• A note on how we used capital \(K\) earlier this semester...

• Assumed capital (essentially machines) has a market price \(“r”\), the “rental rate of capital”

  • Most firms purchase machines outright, rather than rent them per period (e.g. per hour)
  • But like any input, we consider the (opportunity) cost of using a marginal unit of an input as its market price if it were exchanged on the market
  • Hence, consider the price of capital the market rate to rent a machine for an hour

• Exact nature & definition remains controversial to economists to this day

• “Capital” is:

  • hard to define or (especially) aggregate
  • necessarily bound up with time and uncertainty

• Historically, the idea came from farmers

• During harvest time, can consume all produce today, or save some for next year

  • The more you save today, the less you can eat now, but the more you will have in the future
  • The more you consume today, the less you will have in the future

• Firms (and households) get money for investment today by participating in capital markets

• The funds in capital markets come from individual savings

• In discussing capital, we are comparing present goods with future goods

• Futures: claims on goods to be delivered at a future date

  • corn futures, oil futures, etc.

• Financial assets: bonds, lottery winnings, loans

• Real goods: immature orchard of fruit trees; durable goods that yield output later

• Interest rate is a price of future goods in terms of present goods

  • How much individuals will pay to receive income now vs. later

• Investment in capital: present consumption can be saved to buy/build machinery that can increase future income flows

• Consider goods-bundles consumed now vs. consumed at later date

  • i.e. not apples vs. oranges, but apples and oranges today vs. apples and oranges next year

• Agent's objective: optimize time-profile of consumption, maximize net present value

^† Or income, or consumption...

• Example: what is the present value of getting $1,000 one year from now at 5% interest?

$$\begin{align*} PV &= \frac{FV}{(1+r)^n}\\ PV &= \frac{1000}{(1+0.05)^1}\\ PV &= \frac{1000}{1.05}\\ PV &= \$952.38\\ \end{align*}$$

• Example: what is the future value of $1,000 lent for one year at 5% interest?

$$\begin{align*} FV &= PV(1+r)^n\\ FV &= 1000(1+0.05)^1\\ FV &= 1000(1.05)\\ FV &= \$1050\\ \end{align*}$$

• A good rule of thumb: number of years for your principal to double:

$$\frac{72}{r}$$

• This is known as the rule of 72¹

¹ Different people use other numbers, like 70. The point is more to make mental calculations easily rather than accurately.

• The Supply of Capital comes from individual decisions to save

• Sacing is considered a disutility (a bad)

  • Opportunity cost of saving is consumption
  • But, saving (and lending) can earn interest

• Tradeoff: if you save more, you consume less today, but can consume more in the future (with interest income)

• Apply our consumer choice model to “intertemporal” choice to consume: $$u(c_1,c_2)$$

  • \(c_1\): consumption today (period 1)
  • \(c_2\): consumption tomorrow (period 2)

• Define amount of saving as: $$s = M - c_1$$

  • where \(M_0\) is today’s income

• Most people follow a consistent “life cycle” of saving decisions

• People like to “smooth” their consumption over time, rather than experience sudden, discontinuous jumps in consumption level

  • When actual income \(<\) preferred consumption: borrow money
  • When actual income \(>\) preferred consumption: save (and lend) money

$$u(c_1,c_2)$$

• Marginal rate of (intertemporal) substitution: rate at which person gives up future consumption \((c_1)\) to obtain more present consumption \((c_0)\)
  • The slope of the indifference curve!

• Opportunity cost of consumption today \((c_0)\) is \(1+r\)

  • Forgo opportunity to save and invest to earn interest (and consume more) next period

• Let the price of future consumption be $1

  • Then the slope of budget constraint is

$$-\frac{p_{c_0}}{p_{c_1}}=-\frac{(1+r)}{1}=-(1+r)$$

• Consumer maximizes utility subject to budget constraint at \(A\): \((c_0^\star, c_1^\star)\)

• Consumer maximizes utility subject to budget constraint at \(A\): \((c_0^\star, c_1^\star)\)

• Consumes \(c_0^\star\) today, saving \(\color{#6A5ACD}{s = M_0 - c_0^\star}\) to consume \(\color{#e64173}{c_1^\star = s(1+r)}\) next period

• (Overall) Price effect: \(A \rightarrow C\)

  • Higher rate \(r\) leads to less consumption today \(c_0\) and therefore, more saving \(s\)

• Upward sloping savings supply curve

• Substitution effect: as interest rate \(r\) increases, the price of present consumption \(c_0\) is increasing, so consume less today

  • Thus, save more

• Graphically: under higher rate \(BC_2\), substitute more \(c_1\) for less \(c_0\) (more saving) holding utility constant

  • \(A \rightarrow B\): more \(c_1\), less \(c_1\) (more \(s)\)

• Real income effect: the higher interest rate makes you wealthier in real terms, so buy more of everything (including \(c_0\), meaning save less!)
  • \(B \rightarrow C\): attain higher indifference curve \(\color{green}{u_2}\)
  • “Inferior” good: higher interest rates induce more consumption today (and less saving)

• Income & substitution effects cut against each other

• If Substitution effect \(>\) Income effect, then we get a positive price effect:

  • Increase in interest rate causes more saving (less present consumption)

• Matches our intuition, upward-sloping savings supply curve

• If Income effect > Substitution effect, leading to a negative price effect:

  • Increase in interest rate causes less saving (more present consumption)
  • “Giffen-style” scenario, but plausible for saving! (unlike consumer goods)

• Intuition: imagine having an savings target (for rainy day, or retirement), and interest rates increase

• In general, an upward sloping market supply curve

• Giving up money today in exchange for claim on future repayment with interest

  • Individuals that loan their savings are called capitalists 🧐

• Individuals supply more (less) savings at higher (lower) interest rates

• As with labor, a Firm's Demand for Capital:

$$MRP_K=MP_K* MR(q)$$

• \(MRP_K\): marginal revenue product of capital
• \(MP_K\): marginal product of capital

• \(MR(q)\): marginal revenue

• Firms borrow money today in exchange for promising future repayment with interest

• Firms borrow more (less) funds at lower (higher) interest rates

• Note in general, firms are not the only borrowers of funds!

• Individuals borrow money to attain higher consumption than their current income

  • Mortgages, auto loans, student loans, etc.

• Governments also borrow money to attain higher spending levels than their current taxation

• Market Demand+ Demand from Firms + Demand from Individuals + Demand from Government

• Again, consider the “life cycle” of decisions

• People like to “smooth” their consumption over time, rather than experience sudden, discontinuous jumps in consumption level

  • When actual income \(<\) preferred consumption: borrow money
  • When actual income \(>\) preferred consumption: save (and lend) money

• Loanable funds market, where savers and borrowers exchange present & future money

• Equilibrium market interest rate \(r^\star\)

• An increase in Demand raises interest rate \(r\) and quantity of funds loaned/borrowed

• An increase in Supply lowers interest rate \(r\) and quantity of funds loaned/borrowed