Consider the case of the following annuities, and the need to compute either their expected rate of return or duration. Joshua inherited an annuity worth $6,830.77 from his uncle. The annuity will pay him eight equal payments of $1,100 at the end of each year. The annuity fund is offering a return Joshua's friend, Willie, has hired a financial planner for advice on retirement. Considering Willie's current expenses and expected future lifestyle changes, the financial planner has stated that once Willie crosses a threshold of $1,387,311 in savings, he will have enough money for retirement. Willie has nothing saved for his retirement yet, so he plans to start depositing $25,000 in a retirement fund at a fixed rate of 6.00% at the end of each year. It Will take for Willie to reach his retirement goal.

6. why the aggregate supply curve slopes upward in the short run in the short run, the quantity of output that firms supply can deviate from the natural level of output if the actual price level in the economy deviates from the expected price level. several theories explain how this might happen. for example, the misperceptions theory asserts that changes in the price level can temporarily mislead firms about what is happening to their output prices. consider a soybean farmer who expects a price level of 100 in the coming year. if the actual price level turns out to be 90, soybean prices will , and if the farmer mistakenly assumes that the price of soybeans declined relative to other prices of goods and services, she will respond by the quantity of soybeans supplied. if other producers in this economy mistake changes in the price level for changes in their relative prices, the unexpected decrease in the price level causes the quantity of output supplied to the natural level of output in the short run.