Unit 22
Mathematics for Economics and Finance
Class 12 Mathematics
Mathematics for Economics and Finance
Class 12 Mathematics – Mathematics for Economics and Finance Notes PDF
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This unit applies mathematics to economics and finance through consumer and producer surplus, quadratic economic functions, input-output analysis, market-price dynamics, difference equations, the cobweb model, and lagged Keynesian macroeconomic models.
1. Consumer and Producer Surplus
Let demand price be p=D(q) and supply price be p=S(q). At market equilibrium D(q₀)=S(q₀)=p₀.
2. Quadratic Functions in Economics
Quadratic functions commonly model revenue, cost, profit, production, or demand relationships.
The turning point occurs at
If a<0, the vertex gives a maximum; if a>0, it gives a minimum.
If P(q)=−2q²+40q−50, maximum profit occurs at q=−40/[2(−2)]=10.
3. Input-Output Analysis
Leontief input-output analysis models interdependence among economic sectors. Let A be the technical-coefficient matrix, x total output, and d final demand.
4. Dynamics of Market Price
Dynamic market models study how price changes over time when supply and demand adjust with delays. A simple adjustment model can be written as
where λ>0 measures adjustment speed.
An equilibrium price p* satisfies D(p*)=S(p*).
5. Difference Equations
A difference equation relates values of a variable at different discrete times.
The equilibrium value, when a≠1, satisfies
The path tends to equilibrium when |a|<1; it diverges in magnitude when |a|>1.
6. Cobweb Model
In the cobweb model, current supply may depend on a previous period’s price while current demand depends on current price. Price can oscillate around equilibrium.
Depending on slopes/parameters, oscillations can converge, remain constant, or diverge.
7. Lagged Keynesian Macroeconomic Model
A simplified lagged model may let consumption depend on prior income:
With constant autonomous investment I, this yields a first-order difference equation in income.
Its equilibrium and stability can be analyzed using the same difference-equation methods.
8. Worked Examples
If demand is p=10−q, supply p=2+q, then equilibrium is q₀=4,p₀=6. Consumer surplus = ∫₀⁴(10−q)dq−24=8.
For x_{t+1}=0.5x_t+10, equilibrium is x*=20; because |0.5|<1, paths converge.
Given A and d, form (I−A)x=d and solve the linear system for total sector output.
9. Common Mistakes
Surplus is an area, so use correct upper/lower functions and equilibrium limits.
In input-output analysis, distinguish total output x from final demand d.
In a difference equation, time is discrete: t,t+1, not a derivative.
Check stability using the relevant coefficient magnitude, not only the equilibrium value.
10. Important Exam Questions
- Compute consumer and producer surplus from demand/supply functions.
- Find maximum/minimum of a quadratic economic function.
- Solve a two-sector input-output problem.
- Find equilibrium and stability of a first-order difference equation.
- Explain/solve a cobweb model.
- Form and analyze a lagged Keynesian model.
Discussion
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