Peter Zhang

EC 101B MT1

These notes are for Economics 101B: Macroeconomics, based on Sanjay K Chugh’s Modern Macroeconomics. Here is a copy of the textbook. If you’re taking it on Monday, good luck!

Chapter 1: Microeconomics of Consumer Theory

Utility functions track the relative well-being that consumers receive from consuming various goods. Indifference curves are sets of goods that possess the same utility. When indifference curves are convex, they experience diminishing marginal utility. The slope of the curve is the marginal rate of substitution. Consumers are limited by a budget constraint of the form \(P_1c_1 + P_2c_2 = Y\).

To maximize utility subject to the budget constraint, construct a Lagrangian of the form \(L(c_1, c_2, \lambda) = u(c_1, c_2) + \lambda[Y - P_1c_1 - P_2c_2]\). It’s equivalent to finding the bundle where the budget constraint is tangential to an indifference curve.

Chapter 3: Dynamic Consumption-Savings Framework

Actions like borrowing (or dissaving) relate to the intertemporal choices of individual. We can consider a model with two time periods and a simple utility function \((c_1, c_2) = \ln c_1 + \ln c_2\).

Income is the receipt of money over a time period, and the main sources are labor income and interest income. We can model a labor income of \(Y_1\) dollars and an interest \(i\) on initial wealth \(A_0\). Letting \(A_1\) denote savings, we have \(P_1c_1 + A_1 = (1+i)A_0 + Y_1\) for period one. Analogously, we have \(P_2c_2 + A_2 = (1+i)A_1 + Y_2\) for period 2, where \(A_2\) is assumed to be \(0\).

An individual’s private savings in a given time period is the difference between income and expenditures over that period. The private savings in period 1 is \(S_1^{priv} = iA_0 + Y_1 - P_1c_1\), and is a measure of change in wealth.

Wealth \(A_1\) links the two periods, and if consumers were rational and had perfect foresight, then an individual in the two-period model decides on consumptions and savings for both periods at the beginning of period 1. We can combine the two budget constraints by rearranging the second budget constraint to get \(A_1 = \frac{P_2c_2}{1+i} - \frac{Y_2}{1+i}\) which we can substitute back in to yield \(P_1c_1 + \frac{P_2c_2}{1+i} = Y_1 + \frac{Y_2}{1+i} + (1+i)A_0\) which is the lifetime budget constraint (LBC). The right hand represents the present discounted value of lifetime resources, and the left hand is the present discounted value of lifetime consumption.

Graphing \(c_1\) against \(c_2\), we get a LBC slope of \(\frac{-P_1(1+i)}{P_2}\), which makes sense since relative prices are adjusted for interest rates. Graphically, you will see that consumption above the labor income in one period must be balanced out by consumption lower than income in another. Moreover, interest income is ignored because wealth is simply used to transfer resources over time.

Similarly, utility is maximized when the budget constraint is tangential to the utility curve.

In a stock and flow model, stock variables are measurements that occur at a particular moment in time while flow variables occur over the course of an interval.

Chapter 4. Inflation and Interest Rates

Inflation is the rise in price overtime and is defined by \(\pi_t = \frac{P_t - P_{t-1}}{P_{t-1}}\). Deflation is a period where \(\pi < 0\) and disinflation happens when \(pi\) is decreasing. Under inflation, a dollar next year has less purchasing power then it has right now. The real interest rate measures this by using an index of goods: the nominal price may be increasing, but goods themselves may be getting cheaper.

The Fisher equation describes the relationship between nominal interest rate \(i\), real interest rate \(r\), and inflation as \((1+i_t) = (1+r_t)(1+\pi_t)\). Use a logarithm approximation trick, \(r = i - \pi\) for small values. Although transactions usually refer to nominal interest rates, real interest rates drive macroeconomic trends

If we take the nominal LBC \(P_1c_1 + \frac{P_2c_2}{1+i} = Y_1 + \frac{Y_2}{1+i} + (1+i)A_0\) and divide by \(P_1\) we get \(c_1 + \frac{P_2}{P_1} \cdot \frac{c_2}{1+i} = y_1 + y_2 \cdot \frac{P_2}{P_1} \cdot \frac{1}{1+i} + (1+i) \cdot \frac{A_0}{P_1}\). Replacing \(\frac{P_1}{P_2}\) with inflation and defining real net wealth \(a_0 = \frac{A_0}{P_0}\), we apply the Fisher equation to yield \(c_1 + \frac{c_2}{1+r} = y_1 + \frac{y_2}{1+r} + (1+r)a_0\) so we have real consumption and real net wealth. Converting to slope intercept form yields \(c_2 = -(1+r)c_1 + (1+r)y_1 + y_2\) and we can also calculate real private savings as \(s_1^{priv} = ra_0 + y_1 - c_1\).

Notice that the slope equation suggests that an increase in the real interest rate increases consumption of future goods and private savings. But, depending on the shape of the indifference curve, a rising interest rate can either increase or decrease private savings. Empirically studies confirm that the actual relationship between the two is weak.

Now, returning to the two-point model, we can build a Lagrangian model \(u(c_1, c_2) + \lambda[Y_1 + \frac{Y_2}{1+i} - P_1c_1 - \frac{P_2c_2}{1+i}]\). Solving, we find that \(\frac{du/dc_1}{du/duc_2} = 1+r\), which is the consumption-savings optimality condition. We could also take a sequential approach and reach the same condition.

Consumption smoothing is an underlying theme of multi-period utility maximization. Since lifetime utility is strictly increasing and concave, individuals prefer consumption to be relatively stable. Moreover, consumption can be similar despite different income streams because of the ability to borrow.

Not in book: The Euler equation relates the MRS between consumption in different time periods to the real interest rate: \(\frac{u_1(c_1^*, c_2^*)}{u_2(c_1^*, c_2^*)} = 1+r\)

Chapter 6. Firms

The firm is a price-taker in goods, labor, and capital markets. Its profit maximizing decisions occur at the start of period 1 and apply for both periods. It must maximize a dynamic profit function. In each period, a firm uses labor and capital to produce final goods.

Capital goods are physical goods like machines, factories, and computers that are used in the production of others goods. They are stock quantities, so they take time to build up. As a result, purchased capital in period 1 will not be ready until period 2. The time-to-build distinguishes capital and labor.

The production function \(y = f(k, n)\) describes the output of labor \(n\) and capital \(k\). We assume strictly increasing production in both inputs and a decreasing rate of transformation from each input into output. The result is a positive marginal product and diminishing marginal product for each input. The life time profit function is: \(P_1f(k_1, n_1) - P_1(k_2-k_1) - P_1w_1n_1 + \frac{P_2f(k_2, n_2)}{1+i} - \frac{P_2(k_3 - k_2)}{1+i} - \frac{P_2w_2n_2}{1+i}\) where \(P_1\) and \(P_2\) correspond to final good prices, \(w_1\) and \(w_2\) correspond to wages, \(i\) is the nominal interest rate, and \(k_1, k_2, k_3\) are the accumulated capital.

Using the first order conditions, we derive that \(w_1 = f_n(k_1, n_1)\) and \(w_2 = f_n(k_2, n_2)\), which means that firms optimize labor such that the marginal product of labor is equal to the market wage. As a result, the marginal product of labor and optimal wage strictly decrease as \(n\) increases.

Net investment is \(k_{n+1} - k_{n}\), the change in capital between two periods. Economic depreciation is the wearing out of capital goods during production (and roughly 8% depreciates each year, so \(\delta = 8\)). Gross investment is \(inv^{gross} = inv + \delta k\) and is used to measure GDP. However, the depreciation rate is constant over time and so qualitative analyses can ignore depreciation.

Although it would be more realistic to have a separate price for capital goods, they can often overlap with final goods and the real price is the real interest rate. A firm decides how much capital it would like to have in the future. The first order condition is \(r = f_k(k_2, n_2) = mpk\). Regardless of whether the firm is borrowing or paying out of hand, short-term spending on capital is decided by the real interest rate. The relationship between \(r\) and investment is the market capital investment function.

The intertemporal price r is the most relevant for investment. Changes in \(P_1\) directly change \(r\) if \(P_2\) and \(i\) are constant.

Chapter 25. Solow Growth Framework

Real GDP per capita is the main metric used to judge standards of living. Convergence suggests that over time, per-capita GDP should equalize over countries, although the empirical evidence is mixed. The Solow framework explains growth by using exogenous factors to model the transformation of inputs into outputs. After de-trending the model, we compute long-run capital stock.

Long-run growth factors are exogenous to the economy. The first main factor is population growth \(gr_{N} = \frac{N_{t+1} - N_t}{N_t}\). The second main factor is productivity, which is considered labor-augmenting, denoted by \(gr_X = \frac{X_{t+1}-X_t}{X_t}\).

Cobbs-Douglas production function is \(Y_t = K_t^\alpha (X_tN_t)^{1-\alpha}\). If \(\alpha\) is close to zero, an economy is labor intensive. If \(\alpha\) is close to one, an economy is capital intensive.

The Solow model is a closed-economy structure, meaning that the GDP account equation is a resource frontier. Government spending is ignored so that the resource constraint is \(Y_t = C_t + I_t\), while savings is the source of investment \(S_t = I_t\). The framework dictates that aggregate saving is a constant fraction \(s\) of GDP so that \(S_t = s \cdot Y_t\). Aggregate gross investment is \(I_t = K_{t+1} - (1-\delta) K_t\), where \(\delta\) is the rate of depreciation.

Notice \(I_t = sY_t\) and substitute to get \(K_{t+1} - (1-\delta)K_t = sY_t\). Scale by denoting per unit of effective labor variables \(y_t\) and \(k_t\). We then write \(y_t = k_t^\alpha\), suggesting that production has diminishing marginal product in capital. The equilibrium law of motion for the (per-capita) capital stock is:

\[k_{t+1} = \frac{s\cdot k_t^\alpha}{(1+gr_X)\cdot (1+gr_N)} + \frac{(1-\delta)k_t}{(1+gr_X)(1+gr_N)}\]

suggesting that capital is completely determined by previous capital. So, we can rewrite: \(k^* = [\frac{s}{(1+gr_X)(1+gr_N)-(1-\delta)}]^{1/(1-\alpha)}\)

In the long term, the economy moves towards the steady state.

The Solow model predicts that countries converge to steady states, but not all economies have the same per-capita capital stocks. It also suggests an eventual zero growth state, but the empirical consensus is that advanced economies have not stopped growing.

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