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Intermediate Macroeconomics:
Notation and Equations
Eric Sims
University of Notre Dame
Fall 2014
1 Introduction
This handout provides a brief, rough, and incomplete review of what we’ve done this semester. I start by listing and defining variables, then parameters, then key equations, and then finally show a couple of graphs. Reviewing this sheet is not a substitute for going back through the course material. A quick word. The timing notation in this course is that period t is the “current” period and time runs discreetly forward from that. With the exception of the Solow model and a couple of places in the two period consumption model, we just focus on two periods (t + 1 is a stand-in for the “future”).
2 Variables
- Yt: output (also equal to income and expenditure)
- Y (^) td : expenditure (which must equal output in equilibrium)
- Ct: consumption
- It: investment
- Nt: labor input (total hours, or if you prefer, total employment... we do not model the extensive and intensive margins separately, so using any term is fine)
- Gt: government spending
- Mt: money supply
- rt: real interest rate (how many goods you receive in period t + 1 for saving one good in period t
- wt: real wage (how many goods a firm pays a worker in exchange for one unit of the worker’s labor effort
- Rt: real rental rate on capital in the Solow model; how many goods a firm must give up to lease one unit of capital for a period
- it: nominal interest rate
- Pt: price of goods in terms of money (price of money in terms of goods is therefore (^) P^1 t )
- πte+1: expected one period ahead inflation rate
- P (^) te : expected within period price level
- At: total factor productivity
- Zt: variable governing trend productivity growth in the Solow model
- q: investment-specific productivity (may also be interpreted as measuring health of financial sector)
- Y (^) tf : the flexible price level of output; the equilibrium level of output that would emerge in the RBC model. Sometimes also called the “natural rate.”
- kt: capital stock per worker in the Solow model (more generally, any lower case variable in the Solow model is per capita)
- ̂kt: capital stock per effective worker in the Solow model (more generally, any lower case variable with a “hat” in the Solow model is per effective/efficiency worker)
3 Parameters
- β: utility discount factor for households
- δ: depreciation rate on capital
- s: saving rate in the Solow model
- α: production function parameter for Cobb-Douglas production function; exponent on capital, corresponds to fraction of income paid out to capital
- γ: parameter governing the “slope” of the Phillips Curve. A measure of price stickiness; the closer γ is to zero, the stickier are prices; as γ → ∞, prices are flexible
- φ′
Mt Pt
= (^) 1+itit u′(Ct): optimality condition for choice of money holdings; implicitly defines a money demand curve
- 1 = (^) 1+^1 rt (qAt+1FK (Kt+1, Nt+1 + (1 − δ)): first order condition for optimal choice of future capital; implicitly defines investment demand
- Ct = C(Yt − Gt, Yt+1 − Gt+1, rt): consumption function; says consumption increasing function of current “perceived” net income, future “perceived” net income, and the real interest rate. Consumption is increasing in the first two arguments and decreasing in the real interest rate; the partial derivative with respect to the first argument, or the MPC, is between 0 and 1 because of households have a desire to smooth consumption. I write “perceived” with quotation marks because we assume that Ricardian equivalence holds, which has the implication that households behave as though the government balances its budget every period, whether it does so or not.
- Nt = N s(wt, rt): labor supply curve, says that labor supply is increasing in the real wage (sub- stitution effect dominates income effect) and increasing in the real interest rate (households want to save more when rt goes up, so they both consume less and work more).
- Nt = N d(wt, At, Kt): labor demand curve. Labor demand is decreasing in the real wage, increasing in At, and increasing in Kt
- It = Id(rt, At+1, q, Kt): investment demand curve. Says investment is decreasing in the real interest rate, increasing in future productivity, At+1, increasing in q, and decreasing in Kt.
- M Ptt = M d(it, Yt): money demand curve. Says demand for real balances is decreasing in the nominal interest rate and increasing in real output. Can be re-written using the approximate Fisher relationship as: M Ptt = M d(rt + πte+1, Yt). Can also be written by multiplying both sides by Pt to isolate Mt on the left hand side.
- Pt = P (^) te + γ(Yt − Y (^) tf ): Phillips Curve/aggregate supply relationship. Says that if Pt > P (^) te (prices higher than expected), then Yt > Y (^) tf (as long as γ isn’t ∞). Reflects the underlying idea that if prices are higher than expected, some firms end up with relative prices that are lower than desired, which means they have to produce more; hence aggregate output ends up higher.
5 Graphs
The subsections below show the main graphs from some different models with which we have worked. These are not complete and I do not re-do derivations.
5.1 Solow Model
𝑘𝑡
𝑘 0 *
𝑘 0 *
𝑘 (^) t+1 = 𝑘 (^) 𝑡
(1 + 𝑔^1 𝑧)(1 + 𝑔𝑛) 𝑠𝐴𝑘^ 𝑡α^ +^1 −^ 𝛿^ 𝑘^ 𝑡
𝑘t+
This plots capital per effective worker in period t + 1 against capital per effective worker in period t. The curve starts in the origin, is upward-sloping, and concave. Given our assumptions, it must cross a 45 degree line showing points where ̂kt+1 = ̂kt exactly once. This is what we call the steady state, which we denote with a ∗ superscript. In version of the model without growth, just set gz = gn = 0 and re-interpret ̂kt as Kt.
5.2 Two Period Consumption Model
C (^) t+
C (^) t
U=U 0
Ct+1^0
Ct^0
u’(Ct) βu’(Ct+1 ) = l + r^ t
This figure plots an indifference curve associated with a particular level of lifetime utility, U 0. The slope of the indifference curves if the ratio of the marginal utilities, − u ′(Ct) βu′(Ct+1).^ Given the assumed concavity of the utility function, this starts out steep and ends up flat. The straight line is a graphical representation of the intertemporal budget constraint which we call a budget line. The slope of the budget line is −(1 + rt). At an optimum you are on the highest possible indifference curve, a necessary condition for which is that the indifference curve and the budget line have the same slope.
5.4 Real Business Cycle Model
Wt
Wt^0
Yt
N (^) t^0 N (^) t
N (^) t Yt
Yt^0
45° Yt
Yt
Yt
rt^0
rt
Ytd
Yd^ (r)
Ys
Yd
Yt =AF(Kt ,N (^) t ) Yt =Yt
N d
N s
The above picture shows the graphs used to describe equilibrium in the RBC model. The Y d curve is functionally the same as in the endowment economy (and its derivation is similar); we just include investment in it now. The Y s^ curve shows the set of (rt, Yt) pairs (i) consistent with household and firm optimization, (ii) the labor market clearing, and (iii) the production function. Households and firms optimizing means that they are on their labor supply and demand curves, respectively (upper left graph). The labor market clearing means that these curves intersect. The production function is shown in the lower left graph; it is increasing in Nt (holding At and Kt fixed), but at a decreasing rate (the production function is concave). The 45 degree line in the lower rate is just a graphical tool to reflect the vertical axis onto a horizontal axis. The Y s^ curve is upward-sloping because higher rt leads households to supply more labor; this results in more Nt
when the labor market clears, which from the production function means more output.
Pt
Mt
Pt^0
Mt^0
Ms^ Md=PtMd(rt+πt+1e, Yt)
In the RBC model with flexible prices, we can determine nominal variables “after” real variables (e.g. the classical dichotomy holds). Money demand is upward-sloping in Pt (recall the price of money in terms of goods is (^) P^1 t , so it being upward-sloping in this graph is not “odd”); the money supply is exogenous. The intersection determines the price level.
5.5 New Keynesian Model
LM
IS
PC
AD
LRPC
rt
Yt
Yt
Pt
Yt^0 =Ytf
Pt^0 =Pte
rt^0
The IS curve is exactly the same as the Y d^ curve above. The LM curve shows the set of (rt, Yt) pairs where the money market is in equilibrium, taking Mt and Pt as given. The AD curve is the set of (Pt, Yt) pairs where we’re on both the IS and the LM curves. The PC curve is just the graphical
