capital by summing the present value of salaries.

A household’s labor supply decision involves education, work location, work hours, and more. The Mincer model presumes that wages scale linearly with the level of skill: $$W_t = p_t^E\cdot S_t$$ where an individual can grow their inherited skill $$S_0$$ at rate $$\alpha$$ while in school $$S_t = (1+\alpha)S_{t-1}$$ so that an individual stays in school until the marginal effect on lifetime income equals the opportunity cost of staying in school. Since $$Pc = Y$$ and $$n = 1-l$$, we have $$Pc + Wl = W$$. Taking the Lagrangian, we can conclude that the $$\text{MRS} = \frac{W}{P}$$.

#### Chapter 11: Labor Supply and Macroeconomic Institutions

Ricardian Equivalence says that Rational consumers understand that a tax cut today means a tax increase in the future so the entire tax cut is saved. It is a benchmark result, similar to perfect competition, relying on lump-sum taxes, where the total incidence doesn’t depend on individual choices.

A proportional tax or distortionary tax taxes based on consumption. If the relative tax on different goods changes, then sales tax distorts consumption between the two goods. The distortion can also prompt consumers to adopt untaxed choices like leisure. The Laffer curve is based on tradeoff between the tax rate and the tax base and is supported by cross-country evidence. But, very few ecnomists think cutting income tax can increase tax revenue.

#### Chapter 15: Money and Bonds

Money is neutral if changes in money supply do not effect the real economy; this is the Classical view. The Keynesian view is that money is non-neutral because prices are sticky.

Money is a medium of exchange, a unit of account, and a store of value. We use the money-in-the-utility-function formulation $$u(c_t, \frac{M^D_t}{P_t})$$ where $$P_t$$ is measured in \$ per consumption.

A bond is a debt obligation over some period that is repaid with interest. The price of a bond has an inverse relationship with the nominal interest rate so that $$P_t^b = \frac{1}{1 + i_t}$$ where $$i$$ is the price of money, or the opportunity cost of holding money. Expansionary monetary policy purchases short bonds by printing new money, which decreases short-term $$i$$. Contractionary monetary policy involves selling short bonds in exchange for money, which increases short-term $$i$$.

In the new framework, we introduce the asset classes of stocks, short-term bonds, and money so that $$P_tc_t + P_t^b B_T + M_t + S_t a_t = Y_t + M_{t-1} + B_{t-1} + S_ta_{t-1} + D_ta_{t-1}$$ which can be used to craft an intertemporal maximization problem. Computing the FOC with respect the price of stock $$a_t$$, we arrive at the stock-pricing equation $$S_t = (\frac{\beta \lambda_{t+1}}{\lambda_t})(S_{t+1} + D_{t+1})$$ Computing the FOC with respect to bond prices yields the bond-pricing equation $$P_t^b = \frac{\beta \lambda_{t+1}}{\lambda_t}$$ Bonds are a riskless asset while stocks are a risky asset. Real money demand is described by $$\frac{M_t}{P_t} = c_t \cdot (\frac{1+i_t}{i_t})$$ Support consumers plan for some level of $$M_t$$. After a money shock, a Keynesian view suggests prices cannot adjust, so consumption must rise. In the long run, the rate of money growth is equal to the rate of inflation so that $$\mu = \pi$$.

#### Chapter 16: Fiscal and Monetary Interactions

The fiscal authorities budget constraint is $$P_tg_t + B_{t-1}^T = T_t + P_t^B B_t^T + \text{RCB}$$ where $$B_t$$ is the total amount that Congress sells in period $$t$$ and RCB is the profits turned over from the central bank. Since 2007, government assets has expanded to include government sponsored enterprises (GSEs) and mortgage backed securities (MBS). The consolidated flow constraint highlights the fiscal ($$g_t, T_t, B_t$$) and monetary ($$M_t$$) tools available to the government. $$P_tg_t + B_{t-1} = T_t + P_T^B B_t + M_t - M_{t-1}$$ An authority is active if every instrument can be freely chosen without concern for the budget constraint; otherwise, it is passive. One of the authorities (fiscal or monetary) has to ensure the GBC holds (usually monetary). Printing money yields seignorage revenue, an important source of revenue in developing countries. The lifetime consolidated GBC is that the real value of government debt must equal the present discounted value of all fiscal surpluses and seignorage revenues.

A Ricardian fiscal policy has no impact on monetary policy. Otherwise, if the authority changes the timing of collection, the adjustment must induce a change in price. Developing countries often experience FTPL, a sudden rise in inflation, while developed countries experience FTI, a long and sustained rise. Inflation and seignorage tend to be lower in countries with greater central bank independence.

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