Praxilla, who lived in ancient Greece, derives utility from reading poems and from eating cucumbers. Praxilla gets 30 units of marginal utility from her first poem, 27 units of marginal utility from her second poem, 24 units of marginal utility from her third poem, and so on, with marginal utility declining by three units for each additional poem. Praxilla gets six units of marginal utility for each of her first three cucumbers consumed, five units of marginal utility for each of her next three cucumbers consumed, four units of marginal utility for each of the following three cucumbers consumed, and so on, with marginal utility declining by one for every three cucumbers consumed. A poem costs three bronze coins but a cucumber costs only one bronze coin. Praxilla has 18 bronze coins. Sketch Praxilla’s budget set between poems and cucumbers, placing poems on the vertical axis and cucumbers on the horizontal axis. Start off with the choice of zero poems and 18 cucumbers, and calculate the changes in marginal utility of moving along the budget line to the next choice of one poem and 15 cucumbers. Using this step-bystep process based on marginal utility, create a table and identify Praxilla’s utility-maximizing choice. Compare the marginal utility of the two goods and the relative prices at the optimal choice to see if the expected relationship holds.

answer: following the publication of mysterium and with the blessing of the graz school inspectors, kepler began an ambitious program to extend and elaborate his work. he planned four additional books: one on the stationary aspects of the universe (the sun and the fixed stars); one on the planets and their motions; one on the physical nature of planets and the formation of geographical features (focused especially on earth); and one on the effects of the heavens on the earth, to include atmospheric optics, meteorology, and astrology.[24]

he also sought the opinions of many of the astronomers to whom he had sent mysterium, among them reimarus ursus (nicolaus reimers bär)—the imperial mathematician to rudolph ii and a bitter rival of tycho brahe. ursus did not reply directly, but republished kepler's flattering letter to pursue his priority dispute over (what is now called) the tychonic system with tycho. despite this black mark, tycho also began corresponding with kepler, starting with a harsh but legitimate critique of kepler's system; among a host of objections, tycho took issue with the use of inaccurate numerical data taken from copernicus. through their letters, tycho and kepler discussed a broad range of astronomical problems, dwelling on lunar phenomena and copernican theory (particularly its theological viability). but without the significantly more accurate data of tycho's observatory, kepler had no way to address many of these issues.[25]

instead, he turned his attention to chronology and "harmony," the numerological relationships among music, mathematics and the physical world, and their astrological consequences. by assuming the earth to possess a soul (a property he would later invoke to explain how the sun causes the motion of planets), he established a speculative system connecting astrological aspects and astronomical distances to weather and other earthly phenomena. by 1599, however, he again felt his work limited by the inaccuracy of available data—just as growing religious tension was also threatening his continued employment in graz. in december of that year, tycho invited kepler to visit him in prague; on january 1, 1600 (before he even received the invitation), kepler set off in the hopes that tycho's patronage could solve his philosophical problems as well as his social and financial one

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