Macroeconomics · Lesson 9 of 9
The Delhi School problems
The DSE entrance asks the same topics as everyone else, but in general form: letters instead of numbers, two groups of spenders instead of one, and “which way does it move?” instead of “how much?”. Every tool here was built in Lessons 4 to 8; this lesson shows how to use them without numbers.
A question to hold on to
Half of all income goes to workers and half to capitalists. Capitalists save 50% of their income and workers 25%. Investment is 100 and output is demand-determined. If workers’ share rises to three-quarters, what happens to equilibrium output?
Delhi School of Economics entrance, 2006Solved at the end ↓
The approach
How these questions differ
Three habits solve most of them.
- Write the equilibrium condition first, in letters. Output equals planned spending; saving equals investment; the budget constraint holds with equality. Everything else is algebra.
- Put every leakage in the denominator. A multiplier is 1 over whatever share of an extra rupee of income is not re-spent at home.
- For “which way” questions, sign the derivative. Differentiate the solution, or ask which group gains and whether it spends more or less than average.
Part 1 · Interactive
Part 1: who spends matters
Lesson 5 showed that the balanced budget multiplier depends on who pays the tax. The Delhi School pushes the same idea into the multiplier itself: if two groups have different propensities to consume, the economy’s propensity is their weighted average, weighted by income shares.
Two groups
Group A gets λY and consumes c_A of it; group B gets (1 − λ)Y and consumes c_B; both have autonomous consumption c.
Preset to DSE 2006: capitalists (A) save half their income, workers (B) save a quarter, investment is 100 and income is split equally. Now give workers three-quarters of income: move λ to 0.25.
Equilibrium output
266.67
Multiplier 1 ÷ (1 − c_B + λ(c_B − c_A)) = 1 ÷ (1 − 0.75 + 0.5 × 0.25) = 2.6667.
A larger λ gives more income to the group that spends less: the multiplier and output fall.
DSE 2010 · four questions on one model
Fixed prices, demand-determined output, investment fixed at I*, no government. Groups A and B get λY and (1 − λ)Y and consume CA = c + cAYA, CB = c + cBYB, with cA < cB. Find the multiplier, then the effect on output of a rise in λ, in cA, and in cB.
- Multiplier.The formula above: 1/(1 − cB + λ(cB − cA)).
- A rise in λ.cB − cA > 0, so the denominator rises: income moves to the group that spends less, and output falls unambiguously.
- A rise in c_A or c_B.Either raises the weighted propensity λcA + (1 − λ)cB, so output rises unambiguously.
Answer
1/(1 − cB + λ(cB − cA)); output falls with λ and rises with either propensity. Whether λ is above or below a half never matters.
DSE 2010 · a transfer between groups
An economy below full employment has a worker and a capitalist; the worker has the higher MPC. Both pay income tax on the same linear schedule and the budget is balanced. A lump-sum transfer is made from the capitalist to the worker. What happens to the budget?
- Spending rises.The worker spends more of the transferred rupee than the capitalist would have: autonomous consumption rises.
- Income and taxes rise.Output rises by the multiplier, and with it tax revenue; government spending hasn’t changed.
Answer
The budget goes into surplus.
Part 1
Part 1: budgets, trade and the multiplier
DSE 2006 · public investment that pulls in private investment
A demand-constrained economy: households consume half their disposable income; the government keeps a balanced budget with lump-sum taxes; each unit of government investment induces 2 units of private investment. Government investment rises by 10. By how much does income rise?
- Changes in autonomous spending.ΔG = 10, induced ΔI = 20, and a tax rise of 10 cuts consumption by 0.5 × 10 = 5.
- Multiply.(10 + 20 − 5)/(1 − 0.5) = 50.
Answer
50 units: the balanced budget multiplier of 1 on the government’s own 10, plus 20 × 2 = 40 from the private investment it attracts.
DSE 2013 · two questions on one open economy
People save half their disposable income; taxes are τ = 2/5 of income; imports are m = 1/10 of income; I = 3,200, G = 4,000, X = 800. Find income. Then: what tax rate would make trade balance, with G and the exchange rate unchanged?
- Income.Y = (I + G + X)/(1 − c(1 − τ) + m) = 8,000/(1 − 0.5 × 0.6 + 0.1) = 8,000/0.8 = 10,000.
- Balanced trade needs M = X.0.1Y = 800, so Y must fall to 8,000.
- The tax rate that gives it.8,000 = 8,000/(1 − 0.5(1 − τ) + 0.1) needs 0.5(1 − τ) = 0.1, so τ = 4/5.
Answer
10,000; and raise the tax rate to 4/5. Cutting income through higher taxes (lower absorption) is the only lever left, and it takes a lot.
Two quicker ones from the same toolkit. DSE 2019: with MPS 0.3 and MPM 0.1, a rise of ₹10 billion in G raises GDP by 10/(0.3 + 0.1) = ₹25 billion. DSE 2008: with autonomous investment and a constant saving rate, a rise in the saving propensity lowers equilibrium output, the paradox of thrift from Lesson 4.
Part 2
Part 2: money and banks
DSE 2013 · standard at DSE
Banks keep a cash reserve ratio of 20%. People hold 25% of their money as currency and the rest as demand deposits. High-powered money rises by 2,000. By how much does the money supply rise?
- Convert the public's habit into cu.Currency is 25% of money and deposits 75%, so cu = 0.25/0.75 = 1/3. The trap is to use 0.25.
- Multiplier.(1 + 1/3)/(1/3 + 0.2) = (4/3)/(8/15) = 2.5.
Answer
5,000. With cu = 0.25 by mistake, the multiplier would be 1.25/0.45 ≈ 2.78 and the answer about 5,556, which would send you to “none of the above”: the question checks that you convert.
DSE 2005 asked for the signs: the money multiplier (1 + cu)/(cu + rr) is decreasing in the reserve–deposit ratio and decreasing in the cash–deposit ratio (because rr < 1, raising cu raises the denominator by more, proportionately, than the numerator).
DSE 2021: if the government finances its spending with taxes, the monetary base is unaffected on net. Tax payments move reserves out of banks into the government’s account at the central bank, and the spending moves them straight back.
The Baumol–Tobin square-root rule
Y: spending financed from money balances; b: cost of a trip to the bank; i: interest rate. The demand for money has income elasticity ½ and interest elasticity −½.
DSE 2009: if income rises by 10%, transactions demand for money rises by about 5%, since √1.1 ≈ 1.049. Economies of scale in cash management mean money demand grows more slowly than income.
Part 3
Part 3: choosing over time
The Euler equation, again
From Lesson 6. With log utility the exponent is 1.
DSE 2020 and 2019: with log utility and r > ρ, c₁ < c₂; with CRRA utility and r ≥ ρ, c₁ ≤ c₂. The ratio (1 + r)/(1 + ρ) is at least one and any positive power keeps it so.
DSE 2018 wrote utility as u(C) = σ/(σ − 1) · (C(σ−1)/σ − 1). This is CRRA with curvature 1/σ, so the intertemporal elasticity of substitution is σ itself. Read which parameter the question uses before quoting “1/σ”.
DSE 2009: a fall in the interest rate reduces saving unambiguously only for a borrower. For her the substitution effect (present consumption is cheaper) and the income effect (she pays less interest, so is better off) both raise c₁. For a lender the income effect goes the other way, so the net effect is ambiguous.
DSE 2005: under the life-cycle and permanent income hypotheses the MPC out of a permanent rise in income is close to one, and out of a temporary rise close to zero.
DSE 2013 · linear utility
An individual consumes rice in two periods, produces 1 unit in period 2 and nothing in period 1, and can borrow z in period 1 by repaying z(1 + r) in period 2. Utility is x₁ + βx₂ with 0 < β < 1. At what interest rate would she borrow exactly ½?
- Compare what a unit of borrowing gives and costs.It adds 1 to utility now and removes β(1 + r) later.
- Linear utility gives corners.If 1 > β(1 + r) she borrows all she can; if 1 < β(1 + r), nothing. An interior amount like ½ is chosen only when she is indifferent.
Answer
r = 1/β − 1.
A key we think is wrong
The follow-up (DSE 2013 Q33) gives N people with β spread uniformly on [½, 1] and asks for the demand for rice in period 1 when r ≤ 1. Those with β < 1/(1 + r) borrow, a fraction 2/(1 + r) − 1 = (1 − r)/(1 + r) of people, and each borrows the most she can repay, 1/(1 + r). Total demand is N(1 − r)/(1 + r)². The key, N(1 − r)/(1 + r), counts the borrowers but not how much each borrows. We have left it out of the practice set.
Investment. The desired capital stock rises with the marginal product of capital, and also with the expected future price of capital, because an expected capital gain lowers the user cost (DSE 2005). A higher depreciation rate raises the user cost and lowers investment (DSE 2009).
Expectations (DSE 2015). With C = αYe and I = γ(Ye)² and static expectations (Yet = Yt−1), output follows Yt = αYt−1 + γYt−1². Its positive steady state (1 − α)/γ is unstable: start above it and output grows without limit, start below and it shrinks to zero, which is the option keyed as correct. With rational expectations, output is always at (1 − α)/γ.
Solving the opening question
The opening question, solved
DSE 2006 · advanced
Workers save 25%, capitalists 50%, I = 100, and workers’ share of income rises from ½ to ¾. What happens to output?
- Average saving rate before.½ × 0.5 + ½ × 0.25 = 0.375, so Y = 100/0.375 = 800/3.
- After.¼ × 0.5 + ¾ × 0.25 = 0.3125, so Y = 100/0.3125 = 320.
- The change.320 − 800/3 = 160/3.
Answer
Output rises by 160/3 ≈ 53.3. Redistribution towards the group that saves less raises demand and output: the heart of Kalecki’s and the Cambridge economists’ demand-led models.
Watch out for
The mistakes examiners are fishing for
- Using the currency share of money as cu. cu is currency over deposits, not over money.
- Averaging propensities without weights. Weight each group’s propensity by its share of income.
- Assuming an interior solution with linear utility. Linear utility gives corners, except when the consumer is exactly indifferent.
- Quoting 1/σ as the elasticity of substitution by habit. Check how the question parameterised utility.
Revise
The lesson on one screen
The lesson on one screen
- Two groups
- Multiplier 1/(1 − c_B + λ(c_B − c_A)). Shifting income to the bigger spender raises output.
- Balanced budgets
- Add up every change in autonomous spending, including induced investment and the consumption cut from taxes.
- Balanced trade
- M = X pins income; solve for the tax rate that delivers it.
- Money
- cu = currency/deposits. Multiplier decreasing in cu and rr. Baumol–Tobin elasticity ½.
- Over time
- Euler: c₂/c₁ = ((1 + r)/(1 + ρ))^(1/σ). Borrowers save less when r falls; lenders are ambiguous.
- Investment
- Desired capital rises with MPK and expected capital gains; falls with depreciation.
Real questions
The Delhi School questions
Every DSE question above Easy on this module, grouped by technique, moderate before hard within each part.
Part 1 · from Lessons 4 and 5
Demand-led models
Multipliers with two classes of spender, balanced budgets that are not neutral, and taxes set to balance trade.
11 questionsPractise →
Part 2 · from Lesson 8
Money and banks
The money multiplier from the public's and the banks' ratios, and what moves the monetary base.
4 questionsPractise →
Part 3 · from Lessons 6 and 7
Choosing over time
Two-period consumers, the Euler equation, the desired capital stock, and expectations that fulfil themselves.
10 questionsPractise →
All 25 DSE questions, moderate before hard within each part: practise the full set →
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
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