ECON90034 Chap.2 Consumption and Investment Under Certainty
Consumption and Investment Under Certainty
Topic 3 shows how resources move between this year and next, by investing in real projects or by lending and borrowing. You build the investment frontier by ranking projects from the highest return down, draw the capital market line at the market interest rate, and find the point where they touch.
The Fisher separation theorem says that, with perfect capital markets, everyone with the same opportunities makes the same production choice, maximising the present value of wealth, and then uses the market to reach the consumption pattern they prefer.
The review lecture expects you to solve a problem like the tutorial: calculate returns, consumption, wealth, borrowing, repayment and the investment in real projects, and to use the Lagrange method where asked. The chapter works one complete example with four projects and two people whose preferences differ, then shows what a lower interest rate does to each.
What this chapter covers
- 01
Moving resources between periods through real investment or financial markets
- 02
Ranking projects by return and building the investment frontier
- 03
Production and consumption without capital markets
- 04
The capital market line and the present value of wealth
- 05
Tangency conditions: marginal rate of substitution equals one plus the interest rate
- 06
Borrowers and lenders, and what a change in the interest rate does to each
- 07
The Fisher separation theorem and why financial markets matter
- 08
The Lagrange method for optimal consumption
Worked example · free
Production, wealth and consumption with three projects
- 1P and Q beat 8%; R does not. Invest $3m, leaving P0 = 2.
- 2Next-year payoff P1 = 1.30 + 2.24 = 3.54, so W0* = 2 + 3.54 ÷ 1.08 = 2 + 3.278 = 5.278.
- 1With U = C0C1, C1 = 1.08C0 and 2C0 = 5.278, so C0 = 2.639 and C1 = 2.850.
- 1C0 = 2.639 exceeds P0 = 2, so the person borrows 0.639 and repays 0.690: 3.54 − 0.690 = 2.850.
Key terms
- Investment frontier
- The set of consumption pairs reachable by investing part of today's endowment in real projects, funded best project first; with fixed-return projects it is a chain of straight segments.
- Present value of wealth
- The production plan valued today, P0 plus P1 discounted at one plus the interest rate; it is the horizontal intercept of the capital market line.
- Marginal rate of substitution
- The rate at which a person will give up next year's consumption for one more unit today; at the optimum it equals one plus the interest rate.
- Lagrange method
- A way to maximise utility subject to the budget line by adding the constraint times a multiplier and setting partial derivatives to zero; it yields the same tangency condition.
Consumption and Investment Under Certainty FAQ
What does the Fisher separation theorem say?
When capital markets are perfect and complete, the investment decision is made by maximising the present value of wealth, independently of preferences. Everyone with the same opportunities invests the same amount, then borrows or lends to reach their preferred consumption.
How do you tell whether a person borrows or lends?
Compare optimal consumption today with what is left after investing. Consuming more than that remainder means borrowing; consuming less means lending, with the loan repaid next year at one plus the interest rate.
Who gains when the interest rate falls?
Borrowers are always better off. Lenders who remain lenders are worse off, while those who switch to borrowing may end up better or worse off, so the effect on lenders is ambiguous as a group.
Why do all investors pick the same production point?
Because with a capital market everyone can lend or borrow at the same rate, the best production plan is the one with the largest present value, and that is an objective comparison of project returns with the interest rate. Personal tastes for consuming now or later only matter afterwards, when each person moves along the line.
Exam move
Write the five-step routine on a card and run it on every tutorial and past problem: rank, cumulate, invest while returns beat r, value the plan, solve each consumer's tangency. After each answer, check that consumption lies exactly on the capital market line and that next year's consumption equals the project payoff plus or minus the loan repayment.
Practise the Lagrange derivation at least twice so it is fluent when a question insists on it, and draw the frontier, the line and the indifference curves on one labelled diagram. When a question gives a graph instead of numbers, read the interest rate from the two intercepts of the capital market line and the production point from where it touches the frontier.
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