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Intermediate Macroeconomics Unit 2: The Classical Model and Loanable Funds
The classical model explains what determines output, how income is divided between workers and capital owners, and how the real interest rate balances saving and investment in the long run.
Covers: Production function and returns to scale, Factor prices and marginal products, Cobb-Douglas and factor shares, Consumption, investment and government purchases, Loanable funds equilibrium, Fiscal policy and crowding out
Study guide
The classical model describes a closed economy in the long run, when prices are flexible and factors are fully employed. Output is produced by a production function Y = F(K, L) using fixed amounts of capital and labor. Most models assume constant returns to scale: multiplying both inputs by z multiplies output by z. Each input also has a diminishing marginal product: adding workers to a fixed capital stock raises output, but by less with each worker. Because K, L and technology are given, output is fixed at Y = F(K, L), no matter what happens to demand.
Competitive firms hire each factor until its marginal product equals its real price. The real wage W/P equals the marginal product of labor, and the real rental price of capital R/P equals the marginal product of capital. With constant returns, paying each factor its marginal product exactly uses up output, leaving zero economic profit (Euler's theorem). The Cobb-Douglas function Y = A K^α L^(1−α) matches U.S. data well. Its marginal products are MPK = αY/K and MPL = (1 − α)Y/L, so capital receives the share α of income and labor receives 1 − α, regardless of how much of each factor there is. A rise in productivity A raises both factor prices.
Output is used for consumption, investment and government purchases: Y = C + I + G. Consumption depends on disposable income, C = C(Y − T), with a marginal propensity to consume between 0 and 1. Investment depends negatively on the real interest rate, I = I(r), because a higher r raises the cost of financing projects. Government purchases and taxes are set by policy. Rearranging, national saving S = Y − C − G must equal investment. Private saving is Y − T − C, public saving is T − G, and their sum is national saving.
The loanable funds market puts these pieces together. With Y, T and G given, national saving is a fixed amount (a vertical line), and the downward-sloping investment curve meets it at the equilibrium real interest rate. Higher government purchases or a tax cut lowers national saving and raises r, which crowds out investment one for one when saving does not depend on r. A shift in investment demand, such as an investment tax credit, raises r but cannot raise equilibrium investment, because investment is limited by saving. If saving rises with the interest rate, crowding out is partial, but the direction of every effect is the same.
The model's main lesson is that, in the long run, demand shifts change the composition of output rather than its level. A consumer spending boom, a military build-up or a tax cut all raise the real interest rate and reduce investment, with output unchanged. Only changes in factor supplies or technology change Y. This sets up the contrast with short-run models later in the course, where sticky prices let demand affect output.
Key ideas
- Output is fixed by capital, labor and technology in the classical long run.
- Each factor's real price equals its marginal product.
- Cobb-Douglas production gives constant factor shares: α for capital and 1 − α for labor.
- National saving S = Y − C − G equals investment in equilibrium.
- The real interest rate clears the loanable funds market.
- Higher G or lower T lowers national saving, raises r and crowds out investment.
- Shifts in investment demand change r, not equilibrium investment, when saving is fixed.
Formulas and rules
Common mistakes
- Applying a Keynesian multiplier to the classical model, where output cannot change with demand.
- Confusing constant returns to scale with constant marginal product.
- Mixing up α (capital's share) and 1 − α (labor's share) in Cobb-Douglas.
- Thinking an investment tax credit raises equilibrium investment when saving is fixed.
- Forgetting that a tax cut lowers national saving by only MPC × ΔT, since part of it is saved.
Before the test, make sure you can
- Explain why output is fixed in the classical model.
- Calculate MPL, MPK and factor incomes from a Cobb-Douglas production function.
- Predict how immigration, capital destruction or technology changes affect factor prices.
- Calculate private, public and national saving.
- Solve for the equilibrium real interest rate and investment in the loanable funds model.
- Analyze the effects of changes in G, T and investment demand on r, I and C.
- Explain why crowding out is complete with fixed saving and partial with interest-sensitive saving.
Words to know
- Production function
- Y = F(K, L): the relationship that turns capital and labor into output, given technology.
- Constant returns to scale
- Scaling all inputs by z scales output by z.
- Diminishing marginal product
- Each extra unit of one input adds less output when other inputs are fixed.
- Marginal product of labor (MPL)
- Extra output from one more unit of labor; equals the real wage in competitive markets.
- Marginal product of capital (MPK)
- Extra output from one more unit of capital; equals the real rental price of capital.
- Real rental price of capital
- R/P, the cost of renting a unit of capital measured in output.
- Euler's theorem
- With constant returns, paying each factor its marginal product exactly exhausts output.
- Economic profit
- Revenue left after paying all factors; zero in the competitive classical model.
- Cobb-Douglas function
- Y = A K^α L^(1−α); factor income shares equal the exponents.
- Total factor productivity
- The A term in the production function; raises output for given inputs.
- Disposable income
- Income after taxes, Y − T.
- Marginal propensity to consume
- The share of an extra dollar of disposable income that is consumed.
- Private saving
- Y − T − C.
- Public saving
- T − G; negative when the government runs a deficit.
- National saving
- Y − C − G, which must equal investment in a closed economy.
- Loanable funds market
- The market where saving supplies funds and investment demands them, cleared by the real interest rate.
- Crowding out
- The reduction in private investment caused by higher government borrowing raising the interest rate.
Sample practice questions
30 more questions like these, from level 1 to 5, are in the practice set.
In the classical model with competitive firms, how much does each firm pay a worker in real terms?
- The marginal product of labor
- The average product of labor
- Whatever wage clears the market for goods
- Total output divided by the capital stock
Show the answer and steps
Answer: A) The marginal product of labor
- A competitive firm hires labor until the extra output from one more worker equals the real wage W/P.
- So in equilibrium the real wage equals MPL.
A production function Y = F(K, L) has constant returns to scale when:
- Each input's marginal product is constant
- Multiplying K and L by z multiplies output by z
- Adding one more worker always adds the same output
- Output stays the same when both inputs double
Show the answer and steps
Answer: B) Multiplying K and L by z multiplies output by z
- Constant returns means F(zK, zL) = zF(K, L) for any z > 0.
- It is a statement about scaling all inputs together, not about adding one input alone.
Classical model: Y = 5,000, C = 250 + 0.75(Y − T), G = T = 1,000, and I = 1,000 − 50r (r in percent). What is the equilibrium real interest rate?
- 2.5%
- 10%
- 5%
- 15%
Show the answer and steps
Answer: C) 5%
- C = 250 + 0.75 × 4,000 = 3,250.
- National saving S = Y − C − G = 5,000 − 3,250 − 1,000 = 750.
- Set I = S: 1,000 − 50r = 750, so r = 5%.