Map of Microeconomic Optimization

Utility Maximization (Household) §

1. Utility Maximization
   1. Uncompensated Demand Function $(p_1,p_2,I)\mapsto x_1$
   2. Indirect Utility Function $(p_1,p_2,I)\mapsto u$
   3. Roy’s Identity
2. Expenditure Minimization
   1. Compensated Demand Function $(p_1,p_2,u)\mapsto x_1$
   2. Expenditure Function $(p_1,p_2,u)\mapsto I$
   3. Shepard’s Lemma
3. Invert between Indirect Utility Function and Expenditure Function by solving for $I$ or $u$ depending on what you want.

Profit Maximization (Firm) §

1. Profit Maximization $ma{x}_{l,k}\pi =px-wl-rk\text{ such that}x=f(l,k)$
   1. Ordinary Input Demand and Output Supply $(w,r,p)\mapsto l,k$
   2. Output Input Demand and Output Supply $(w,r,p)\mapsto x$
   3. Profit Function $(w,r,p)\mapsto \pi$
      • Hotelling’s Lemma back to input demand
2. Cost Minimization $mi{n}_{l,k}c=wl+rk\text{such that}x=f(l,k)$
   1. Conditional Input Demand and Output Supply $(w,r,\bar{x})\mapsto l,k$
   2. Conditional Cost Function $(w,r,\bar{x})\mapsto C$
      • Shepard’s Lemma back to conditional input demand
3. $ma{x}_{x}px-C(w,r,\bar{x})$ for cost function to output supply
4. substitute Input Demand and Output Supply into Conditional Input Demand to get Ordinary Input Demand and Output Supply